Bessel and Grüss Type Inequalities in Inner Product Modules over Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq1_HTML.gif -Algebras

  • A G Ghazanfari1Email author and

    Affiliated with

    • S S Dragomir2, 3

      Affiliated with

      Journal of Inequalities and Applications20112011:562923

      DOI: 10.1155/2011/562923

      Received: 11 January 2011

      Accepted: 1 March 2011

      Published: 14 March 2011

      Abstract

      We give an analogue of the Bessel inequality and we state a simple formulation of the Grüss type inequality in inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq2_HTML.gif -modules, which is a refinement of it. We obtain some further generalization of the Grüss type inequalities in inner product modules over proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq3_HTML.gif -algebras and unital Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq4_HTML.gif -algebras for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq5_HTML.gif -seminorms and positive linear functionals.

      1. Introduction

      A proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq6_HTML.gif -algebra is a complex Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq7_HTML.gif -algebra http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq8_HTML.gif where the underlying Banach space is a Hilbert space with respect to the inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq9_HTML.gif satisfying the properties http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq11_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq12_HTML.gif . A http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq13_HTML.gif -algebra is a complex Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq14_HTML.gif -algebra http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq15_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq16_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq17_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq18_HTML.gif is a proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq19_HTML.gif -algebra or a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq20_HTML.gif -algebra and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq21_HTML.gif is such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq22_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq23_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq24_HTML.gif .

      For a proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq25_HTML.gif -algebra http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq26_HTML.gif , the trace class associated with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq27_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq28_HTML.gif . For every positive http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq29_HTML.gif there exists the square root of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq30_HTML.gif , that is, a unique positive http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq31_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq32_HTML.gif , the square root of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq33_HTML.gif is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq34_HTML.gif . There are a positive linear functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq35_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq36_HTML.gif and a norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq37_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq38_HTML.gif , related to the norm of A by the equality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq39_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq40_HTML.gif .

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq41_HTML.gif be a proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq42_HTML.gif -algebra or a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq43_HTML.gif -algebra. A semi-inner product module over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq44_HTML.gif is a right module http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq45_HTML.gif over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq46_HTML.gif together with a generalized semi-inner product, that is with a mapping http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq47_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq48_HTML.gif , which is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq49_HTML.gif -valued if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq50_HTML.gif is a proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq51_HTML.gif -algebra, or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq52_HTML.gif -valued if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq53_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq54_HTML.gif -algebra, having the following properties:

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq55_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq56_HTML.gif ,

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq57_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq58_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq59_HTML.gif ,

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq60_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq61_HTML.gif ,

      (iv) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq62_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq63_HTML.gif .

      We will say that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq64_HTML.gif is a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq65_HTML.gif -module if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq66_HTML.gif is a proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq67_HTML.gif -algebra and that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq68_HTML.gif is a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq69_HTML.gif -module if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq70_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq71_HTML.gif -algebra.

      If, in addition,

      (v) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq72_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq73_HTML.gif ,

      then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq74_HTML.gif is called an inner product module over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq75_HTML.gif . The absolute value of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq76_HTML.gif is defined as the square root of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq77_HTML.gif and it is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq78_HTML.gif .

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq79_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq80_HTML.gif -algebra. A seminorm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq81_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq82_HTML.gif is a real-valued function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq83_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq85_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq86_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq87_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq88_HTML.gif . A seminorm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq89_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq90_HTML.gif is called a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq91_HTML.gif -seminorm if it satisfies the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq92_HTML.gif -condition: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq93_HTML.gif . By Sebestyen's theorem [1, Theorem  38.1] every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq94_HTML.gif -seminorm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq95_HTML.gif on a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq96_HTML.gif -algebra http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq97_HTML.gif is submultiplicative, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq98_HTML.gif , and by [2, Section  39, Lemma  2(i)] http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq99_HTML.gif . For every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq100_HTML.gif , the spectral radius of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq101_HTML.gif is defined to be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq102_HTML.gif .

      The Pták function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq103_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq104_HTML.gif -algebra http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq105_HTML.gif is defined to be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq106_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq107_HTML.gif . This function has important roles in Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq108_HTML.gif -algebras, for example, on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq109_HTML.gif -algebras, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq110_HTML.gif is equal to the norm and on Hermitian Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq111_HTML.gif -algebras http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq112_HTML.gif is the greatest http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq113_HTML.gif -seminorm. By utilizing properties of the spectral radius and the Pták function, Pták [3] showed in 1970 that an elegant theory for Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq114_HTML.gif -algebras arises from the inequality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq115_HTML.gif .

