# Some Vector Inequalities for Continuous Functions of Self-Adjoint Operators in Hilbert Spaces

- SS Dragomir
^{1, 2}Email author

**2011**:564836

**DOI: **10.1155/2011/564836

© S. S. Dragomir. 2011

**Received: **24 November 2010

**Accepted: **21 February 2011

**Published: **15 March 2011

## Abstract

On utilizing the spectral representation of self-adjoint operators in Hilbert spaces, some inequalities for the composite operator , where and for various classes of continuous functions are given. Applications for the power function and the logarithmic function are also provided.

## 1. Introduction

*spectral family*. Then, for any continuous function , it is well known that we have the following

*spectral representation in terms of the Riemann-Stieltjes integral*:

*bounded variation*on the interval and

for any
. It is also well known that
is *monotonic nondecreasing* and *right continuous* on
.

Utilising the spectral representation from (1.2), we have established the following Ostrowski-type vector inequality [1].

Theorem 1.1.

for any and for any .

Another result that compares the function of a self-adjoint operator with the integral mean is embodied in the following theorem [2].

Theorem 1.2.

for any .

The trapezoid version of the above result has been obtained in [3] and is as follows.

Theorem 1.3.

for any .

For various inequalities for functions of self-adjoint operators, see [4–8]. For recent results see [1, 9–12].

where are vectors in the Hilbert space and is a self-adjoint operator with , and provide different bounds for some classes of continuous functions . Applications for some particular cases including the power and logarithmic functions are provided as well.

## 2. Some Vector Inequalities

The following representation in terms of the spectral family is of interest in itself.

Lemma 2.1.

Proof.

which is an equality of interest in itself.

which together with (2.3) and (2.2) produce the desired result (2.1).

The following vector version may be stated as well.

Corollary 2.2.

for any .

The following result that provides some bounds for continuous functions of bounded variation may be stated as well.

Theorem 2.3.

for any .

Proof.

where denotes the total variation of on .

for any .

then from (2.8), we deduce the first part of (2.6).

for any and .

Since in the operator order, then which gives that , that is, for any , which implies that for any . Therefore, which together with (2.10) prove the last part of (2.6).

The case of Lipschitzian functions is as follows.

Theorem 2.4.

for any .

Proof.

for any and the first inequality in (2.11) is proved.

for any .

for any and the second part of (2.11) is proved.

for any .

for any .

for any .

On making use of (2.20) and (2.17), we then deduce the third part of (2.11).

for any , which proves the last inequality in (2.11).

The case of nondecreasing monotonic functions is as follows.

Theorem 2.5.

for any .

Proof.

for any , which proves the first inequality in (2.23).

for any .

for any .

for any .

for any

On making use of the inequality (2.26), we deduce the second inequality in (2.23).

The last part follows by (2.21), and the details are omitted.

## 3. Applications

We consider the power function , where and . The following power inequalities hold.

Proposition 3.1.

Let be a self-adjoint operator in the Hilbert space with the spectrum for some real numbers with .

The proof follows from Theorem 2.4 applied for the power function.

Proposition 3.2.

Let be a self-adjoint operator in the Hilbert space with the spectrum for some real numbers with .

The proof follows from Theorem 2.5.

Now, consider the logarithmic function . We have the following

Proposition 3.3.

The proof follows from Theorems 2.4 and 2.5 applied for the logarithmic function.

## Authors’ Affiliations

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## Copyright

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