Applications of Kato’s inequality for n-tuples of operators in Hilbert spaces, (I)

Journal of Inequalities and Applications20132013:21

DOI: 10.1186/1029-242X-2013-21

Received: 21 August 2012

Accepted: 28 December 2012

Published: 16 January 2013

Abstract

In this paper, by the use of the famous Kato’s inequality for bounded linear operators, we establish some inequalities for n-tuples of operators and apply them for functions of normal operators defined by power series as well as for some norms and numerical radii that arise in multivariate operator theory.

MSC:47A63, 47A99.

Keywords

bounded linear operators functions of normal operators inequalities for operators norm and numerical radius inequalities Kato’s inequality

1 Introduction

The ‘square root’ of a positive bounded self-adjoint operator on H can be defined as follows (see, for instance, [[1], p.240]).

If the operator A B ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq1_HTML.gif is self-adjoint and positive, then there exists a unique positive self-adjoint operator B : = A B ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq2_HTML.gif such that B 2 = A http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq3_HTML.gif. If A is invertible, then so is B.

If A B ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq1_HTML.gif, then the operator A A http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq4_HTML.gif is self-adjoint and positive. Define the ‘absolute value’ operator by | A | : = A A http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq5_HTML.gif.

In 1952, Kato [2] proved the following generalization of Schwarz inequality:
| T x , y | 2 ( T T ) α x , x ( T T ) 1 α y , y , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ1_HTML.gif
(1.1)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif, α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif and T is a bounded linear operator on H.

Utilizing the modulus notation introduced before, we can write (1.1) as follows:
| T x , y | 2 | T | 2 α x , x | T | 2 ( 1 α ) y , y . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ2_HTML.gif
(1.2)

For results related to the Kato’s inequality, see [218] and [19].

In the recent paper [20], by employing Kato’s inequality (1.2), Dragomir established the following results for sequences of bonded linear operators on complex Hilbert spaces.

Theorem 1.1 Let ( T 1 , , T n ) B ( H ) × × B ( H ) : = B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq8_HTML.gif be an n-tuple of bounded linear operators on the Hilbert space ( H ; , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq9_HTML.gif and ( p 1 , , p n ) R + n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq10_HTML.gif be an n-tuple of nonnegative weights not all of them equal to zero. Then we have
j = 1 n p j | T j x , y | 2 j = 1 n p j | T j | 2 x , x α j = 1 n p j | T j | 2 y , y 1 α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ3_HTML.gif
(1.3)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif and α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif.

He also obtained the following result.

Theorem 1.2 With the assumptions in Theorem  1.1, we have
j = 1 n p j | T j x , y | j = 1 n p j | T j | 2 α x , x 1 / 2 j = 1 n p j | T j | 2 ( 1 α ) y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ4_HTML.gif
(1.4)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif.

For various related results, see the papers [2131].

Motivated by the above results, we establish in this paper other similar inequalities for n-tuples of bounded linear operators that can be obtained from Kato’s result (1.2) and apply them to functions of normal operators defined by power series as well as to some norms and numerical radii that can be associated with these n-tuples of bonded linear operators on Hilbert spaces.

2 Some inequalities for an n-tuple of linear operators

Employing Kato’s inequality (1.2), we can state the following new result.

Theorem 2.1 Let ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq12_HTML.gif be an n-tuple of bounded linear operators on the Hilbert space ( H ; , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq9_HTML.gif and ( p 1 , , p n ) R + n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq10_HTML.gif be an n-tuple of nonnegative weights, not all of them equal to zero. Then we have
j = 1 n p j | T j x , y | j = 1 n p j ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) x , x 1 / 2 × j = 1 n p j ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ5_HTML.gif
(2.1)
for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif, α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif and, in particular, for α = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq13_HTML.gif
j = 1 n p j | T j x , y | j = 1 n p j | T j | x , x 1 / 2 j = 1 n p j | T j | y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ6_HTML.gif
(2.2)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif.