      This inequality characterizes Hermitian (and symmetric) Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq116_HTML.gif -algebras, and further characterizations of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq117_HTML.gif -algebras follow as a result of Pták theory.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq118_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq119_HTML.gif -algebra. We define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq120_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ1_HTML.gif
      (1.1)

      and call the elements of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq121_HTML.gif positive.

      The set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq122_HTML.gif of positive elements is obviously a convex cone (i.e., it is closed under convex combinations and multiplication by positive constants). Hence we call http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq123_HTML.gif the positive cone. By definition, zero belongs to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq124_HTML.gif . It is also clear that each positive element is Hermitian.

      We recall that a Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq125_HTML.gif -algebra http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq126_HTML.gif is said to be an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq127_HTML.gif -algebra provided there exists on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq128_HTML.gif a second norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq129_HTML.gif , not necessarily complete, which is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq130_HTML.gif -norm. The second norm will be called an auxiliary norm.

      Definition 1.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq131_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq132_HTML.gif -algebra. A semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq133_HTML.gif -module (or semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq134_HTML.gif -module) is a complex vector space which is also a right http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq135_HTML.gif -module http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq136_HTML.gif with a sesquilinear semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq137_HTML.gif , fulfilling
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ2_HTML.gif
      (1.2)
      for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq138_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq139_HTML.gif . Furthermore, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq140_HTML.gif satisfies the strict positivity condition
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ3_HTML.gif
      (1.3)

      then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq141_HTML.gif is called an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq142_HTML.gif -module (or inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq143_HTML.gif -module).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq144_HTML.gif be a seminorm or a positive linear functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq146_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq147_HTML.gif is a seminorm on a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq148_HTML.gif -module http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq149_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq150_HTML.gif is said to be a semi-Hilbert http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq151_HTML.gif -module.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq152_HTML.gif is a norm on an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq153_HTML.gif -module http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq154_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq155_HTML.gif is said to be a pre-Hilbert http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq156_HTML.gif -module.

      A pre-Hilbert http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq157_HTML.gif -module which is complete with respect to its norm is called a Hilbert http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq158_HTML.gif -module.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq159_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq160_HTML.gif are self adjoint, therefore we get the following Corollary.

      Corollary 1.2.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq161_HTML.gif is a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq162_HTML.gif -module, then the following symmetry condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ4_HTML.gif
      (1.4)
      Example 1.3.
      1. (a)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq163_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq164_HTML.gif -algebra and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq165_HTML.gif a positive linear functional or a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq166_HTML.gif -seminorm on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq167_HTML.gif . It is known that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq168_HTML.gif is a semi-Hilbert http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq169_HTML.gif -module over itself with the inner product defined by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq170_HTML.gif , in this case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq171_HTML.gif .

         
      2. (b)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq172_HTML.gif be a Hermitian Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq173_HTML.gif -algebra and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq174_HTML.gif be the Pták function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq175_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq176_HTML.gif is a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq177_HTML.gif -module and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq178_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq179_HTML.gif is a semi-Hilbert http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq180_HTML.gif -module.

         
      3. (c)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq181_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq182_HTML.gif -algebra and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq183_HTML.gif be the auxiliary norm on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq184_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq185_HTML.gif is an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq186_HTML.gif -module and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq187_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq188_HTML.gif is a pre-Hilbert http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq189_HTML.gif -module.

         
      4. (d)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq190_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq191_HTML.gif -algebra and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq192_HTML.gif (a semi-inner product) an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq193_HTML.gif -module. Since tr is a positive linear functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq194_HTML.gif and for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq195_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq196_HTML.gif ; therefore http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq197_HTML.gif is a (semi-Hilbert) pre-Hilbert http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq198_HTML.gif -module.

         

      In the present paper, we give an analogue of the Bessel inequality (2.7) and we obtain some further generalization and a simple form for the Grüss type inequalities in inner product modules over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq199_HTML.gif -algebras, proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq200_HTML.gif -algebras, and unital Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq201_HTML.gif -algebras.

      2. Schwarz and Bessel Inequality

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq202_HTML.gif is a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq203_HTML.gif -module, then the following Schwarz inequality holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ5_HTML.gif
      (2.1)

      (e.g. [4, Lemma  15.1.3]).

