Open Access

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

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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq3_HTML.gif -algebras and unital Banach https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq4_HTML.gif -algebras for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq6_HTML.gif -algebra is a complex Banach https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq7_HTML.gif -algebra https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq9_HTML.gif satisfying the properties https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq11_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq12_HTML.gif . A https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq13_HTML.gif -algebra is a complex Banach https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq14_HTML.gif -algebra https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq15_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq16_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq17_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq18_HTML.gif is a proper https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq19_HTML.gif -algebra or a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq20_HTML.gif -algebra and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq21_HTML.gif is such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq22_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq23_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq24_HTML.gif .

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq41_HTML.gif be a proper https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq42_HTML.gif -algebra or a https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq44_HTML.gif is a right module https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq45_HTML.gif over https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq47_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq48_HTML.gif , which is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq49_HTML.gif -valued if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq50_HTML.gif is a proper https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq51_HTML.gif -algebra, or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq52_HTML.gif -valued if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq53_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq54_HTML.gif -algebra, having the following properties:

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

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

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

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

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

If, in addition,

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

then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq75_HTML.gif . The absolute value of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq77_HTML.gif and it is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq78_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq79_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq80_HTML.gif -algebra. A seminorm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq81_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq82_HTML.gif is a real-valued function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq83_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq84_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq85_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq86_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq87_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq88_HTML.gif . A seminorm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq89_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq90_HTML.gif is called a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq91_HTML.gif -seminorm if it satisfies the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq92_HTML.gif -condition: https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq94_HTML.gif -seminorm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq95_HTML.gif on a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq96_HTML.gif -algebra https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq97_HTML.gif is submultiplicative, that is, https://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)] https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq99_HTML.gif . For every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq100_HTML.gif , the spectral radius of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq101_HTML.gif is defined to be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq102_HTML.gif .

The Pták function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq103_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq104_HTML.gif -algebra https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq105_HTML.gif is defined to be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq106_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq108_HTML.gif -algebras, for example, on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq109_HTML.gif -algebras, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq111_HTML.gif -algebras https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq112_HTML.gif is the greatest https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq114_HTML.gif -algebras arises from the inequality https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq116_HTML.gif -algebras, and further characterizations of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq118_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq119_HTML.gif -algebra. We define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq120_HTML.gif by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq121_HTML.gif positive.

The set https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq125_HTML.gif -algebra https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq126_HTML.gif is said to be an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq127_HTML.gif -algebra provided there exists on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq128_HTML.gif a second norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq129_HTML.gif , not necessarily complete, which is a https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq131_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq132_HTML.gif -algebra. A semi-inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq133_HTML.gif -module (or semi-inner product https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq135_HTML.gif -module https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq136_HTML.gif with a sesquilinear semi-inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq137_HTML.gif , fulfilling
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ2_HTML.gif
(1.2)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq139_HTML.gif . Furthermore, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq140_HTML.gif satisfies the strict positivity condition
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ3_HTML.gif
(1.3)

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

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq146_HTML.gif . If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq148_HTML.gif -module https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq149_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq151_HTML.gif -module.

If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq153_HTML.gif -module https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq154_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq156_HTML.gif -module.

A pre-Hilbert https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq158_HTML.gif -module.

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq159_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq161_HTML.gif is a semi-inner product https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq163_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq164_HTML.gif -algebra and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq165_HTML.gif a positive linear functional or a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq166_HTML.gif -seminorm on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq167_HTML.gif . It is known that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq168_HTML.gif is a semi-Hilbert https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq170_HTML.gif , in this case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq171_HTML.gif .

     
  2. (b)

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

     
  3. (c)

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

     
  4. (d)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq190_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq191_HTML.gif -algebra and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq194_HTML.gif and for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq195_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq196_HTML.gif ; therefore https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq197_HTML.gif is a (semi-Hilbert) pre-Hilbert https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq199_HTML.gif -algebras, proper https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq200_HTML.gif -algebras, and unital Banach https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq202_HTML.gif is a semi-inner product https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq204_HTML.gif is a semi-inner product https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq206_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ6_HTML.gif
(2.2)
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq208_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq209_HTML.gif -algebra, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq210_HTML.gif a positive linear functional on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq211_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq212_HTML.gif be a semi-inner https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq214_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq215_HTML.gif ; the Schwarz inequality for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq216_HTML.gif implies that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq217_HTML.gif -module https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq218_HTML.gif , one for positive linear functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq219_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq220_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ9_HTML.gif
(2.5)
and another one for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq221_HTML.gif -seminorm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq222_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq223_HTML.gif :
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq225_HTML.gif , then
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq226_HTML.gif -modules and inner product https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq229_HTML.gif -modules.