Proof Utilizing Kato’s inequality, we have
| T j x , y | | T j | 2 α x , x 1 / 2 | T j | 2 ( 1 α ) y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equa_HTML.gif
and by replacing α with 1 α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq14_HTML.gif,
| T j x , y | | T j | 2 ( 1 α ) x , x 1 / 2 | T j | 2 α y , y 1 / 2 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equb_HTML.gif
which by summation gives
| T j x , y | 1 2 [ | T j | 2 α x , x 1 / 2 | T j | 2 ( 1 α ) y , y 1 / 2 + | T j | 2 ( 1 α ) x , x 1 / 2 | T j | 2 α y , y 1 / 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ7_HTML.gif
(2.3)
for any j { 1 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq15_HTML.gif and x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif. By the elementary inequality
a b + c d ( a 2 + c 2 ) 1 / 2 ( b 2 + d 2 ) 1 / 2 , a , b , c , d 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ8_HTML.gif
(2.4)
we have
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equc_HTML.gif
which by (2.3) produces
| T j x , y | ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) x , x 1 / 2 ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ9_HTML.gif
(2.5)
for any j { 1 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq15_HTML.gif and x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif. Multiplying the inequalities (2.5) with the positive weights p j http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq16_HTML.gif, summing over j from 1 to n and utilizing the weighted Cauchy-Buniakowski-Schwarz inequality
j = 1 n p j a j b j ( j = 1 n p j a j 2 ) 1 / 2 ( j = 1 n p j b j 2 ) 1 / 2 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equd_HTML.gif
where ( a 1 , , a n ) , ( b 1 , , b n ) R + n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq17_HTML.gif, we have
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ10_HTML.gif
(2.6)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif, and the inequality in (2.1) is proved. □

Remark 2.1 In order to provide some applications for functions of normal operators defined by power series, we need to state the inequality (2.1) for normal operators N j http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq18_HTML.gif, j { 1 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq15_HTML.gif, namely,
j = 1 n p j | N j x , y | j = 1 n p j ( | N j | 2 α + | N j | 2 ( 1 α ) 2 ) x , x 1 / 2 × j = 1 n p j ( | N j | 2 α + | N j | 2 ( 1 α ) 2 ) y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ11_HTML.gif
(2.7)

for any α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif and for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif.

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

Theorem 2.2 Let ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq12_HTML.gif be an n-tuple of bounded linear operators on the Hilbert space ( H ; , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq9_HTML.gif and ( p 1 , , p n ) R + n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq10_HTML.gif be an n-tuple of nonnegative weights, not all of them equal to zero. Then we have
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ12_HTML.gif
(2.8)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif and α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif.

Proof We must prove the inequalities only in the case α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq19_HTML.gif, since the case α = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq20_HTML.gif or α = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq21_HTML.gif follows directly from the corresponding case of Kato’s inequality.

Utilizing Kato’s inequality for the operator T j http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq22_HTML.gif, j { 1 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq23_HTML.gif, we have
| T j x , y | 2 | T j | 2 α x , x | T j | 2 ( 1 α ) y , y http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ13_HTML.gif
(2.9)
and, by replacing α with 1 α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq14_HTML.gif,
| T j x , y | 2 | T j | 2 ( 1 α ) x , x | T j | 2 α y , y http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ14_HTML.gif
(2.10)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif.

By the Hölder-McCarthy inequality P r x , x P x , x r http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq24_HTML.gif that holds for the positive operator P, for r ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq25_HTML.gif and x H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq26_HTML.gif with x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq27_HTML.gif, we also have
| T j | 2 α x , x | T j | 2 ( 1 α ) y , y | T j | 2 x , x α | T j | 2 y , y 1 α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ15_HTML.gif
(2.11)
and
| T j | 2 ( 1 α ) x , x | T j | 2 α y , y | T j | 2 x , x 1 α | T j | 2 y , y α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ16_HTML.gif
(2.12)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif, j { 1 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq15_HTML.gif and α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq28_HTML.gif.

If we add (2.9) with (2.10) and make use of (2.11) and (2.12), we deduce
2 | T j x , y | 2 | T j | 2 x , x α | T j | 2 y , y 1 α + | T j | 2 y , y α | T j | 2 x , x 1 α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ17_HTML.gif
(2.13)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif, j { 1 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq15_HTML.gif and α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq28_HTML.gif.