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq204_HTML.gif is a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq205_HTML.gif -module, then there are two forms of the Schwarz inequality: for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq206_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ6_HTML.gif
      (2.2)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ7_HTML.gif
      (2.3)

      First Saworotnow in [5] proved the strong Schwarz inequality, but the direct proof of that for a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq207_HTML.gif -module can be found in [6].

      Now let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq208_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq209_HTML.gif -algebra, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq210_HTML.gif a positive linear functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq211_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq212_HTML.gif be a semi-inner http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq213_HTML.gif -module. We can define a sesquilinear form on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq214_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq215_HTML.gif ; the Schwarz inequality for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq216_HTML.gif implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ8_HTML.gif
      (2.4)
      In [7, Proposition  1, Remark  1] the authors present two other forms of the Schwarz inequality in semi-inner http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq217_HTML.gif -module http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq218_HTML.gif , one for positive linear functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq219_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq220_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ9_HTML.gif
      (2.5)
      and another one for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq221_HTML.gif -seminorm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq222_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq223_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ10_HTML.gif
      (2.6)
      The classical Bessel inequality states that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq224_HTML.gif is a family of orthonormal vectors in a Hilbert space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq225_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ11_HTML.gif
      (2.7)
      Furthermore, some results concerning upper bounds for the expression
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ12_HTML.gif
      (2.8)
      and for the expression related to the Grüss-type inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ13_HTML.gif
      (2.9)

      have been proved in [8]. A version of the Bessel inequality for inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq226_HTML.gif -modules and inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq227_HTML.gif -modules can be found in [9], also there is a version of it for Hilbert http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq228_HTML.gif -modules in [10, Theorem  3.1]. We provide here an analogue of the Bessel inequality for inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq229_HTML.gif -modules.

      Lemma 2.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq230_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq231_HTML.gif -algebra, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq232_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq233_HTML.gif -module, and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq234_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq235_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq236_HTML.gif are idempotent. Then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ14_HTML.gif
      (2.10)

      Proof.

      By [11, Lemma  1] or a straightforward calculation shows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ15_HTML.gif
      (2.11)

      3. Grüss Type Inequalities

      Before stating the main results, let us fix the rest of our notation. We assume, unless stated otherwise, throughout this section that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq237_HTML.gif is a unital Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq238_HTML.gif -algebra. Also if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq239_HTML.gif is a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq240_HTML.gif -module and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq241_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq242_HTML.gif -seminorm on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq243_HTML.gif , we put http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq244_HTML.gif , and if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq245_HTML.gif is a positive linear functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq246_HTML.gif , we put http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq247_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq248_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq249_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq250_HTML.gif be idempotent, we set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq251_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq252_HTML.gif .

      Dragomir in [8, Lemma  4] shows that in a Hilbert space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq253_HTML.gif , the condition
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ16_HTML.gif
      (3.1)
      is equivalent to the condition
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ17_HTML.gif
      (3.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq254_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq255_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq256_HTML.gif . But for semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq257_HTML.gif -modules we have the following lemma, which is a generalization of [7, Lemma  1].

      Lemma 3.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq258_HTML.gif be a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq259_HTML.gif -module and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq260_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq261_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq262_HTML.gif . Then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ18_HTML.gif
      (3.3)
      if and only if
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ19_HTML.gif
      (3.4)

      Proof.

      Follows from the equalities:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ20_HTML.gif
      (3.5)

      Remark 3.2.

      By making use of the previous Lemma 3.1, we may conclude the following statements.

      (i)Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq263_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq264_HTML.gif -module and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq265_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq266_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq267_HTML.gif are idempotent, then inequality (3.3) implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ21_HTML.gif
      (3.6)
      (ii)Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq268_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq269_HTML.gif -module and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq270_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq271_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq272_HTML.gif are idempotent. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq273_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq274_HTML.gif -seminorm on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq275_HTML.gif then inequality (3.3) implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ22_HTML.gif
      (3.7)
      and if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq276_HTML.gif is a positive linear functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq277_HTML.gif from inequality (3.3) and [2, Section  37 Lemma  6(iii)], we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ23_HTML.gif
      (3.8)

      (iii)Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq278_HTML.gif be a proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq279_HTML.gif -algebra, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq280_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq281_HTML.gif -module, and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq282_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq283_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq284_HTML.gif are idempotent. Since for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq285_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq286_HTML.gif inequality (3.3) is valid only if

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ24_HTML.gif
      (3.9)

      We are able now to state our first main result.