Lemma 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq230_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq231_HTML.gif -algebra, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq232_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq233_HTML.gif -module, and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq235_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq236_HTML.gif are idempotent. Then
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq237_HTML.gif is a unital Banach https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq238_HTML.gif -algebra. Also if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq239_HTML.gif is a semi-inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq240_HTML.gif -module and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq241_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq242_HTML.gif -seminorm on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq243_HTML.gif , we put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq244_HTML.gif , and if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq245_HTML.gif is a positive linear functional on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq246_HTML.gif , we put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq247_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq249_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq250_HTML.gif be idempotent, we set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq251_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq253_HTML.gif , the condition
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ17_HTML.gif
(3.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq254_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq255_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq256_HTML.gif . But for semi-inner product https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq258_HTML.gif be a semi-inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq259_HTML.gif -module and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq260_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq261_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq262_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ18_HTML.gif
(3.3)
if and only if
https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq263_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq264_HTML.gif -module and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq266_HTML.gif such that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ21_HTML.gif
(3.6)
(ii)Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq268_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq269_HTML.gif -module and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq271_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq272_HTML.gif are idempotent. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq273_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq274_HTML.gif -seminorm on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq275_HTML.gif then inequality (3.3) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ22_HTML.gif
(3.7)
and if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq276_HTML.gif is a positive linear functional on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ23_HTML.gif
(3.8)

(iii)Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq278_HTML.gif be a proper https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq279_HTML.gif -algebra, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq280_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq281_HTML.gif -module, and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq283_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq284_HTML.gif are idempotent. Since for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq285_HTML.gif , https://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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq287_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq288_HTML.gif -module and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq290_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq291_HTML.gif are idempotent. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq292_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq293_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq294_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq295_HTML.gif are real numbers and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq296_HTML.gif such that
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq297_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ27_HTML.gif
(3.12)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ28_HTML.gif
(3.13)
Analogously, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq298_HTML.gif , we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ30_HTML.gif
(3.15)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ31_HTML.gif
(3.16)
Since
https://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,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ34_HTML.gif
(3.19)
on
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ35_HTML.gif
(3.20)
we get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq299_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq300_HTML.gif -module and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq302_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq303_HTML.gif are idempotent. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq304_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq305_HTML.gif are such that
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ37_HTML.gif
    (3.22)
     
and if we put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq306_HTML.gif , and https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq308_HTML.gif , we set
    https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq309_HTML.gif be a Banach https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq310_HTML.gif -algebra, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq311_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq312_HTML.gif -module, and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq314_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq315_HTML.gif are idempotent. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq316_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq317_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq318_HTML.gif are real numbers and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq319_HTML.gif such that
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq320_HTML.gif be a Hermitian Banach https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq321_HTML.gif -algebra and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq322_HTML.gif be the Pták function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq323_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq324_HTML.gif is a semi-inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq325_HTML.gif -module and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq326_HTML.gif with the properties that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ44_HTML.gif
(3.29)
then we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq327_HTML.gif be a proper https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq328_HTML.gif -algebra, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq329_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq330_HTML.gif -module, and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq332_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq333_HTML.gif are idempotent. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq334_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq335_HTML.gif are real numbers and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq336_HTML.gif such that
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq337_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq338_HTML.gif -module, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq339_HTML.gif be a positive linear functional on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq340_HTML.gif , and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq342_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq343_HTML.gif are idempotent. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq344_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq345_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq346_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq347_HTML.gif are real numbers and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq348_HTML.gif such that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ50_HTML.gif
(3.35)

Proof.

By taking https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ51_HTML.gif
(3.36)
Analogously
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ52_HTML.gif
(3.37)
Now, using Acz https://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
https://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,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq351_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq352_HTML.gif -module and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq354_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq355_HTML.gif are idempotent. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq356_HTML.gif and if we define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ55_HTML.gif
(4.1)
then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ56_HTML.gif
(4.2)

Proof.

For every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq357_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ57_HTML.gif
(4.3)
Therefore
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ59_HTML.gif
(4.5)
on
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ60_HTML.gif
(4.6)
we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ61_HTML.gif
(4.7)

Corollary 4.2.

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

Corollary 4.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq367_HTML.gif be a proper https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq368_HTML.gif -algebra, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq369_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq370_HTML.gif -module, and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq372_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq373_HTML.gif are idempotent. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq374_HTML.gif and if we consider
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ64_HTML.gif
(4.10)
then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq375_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq376_HTML.gif -module and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq378_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq379_HTML.gif are idempotent. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq380_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq381_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq382_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq383_HTML.gif such that
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq384_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq385_HTML.gif one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ68_HTML.gif
(4.14)
Put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq386_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq387_HTML.gif , and since
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq388_HTML.gif be an inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq389_HTML.gif -module, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq390_HTML.gif be a positive linear functional on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq391_HTML.gif , and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq393_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq394_HTML.gif are idempotent. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq395_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq396_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq397_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq398_HTML.gif are such that
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq399_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq400_HTML.gif implies that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq401_HTML.gif instead of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq402_HTML.gif and taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq403_HTML.gif on both sides, we have
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq405_HTML.gif generated by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq406_HTML.gif is not equal to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq407_HTML.gif , then there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq408_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq409_HTML.gif . We put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq410_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq411_HTML.gif and for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq412_HTML.gif , we have
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ75_HTML.gif
    (4.21)
     
For every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq413_HTML.gif , if we put
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ76_HTML.gif
(4.22)
then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ77_HTML.gif
(4.23)
therefore
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_Equ78_HTML.gif
(4.24)
Now if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq414_HTML.gif is a constant such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq415_HTML.gif , then there is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq416_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq417_HTML.gif ; therefore
https://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 https://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
    https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq419_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq420_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq421_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq422_HTML.gif (resp. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F562923/MediaObjects/13660_2011_Article_2349_IEq423_HTML.gif ) then the constant 1 coefficient of https://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.