Now, if we multiply (2.13) with p j 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq29_HTML.gif, sum over j from 1 to n, we get
2 j = 1 n p j | T j x , y | 2 j = 1 n p j | T j | 2 x , x α | T j | 2 y , y 1 α + j = 1 n p j | T j | 2 y , y α | T j | 2 x , x 1 α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ18_HTML.gif
(2.14)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif and α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq28_HTML.gif.

Since | T j | 2 x , x = T j x 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq30_HTML.gif and | T j | 2 y , y = T j y 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq31_HTML.gif, j { 1 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq15_HTML.gif, then we get from (2.14) the first inequality in (2.8).

Now, on making use of the weighted Hölder discrete inequality
j = 1 n p j a j b j ( j = 1 n p j a j p ) 1 / p ( j = 1 n p j b j q ) 1 / q , p , q > 1 , 1 p + 1 q = 1 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Eque_HTML.gif
where ( a 1 , , a n ) , ( b 1 , , b n ) R + n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq17_HTML.gif, we also have
j = 1 n p j T j x 2 α T j y 2 ( 1 α ) ( j = 1 n p j T j x 2 ) α ( j = 1 n p j T j y 2 ) 1 α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equf_HTML.gif
and
j = 1 n p j T j y 2 α T j x 2 ( 1 α ) ( j = 1 n p j T j y 2 ) α ( j = 1 n p j T j x 2 ) 1 α . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equg_HTML.gif

Summing these two inequalities, we deduce the second inequality in (2.8).

Finally, on utilizing the Hölder inequality
a b + c d ( a p + c p ) 1 / p ( b q + d q ) 1 / q , a , b , c , d 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equh_HTML.gif
where p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq32_HTML.gif and 1 p + 1 q = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq33_HTML.gif, we have
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equi_HTML.gif

and the proof is concluded. □

Remark 2.2 For α = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq13_HTML.gif, we get from (2.8) that
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ19_HTML.gif
(2.15)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif.

3 Inequalities for functions of normal operators

Now, by the help of power series f ( z ) = n = 0 a n z n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq34_HTML.gif, we can naturally construct another power series which will have as coefficients the absolute values of the coefficient of the original series, namely, f A ( z ) : = n = 0 | a n | z n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq35_HTML.gif. It is obvious that this new power series will have the same radius of convergence as the original series. We also notice that if all coefficients a n 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq36_HTML.gif, then f A = f http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq37_HTML.gif.

As some natural examples that are useful for applications, we can point out that if
f ( z ) = n = 1 ( 1 ) n n z n = ln 1 1 + z , z D ( 0 , 1 ) ; g ( z ) = n = 0 ( 1 ) n ( 2 n ) ! z 2 n = cos z , z C ; h ( z ) = n = 0 ( 1 ) n ( 2 n + 1 ) ! z 2 n + 1 = sin z , z C ; l ( z ) = n = 0 ( 1 ) n z n = 1 1 + z , z D ( 0 , 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ20_HTML.gif
(3.1)
then the corresponding functions constructed by the use of the absolute values of the coefficients are as follows:
f A ( z ) = n = 1 1 n z n = ln 1 1 z , z D ( 0 , 1 ) ; g A ( z ) = n = 0 1 ( 2 n ) ! z 2 n = cosh z , z C ; h A ( z ) = n = 0 1 ( 2 n + 1 ) ! z 2 n + 1 = sinh z , z C ; l A ( z ) = n = 0 z n = 1 1 z , z D ( 0 , 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ21_HTML.gif
(3.2)

The following result is a functional inequality for normal operators that can be obtained from (2.1).

Theorem 3.1 Let f ( z ) = n = 0 a n z n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq38_HTML.gif be a function defined by power series with complex coefficients and convergent on the open disk D ( 0 , R ) C http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq39_HTML.gif, R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq40_HTML.gif. If N is a normal operator on the Hilbert space H, for α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq19_HTML.gif, we have that N 2 α , N 2 ( 1 α ) < R http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq41_HTML.gif, then we have the inequality
| f ( N ) x , y | 1 2 [ f A ( | N | 2 α ) + f A ( | N | 2 ( 1 α ) ) ] x , x 1 / 2 × [ f A ( | N | 2 α ) + f A ( | N | 2 ( 1 α ) ) ] y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ22_HTML.gif
(3.3)
for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif. In particular, if N < R http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq42_HTML.gif, then
| f ( N ) x , y | f A ( | N | ) x , x 1 / 2 f A ( | N | ) y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ23_HTML.gif
(3.4)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif.