      Theorem 3.3.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq287_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq288_HTML.gif -module and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq289_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq290_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq291_HTML.gif are idempotent. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq292_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq293_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq294_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq295_HTML.gif are real numbers and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq296_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ25_HTML.gif
      (3.10)
      hold, then one has the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ26_HTML.gif
      (3.11)

      Proof.

      By [11, Lemma  2] or, a straightforward calculation shows that for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq297_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ27_HTML.gif
      (3.12)
      Therefore
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ28_HTML.gif
      (3.13)
      Analogously, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq298_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ29_HTML.gif
      (3.14)
      The equalities (3.10), (3.13), and (3.14) imply that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ30_HTML.gif
      (3.15)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ31_HTML.gif
      (3.16)
      Since
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ32_HTML.gif
      (3.17)
      therefore the Schwarz's inequality (2.1) holds, that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ33_HTML.gif
      (3.18)
      Finally, using the elementary inequality for real numbers
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ34_HTML.gif
      (3.19)
      on
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ35_HTML.gif
      (3.20)
      we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ36_HTML.gif
      (3.21)
      Remark 3.4.
      1. (i)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq299_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq300_HTML.gif -module and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq301_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq302_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq303_HTML.gif are idempotent. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq304_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq305_HTML.gif are such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ37_HTML.gif
        (3.22)
         
      and if we put http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq306_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq307_HTML.gif , then, by (3.15) and (3.16), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ38_HTML.gif
      (3.23)
      These and (3.11) imply that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ39_HTML.gif
      (3.24)
      Therefore, (3.11) is a refinement and a simple formulation of [9, Theorem  4.1.].
      1. (ii)
        If for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq308_HTML.gif , we set
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ40_HTML.gif
        (3.25)
         

      then similarly (3.11) is a refinement and a simple form of [9, Corollary  4.3].

      Corollary 3.5.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq309_HTML.gif be a Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq310_HTML.gif -algebra, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq311_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq312_HTML.gif -module, and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq313_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq314_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq315_HTML.gif are idempotent. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq316_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq317_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq318_HTML.gif are real numbers and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq319_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ41_HTML.gif
      (3.26)
      hold, then one has the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ42_HTML.gif
      (3.27)

      Proof.

      Using the schwarz's inequality (2.6), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ43_HTML.gif
      (3.28)

      The assumptions (3.26) and the elementary inequality for real numbers (3.19) will provide the desired result (3.27).

      Example 3.6.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq320_HTML.gif be a Hermitian Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq321_HTML.gif -algebra and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq322_HTML.gif be the Pták function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq323_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq324_HTML.gif is a semi-inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq325_HTML.gif -module and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq326_HTML.gif with the properties that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ44_HTML.gif
      (3.29)
      then we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ45_HTML.gif
      (3.30)

      That is interesting in its own right.

      Corollary 3.7.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq327_HTML.gif be a proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq328_HTML.gif -algebra, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq329_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq330_HTML.gif -module, and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq331_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq332_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq333_HTML.gif are idempotent. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq334_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq335_HTML.gif are real numbers and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq336_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ46_HTML.gif
      (3.31)
      hold, then one has the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ47_HTML.gif
      (3.32)

      Proof.

      Using the strong Schwarz's inequality (2.3), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ48_HTML.gif
      (3.33)

      The assumptions (3.31) and the elementary inequality for real numbers (3.19) will provide (3.32).

      The following companion of the Grüss inequality for positive linear functionals holds.

      Theorem 3.8.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq337_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq338_HTML.gif -module, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq339_HTML.gif be a positive linear functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq340_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq341_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq342_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq343_HTML.gif are idempotent. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq344_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq345_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq346_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq347_HTML.gif are real numbers and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq348_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ49_HTML.gif
      (3.34)
      hold, then one has the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ50_HTML.gif
      (3.35)

      Proof.

      By taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq349_HTML.gif on both sides of (3.12), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ51_HTML.gif
      (3.36)
      Analogously
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ52_HTML.gif
      (3.37)
      Now, using Acz http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq350_HTML.gif l's inequality for real numbers, that is, we recall that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ53_HTML.gif
      (3.38)
      and the Schwarz's inequality for positive linear functionals, that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ54_HTML.gif
      (3.39)

      we deduce (3.35).

      4. Some Related Results

      Theorem 4.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq351_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq352_HTML.gif -module and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq353_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq354_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq355_HTML.gif are idempotent. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq356_HTML.gif and if we define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ55_HTML.gif
      (4.1)
      then we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ56_HTML.gif
      (4.2)

      Proof.

      For every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq357_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq358_HTML.gif , by (3.13) and (3.14), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ57_HTML.gif
      (4.3)
      Therefore
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ58_HTML.gif
      (4.4)
      Now, using the elementary inequality for real numbers
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ59_HTML.gif
      (4.5)
      on
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ60_HTML.gif
      (4.6)
      we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ61_HTML.gif
      (4.7)

      Corollary 4.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq359_HTML.gif be a Banach http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq360_HTML.gif -algebra, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq361_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq362_HTML.gif -module, and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq363_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq364_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq365_HTML.gif are idempotent. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq366_HTML.gif and put
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ62_HTML.gif
      (4.8)
      then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ63_HTML.gif
      (4.9)

      Corollary 4.3.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq367_HTML.gif be a proper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq368_HTML.gif -algebra, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq369_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq370_HTML.gif -module, and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq371_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq372_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq373_HTML.gif are idempotent. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq374_HTML.gif and if we consider
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ64_HTML.gif
      (4.10)
      then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ65_HTML.gif
      (4.11)

      From a different perspective, we can state the following result as well.

      Theorem 4.4.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq375_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq376_HTML.gif -module and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq377_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq378_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq379_HTML.gif are idempotent. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq380_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq381_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq382_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq383_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ66_HTML.gif
      (4.12)
      then we have the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ67_HTML.gif
      (4.13)

      Proof.

      We know that for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq384_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq385_HTML.gif one has
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ68_HTML.gif
      (4.14)
      Put http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq386_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq387_HTML.gif , and since
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ69_HTML.gif
      (4.15)
      using (4.14), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ70_HTML.gif
      (4.16)

      Now, inequality (4.13) follows from inequalities (3.15) and (4.16).

      The following companion of the Grüss inequality for positive linear functionals holds.

      Theorem 4.5.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq388_HTML.gif be an inner product http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq389_HTML.gif -module, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq390_HTML.gif be a positive linear functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq391_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq392_HTML.gif be a finite set of orthogonal elements in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq393_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq394_HTML.gif are idempotent. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq395_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq396_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq397_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq398_HTML.gif are such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ71_HTML.gif
      (4.17)
      then we have the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ72_HTML.gif
      (4.18)

      Proof.

      The inequality (4.14) for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq399_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq400_HTML.gif implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ73_HTML.gif
      (4.19)
      By making use of inequality (3.12) for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq401_HTML.gif instead of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq402_HTML.gif and taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq403_HTML.gif on both sides, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ74_HTML.gif
      (4.20)

      From (4.19) and (4.20), we easily deduce (4.18).

      Remark 4.6.
      1. (i)
        The constant 1 coefficient of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq404_HTML.gif in (3.11) is sharp, in the sense that it cannot be replaced by a smaller quantity. If the submodule of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq405_HTML.gif generated by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq406_HTML.gif is not equal to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq407_HTML.gif , then there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq408_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq409_HTML.gif . We put http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq410_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq411_HTML.gif and for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq412_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ75_HTML.gif
        (4.21)
         
      For every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq413_HTML.gif , if we put
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ76_HTML.gif
      (4.22)
      then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ77_HTML.gif
      (4.23)
      therefore
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ78_HTML.gif
      (4.24)
      Now if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq414_HTML.gif is a constant such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq415_HTML.gif , then there is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq416_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq417_HTML.gif ; therefore
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ79_HTML.gif
      (4.25)
      1. (ii)
        Similarly, the constant 1 coefficient of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq418_HTML.gif in (3.32) is best possible, it is sufficient instead of (4.22) to put
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ80_HTML.gif
        (4.26)
         
      1. (iii)

        If there is a nonzero element http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq419_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq420_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq421_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq422_HTML.gif (resp. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq423_HTML.gif ) then the constant 1 coefficient of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq424_HTML.gif in (3.27) (resp. (3.35)) is best possible. Also similarly, the inequalities in Theorem 4.1, Corollaries 4.2 and 4.3, and Theorems 4.4 and 4.5 are sharp. However, the details are omitted.

         

      Authors’ Affiliations

      (1)
      Department of Mathematics, Lorestan University
      (2)
      School of Engineering and Science, Victoria University
      (3)
      School of Computational and Applied Mathematics, University of the Witwatersrand

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      Copyright

      © A. G. Ghazanfari and S. S. Dragomir. 2011

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.