Proof If N is a normal operator, then for any j N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq43_HTML.gif, we have that
| N j | 2 = ( N N ) j = | N | 2 j . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equj_HTML.gif
Now, utilizing the inequality (2.9), we can write
| j = 0 n a j N j x , y | j = 0 n | a j | | N j x , y | j = 0 n | a j | ( | N | 2 j α + | N | 2 j ( 1 α ) 2 ) x , x 1 / 2 × j = 0 n | a j | ( | N | 2 j α + | N | 2 j ( 1 α ) 2 ) y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ24_HTML.gif
(3.5)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif and n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq44_HTML.gif. Since N 2 α , N 2 ( 1 α ) < R http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq41_HTML.gif, then it follows that the series j = 0 | a j | ( | N | 2 α ) j http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq45_HTML.gif and j = 0 | a j | ( | N | 2 ( 1 α ) ) j http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq46_HTML.gif are absolute convergent in B ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq47_HTML.gif, and by taking the limit over n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq48_HTML.gif in (3.5), we deduce the desired result (3.3). □

Remark 3.1 With the assumptions in Theorem 3.1, if we take the supremum over y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq49_HTML.gif, y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq50_HTML.gif, then we get the vector inequality
f ( N ) x 1 2 [ f A ( | N | 2 α ) + f A ( | N | 2 ( 1 α ) ) ] x , x 1 / 2 × f A ( | N | 2 α ) + f A ( | N | 2 ( 1 α ) ) 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ25_HTML.gif
(3.6)
for any x H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq26_HTML.gif, which in its turn produces the norm inequality
f ( N ) 1 2 f A ( | N | 2 α ) + f A ( | N | 2 ( 1 α ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ26_HTML.gif
(3.7)
for any α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif. Making use of the examples in (3.1) and (3.2), we can state the vector inequalities
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ27_HTML.gif
(3.8)
and
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ28_HTML.gif
(3.9)
for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif and N < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq51_HTML.gif. We also have the inequalities
| sin ( N ) x , y | 1 2 [ sinh ( | N | 2 α ) + sinh ( | N | 2 ( 1 α ) ) ] x , x 1 / 2 × [ sinh ( | N | 2 α ) + sinh ( | N | 2 ( 1 α ) ) ] y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ29_HTML.gif
(3.10)
and
| cos ( N ) x , y | 1 2 [ cosh ( | N | 2 α ) + cosh ( | N | 2 ( 1 α ) ) ] x , x 1 / 2 × [ cosh ( | N | 2 α ) + cosh ( | N | 2 ( 1 α ) ) ] y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ30_HTML.gif
(3.11)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif and N a normal operator.

If we utilize the following function as power series representations with nonnegative coefficients:
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ31_HTML.gif
(3.12)
where Γ is the gamma function, then we can state the following vector inequalities:
| exp ( N ) x , y | 1 2 [ exp ( | N | 2 α ) + exp ( | N | 2 ( 1 α ) ) ] x , x 1 / 2 × [ exp ( | N | 2 α ) + exp ( | N | 2 ( 1 α ) ) ] y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ32_HTML.gif
(3.13)
for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif and N a normal operator. If N < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq51_HTML.gif, then we also have the inequalities
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ33_HTML.gif
(3.14)
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ34_HTML.gif
(3.15)
and
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ35_HTML.gif
(3.16)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif. From a different perspective, we also have

Theorem 3.2 With the assumption of Theorem  3.1 and if N is a normal operator on the Hilbert space H and z C http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq52_HTML.gif such that N 2 , | z | 2 < R http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq53_HTML.gif, then we have the inequalities
| f ( z N ) x , y | 2 1 2 f A ( | z | 2 ) [ f A ( | N | 2 ) x , x α f A ( | N | 2 ) y , y 1 α + f A ( | N | 2 ) x , x 1 α f A ( | N | 2 ) y , y α ] 1 2 f A ( | z | 2 ) ( f A ( | N | 2 ) x , x + f A ( | N | 2 ) y , y ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ36_HTML.gif
(3.17)
for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif and α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif. In particular, for α = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq13_HTML.gif, we have
| f ( z N ) x , y | 2 f A ( | z | 2 ) f A ( | N | 2 ) x , x 1 / 2 f A ( | N | 2 ) y , y 1 / 2 1 2 f A ( | z | 2 ) ( f A ( | N | 2 ) x , x + f A ( | N | 2 ) y , y ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ37_HTML.gif
(3.18)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif.

Proof If we use the second and third inequality from (2.8) for powers of operators, we have
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ38_HTML.gif
(3.19)
for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif and α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif. Since N is a normal operator on the Hilbert space H, then
N j x 2 = | N j | 2 x , x = | N | 2 j x , x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equk_HTML.gif
and
( N ) j y 2 = | ( N ) j | 2 y , y = | N | 2 j y , y = | N | 2 j y , y http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equl_HTML.gif
for any j { 0 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq54_HTML.gif and for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif. Then from (3.19), we have
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ39_HTML.gif
(3.20)
for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif and α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif. By the weighted Cauchy-Buniakowski-Schwarz inequality, we also have
| j = 0 n a j z j N j x , y | 2 j = 0 n | a j | | z | 2 j j = 0 n | a j | | N j x , y | 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ40_HTML.gif
(3.21)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif.

Now, since the series j = 0 a j z j N j http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq55_HTML.gif, j = 0 | a j | | z | 2 j http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq56_HTML.gif, j = 0 | a j | | N | 2 j http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq57_HTML.gif are convergent, then by (3.20) and (3.21), on letting n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq48_HTML.gif, we deduce the desired result (3.17). □

Similar inequalities for some particular functions of interest can be stated. However, the details are left to the interested reader.

4 Applications for the Euclidean norm

In [29], the author has introduced the following norm on the Cartesian product B ( n ) ( H ) : = B ( H ) × × B ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq58_HTML.gif, where B ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq47_HTML.gif denotes the Banach algebra of all bounded linear operators defined on the complex Hilbert space H:
( T 1 , , T n ) e : = sup ( λ 1 , , λ n ) B n λ 1 T 1 + + λ n T n , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ41_HTML.gif
(4.1)

where ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq59_HTML.gif and B n : = { ( λ 1 , , λ n ) C n | j = 1 n | λ j | 2 1 } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq60_HTML.gif is the Euclidean closed ball in C n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq61_HTML.gif.

It is clear that e http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq62_HTML.gif is a norm on B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq63_HTML.gif and, for any ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq64_HTML.gif, we have
( T 1 , , T n ) e = ( T 1 , , T n ) e , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equm_HTML.gif

where T j http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq65_HTML.gif is the adjoint operator of T j http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq22_HTML.gif, j { 1 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq23_HTML.gif. We call this the Euclidean norm of an n-tuple of operators ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq66_HTML.gif.

It has been shown in [29] that the following basic inequality for theEuclidean norm holds true:
1 n j = 1 n | T j | 2 1 2 ( T 1 , , T n ) e j = 1 n | T j | 2 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ42_HTML.gif
(4.2)

for any n-tuple ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq66_HTML.gif and the constants 1 n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq67_HTML.gif and 1 are best possible.

In the same paper [29], the author has introduced the Euclidean operator radius of an n-tuple of operators ( T 1 , , T n ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq68_HTML.gif by
w e ( T 1 , , T n ) : = sup x = 1 ( j = 1 n | T j x , x | 2 ) 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ43_HTML.gif
(4.3)
and proved that w e ( ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq69_HTML.gif is a norm on B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq63_HTML.gif and satisfies the double inequality
1 2 ( T 1 , , T n ) e w e ( T 1 , , T n ) ( T 1 , , T n ) e http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ44_HTML.gif
(4.4)

for each n-tuple ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq66_HTML.gif.

As pointed out in [29], the Euclidean numerical radius also satisfies the double inequality
1 2 n j = 1 n | T j | 2 1 2 w e ( T 1 , , T n ) j = 1 n | T j | 2 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ45_HTML.gif
(4.5)

for any ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq70_HTML.gif and the constants 1 2 n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq71_HTML.gif and 1 are best possible.

In [30], by utilizing the concept of hypo-Euclidean norm on H n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq72_HTML.gif, we obtained the following representation for the Euclidean norm.

Proposition 4.1 For any ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq73_HTML.gif, we have
( T 1 , , T n ) e = sup y = 1 , x = 1 ( j = 1 n | T j y , x | 2 ) 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ46_HTML.gif
(4.6)

We can state now the following result.

Theorem 4.1 For any ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq73_HTML.gif, we have
( T 1 , , T n ) e 2 1 2 [ ( j = 1 n | T j | 2 ) α ( j = 1 n | T j | 2 ) 1 α + ( j = 1 n | T j | 2 ) 1 α ( j = 1 n | T j | 2 ) α ] 1 2 [ j = 1 n | T j | 2 + j = 1 n | T j | 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ47_HTML.gif
(4.7)
and
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ48_HTML.gif
(4.8)

for any α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif.

Proof We have from the second inequality in (2.8)
j = 1 n | T j x , y | 2 1 2 [ ( j = 1 n | T j | 2 x , x ) α ( j = 1 n | T j | 2 y , y ) 1 α + ( j = 1 n | T j | 2 x , x ) 1 α ( j = 1 n | T j | 2 y , y ) α ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ49_HTML.gif
(4.9)
for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif with x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq11_HTML.gif and α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif. Taking the supremum over x = y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq74_HTML.gif, we have
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equn_HTML.gif
which proves the first part of (4.7). The second part follows by the elementary inequality
a α b 1 α α a + ( 1 α ) b http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equo_HTML.gif

for a , b 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq75_HTML.gif and α [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq7_HTML.gif. The inequality (4.8) follows from (4.9) by taking y = x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq76_HTML.gif and then the supremum over x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq27_HTML.gif. □

5 Applications for s-1-norm and s-1-numerical radius

Following [20], we consider the s-p-norm of the n-tuple of operators ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq66_HTML.gif by
( T 1 , , T n ) s , p : = sup y = 1 , x = 1 [ ( j = 1 n | T j y , x | p ) 1 p ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ50_HTML.gif
(5.1)
For p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq77_HTML.gif, we get
( T 1 , , T n ) s , 2 = ( T 1 , , T n ) e . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equp_HTML.gif
We are interested in this section in the case p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq78_HTML.gif, namely, on the s-1-norm defined by
( T 1 , , T n ) s , 1 : = sup y = 1 , x = 1 j = 1 n | T j y , x | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equq_HTML.gif
Since for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif we have j = 1 n | T j y , x | | j = 1 n T j y , x | http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq79_HTML.gif, then by the properties of the supremum, we get the basic inequality
j = 1 n T j ( T 1 , , T n ) s , 1 j = 1 n T j . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ51_HTML.gif
(5.2)
Similarly, we can also consider the s-p-numerical radius of the n-tuple of operators ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq73_HTML.gif by [20]
w s , p ( T 1 , , T n ) : = sup x = 1 [ ( j = 1 n | T j x , x | p ) 1 p ] , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ52_HTML.gif
(5.3)

which for p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq77_HTML.gif reduces to the Euclidean operator radius introduced previously.

We observe that the s-p-numerical radius is also a norm on B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq80_HTML.gif for p 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq81_HTML.gif, and for p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq78_HTML.gif it satisfies the basic inequality
w ( j = 1 n T j ) w s , 1 ( T 1 , , T n ) j = 1 n w ( T j ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ53_HTML.gif
(5.4)

We can state the following result.

Theorem 5.1 For any ( T 1 , , T n ) B ( n ) ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq73_HTML.gif, we have
http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ54_HTML.gif
(5.5)
and
w s , 1 ( T 1 , , T n ) j = 1 n ( | T j | 2 α + | T j | 2 ( 1 α ) + | T j | 2 α + | T j | 2 ( 1 α ) 4 ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ55_HTML.gif
(5.6)
Proof From (2.1) we have
j = 1 n | T j x , y | j = 1 n ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) x , x 1 / 2 × j = 1 n ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) y , y 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ56_HTML.gif
(5.7)

for any x , y H http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq6_HTML.gif.

Taking the supremum over y = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq50_HTML.gif, x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq27_HTML.gif in (5.7), we have
( T 1 , , T n ) s , 1 [ sup x = 1 j = 1 n ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) x , x ] 1 / 2 × [ sup y = 1 j = 1 n ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) y , y ] 1 / 2 = j = 1 n ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) 1 / 2 × j = 1 n ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equr_HTML.gif

and the first inequality in (5.5) is proved. The second part follows by the arithmetic mean-geometric mean inequality.

Now, if we take y = x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq76_HTML.gif in (5.7), then we get
j = 1 n | T j x , x | j = 1 n ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) x , x 1 / 2 × j = 1 n ( | T j | 2 α + | T j | 2 ( 1 α ) 2 ) x , x 1 / 2 1 2 j = 1 n ( | T j | 2 α + | T j | 2 ( 1 α ) + | T j | 2 α + | T j | 2 ( 1 α ) 2 ) x , x . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equs_HTML.gif

Taking the supremum over x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq27_HTML.gif, we deduce the desired result (5.6). □

Remark 5.1 If we take α = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq13_HTML.gif in the first inequality in (5.5), then we deduce
( T 1 , , T n ) s , 1 j = 1 n | T j | 1 / 2 j = 1 n | T j | 1 / 2 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ57_HTML.gif
(5.8)
and then we get the following refinement of the generalized triangle inequality:
j = 1 n T j ( T 1 , , T n ) s , 1 j = 1 n | T j | 1 / 2 j = 1 n | T j | 1 / 2 1 2 [ j = 1 n | T j | + j = 1 n | T j | ] j = 1 n T j . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equt_HTML.gif
From (5.6) we also have, for α = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq13_HTML.gif,
w s , 1 ( T 1 , , T n ) j = 1 n ( | T j | + | T j | 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_Equ58_HTML.gif
(5.9)

Declarations

Acknowledgements

The authors wish to thank the anonymous referees for their valuable comments. Also, this research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0008474).

Authors’ Affiliations

(1)
School of Computer Science and Mathematics, MCMC, Victoria University of Technology
(2)
School of Computational & Applied Mathematics, University of the Witwatersrand
(3)
Department of Mathematics Education and the RINS, Gyeongsang National University
(4)
Department of Mathematics, Changwon National University

References

  1. Helmberg G: Introduction to Spectral Theory in Hilbert Space. Wiley, New York; 1969.MATH
  2. Kato T: Notes on some inequalities for linear operators. Math. Ann. 1952, 125: 208–212. 10.1007/BF01343117MATHMathSciNetView Article
  3. Fujii M, Lin C-S, Nakamoto R: Alternative extensions of Heinz-Kato-Furuta inequality. Sci. Math. 1999, 2(2):215–221.MATHMathSciNet
  4. Fujii M, Furuta T, Löwner-Heinz C: Heinz-Kato inequalities. Math. Jpn. 1993, 38(1):73–78.MATH
  5. Fujii M, Kamei E, Kotari C, Yamada H: Furuta’s determinant type generalizations of Heinz-Kato inequality. Math. Jpn. 1994, 40(2):259–267.MATHMathSciNet
  6. Fujii M, Kim YO, Seo Y: Further extensions of Wielandt type Heinz-Kato-Furuta inequalities via Furuta inequality. Arch. Inequal. Appl. 2003, 1(2):275–283.MATHMathSciNet
  7. Fujii M, Kim YO, Tominaga M: Extensions of the Heinz-Kato-Furuta inequality by using operator monotone functions. Far East J. Math. Sci.: FJMS 2002, 6(3):225–238.MATHMathSciNet
  8. Fujii M, Nakamoto R: Extensions of Heinz-Kato-Furuta inequality. Proc. Am. Math. Soc. 2000, 128(1):223–228. 10.1090/S0002-9939-99-05242-9MATHMathSciNetView Article
  9. Fujii M, Nakamoto R: Extensions of Heinz-Kato-Furuta inequality. II. J. Inequal. Appl. 1999, 3(3):293–302.MATHMathSciNet
  10. Furuta T: Equivalence relations among Reid, Löwner-Heinz and Heinz-Kato inequalities, and extensions of these inequalities. Integral Equ. Oper. Theory 1997, 29(1):1–9. 10.1007/BF01191475MATHMathSciNetView Article
  11. Furuta T: Determinant type generalizations of Heinz-Kato theorem via Furuta inequality. Proc. Am. Math. Soc. 1994, 120(1):223–231.MATHMathSciNet
  12. Furuta T: An extension of the Heinz-Kato theorem. Proc. Am. Math. Soc. 1994, 120(3):785–787. 10.1090/S0002-9939-1994-1169027-6MATHMathSciNetView Article
  13. Kittaneh F: Notes on some inequalities for Hilbert space operators. Publ. Res. Inst. Math. Sci. 1988, 24(2):283–293. 10.2977/prims/1195175202MATHMathSciNetView Article
  14. Kittaneh F: Norm inequalities for fractional powers of positive operators. Lett. Math. Phys. 1993, 27(4):279–285. 10.1007/BF00777375MATHMathSciNetView Article
  15. Lin C-S: On Heinz-Kato-Furuta inequality with best bounds. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 2008, 15(1):93–101.MATHMathSciNet
  16. Lin C-S: On chaotic order and generalized Heinz-Kato-Furuta-type inequality. Int. Math. Forum 2007, 2(37–40):1849–1858.MATHMathSciNet
  17. Lin C-S: On inequalities of Heinz and Kato, and Furuta for linear operators. Math. Jpn. 1999, 50(3):463–468.MATH
  18. Lin C-S: On Heinz-Kato type characterizations of the Furuta inequality. II. Math. Inequal. Appl. 1999, 2(2):283–287.MATHMathSciNet
  19. Uchiyama M: Further extension of Heinz-Kato-Furuta inequality. Proc. Am. Math. Soc. 1999, 127(10):2899–2904. 10.1090/S0002-9939-99-05266-1MATHMathSciNetView Article
  20. Dragomir SS: Some inequalities of Kato’s type for sequences of operators in Hilbert spaces. Publ. Res. Inst. Math. Sci. 2012, 48: 937–955. 10.2977/PRIMS/92MATHMathSciNetView Article
  21. Cho YJ, Dragomir SS, Pearce CEM, Kim SS: Cauchy-Schwarz functionals. Bull. Aust. Math. Soc. 2000, 62: 479–491. 10.1017/S0004972700019006MATHMathSciNetView Article
  22. Dragomir SS, Cho YJ, Kim SS: Some inequalities in inner product spaces related to the generalized triangle inequality. Appl. Math. Comput. 2011, 217: 7462–7468. 10.1016/j.amc.2011.02.046MATHMathSciNetView Article
  23. Dragomir SS, Cho YJ, Kim JK: Subadditivity of some functionals associated to Jensen’s inequality with applications. Taiwan. J. Math. 2011, 15: 1815–1828.MATHMathSciNet
  24. Lin C-S, Cho YJ: On Hölder-McCarthy-type inequalities with power. J. Korean Math. Soc. 2002, 39: 351–361.MATHMathSciNetView Article
  25. Lin C-S, Cho YJ: On norm inequalities of operators on Hilbert spaces. 2. In Inequality Theory and Applications. Edited by: Cho YJ, Kim JK, Dragomir SS. Nova Science Publishers, New York; 2003:165–173.
  26. Lin C-S, Cho YJ: On Kantorovich inequality and Hölder-McCarthy inequalities. Dyn. Contin. Discrete Impuls. Syst. 2004, 11: 481–490.MATHMathSciNet
  27. Lin C-S, Cho YJ: Characteristic property for inequalities of bounded linear operators. 4. In Inequality Theory and Applications Edited by: Cho YJ, Kim JK, Dragomir SS. 2007, 85–92.
  28. Lin C-S, Cho YJ:Characterizations of operator inequality A B C http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq82_HTML.gif. Math. Inequal. Appl. 2011, 14: 575–580.MATHMathSciNet
  29. Popescu G: Unitary invariants in multivariable operator theory. Mem. Am. Math. Soc. 2009., 200: Article ID 941
  30. Dragomir SS: The hypo-Euclidean norm of an n -tuple of vectors in inner product spaces and applications. J. Inequal. Pure Appl. Math. 2007., 8(2): Article ID 52MathSciNet
  31. McCarthy CA: c p http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-21/MediaObjects/13660_2012_Article_454_IEq83_HTML.gif. Isr. J. Math. 1967, 5: 249–271. 10.1007/BF02771613MATHView Article

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