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

Let be a self-adjoint operator on the complex Hilbert space with the spectrum included in the interval for some real numbers and let be its 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:

(1.1)

which in terms of vectors can be written as

(1.2)

for any . The function is of bounded variation on the interval and

(1.3)

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.

Let be a self-adjoint operator in the Hilbert space with the spectrum for some real numbers and let be its spectral family. If is a continuous function of bounded variation on , then one has the inequality

(1.4)

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.

With the assumptions in Theorem 1.1 one has the inequalities

(1.5)

for any .

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

Theorem 1.3.

With the assumptions in Theorem 1.1 one has the inequalities

(1.6)

for any .

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

In this paper, we investigate the quantity

(1.7)

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.

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

Lemma 2.1.

Let be a self-adjoint operator in the Hilbert space with the spectrum for some real numbers and let be its spectral family. If is a continuous function on with , then one has the representation

(2.1)

Proof.

We observe

(2.2)

which is an equality of interest in itself.

Since are projections, we have for any and then we can write

(2.3)

Integrating by parts in the Riemann-Stieltjes integral and utilizing the spectral representation (1.1), we have

(2.4)

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.

With the assumptions of Lemma 2.1 one has the equality

(2.5)

for any .

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

Theorem 2.3.

Let be a self-adjoint operator in the Hilbert space with the spectrum for some real numbers , and let be its spectral family. If is a continuous function of bounded variation on with , then we have the inequality

(2.6)

for any .

Proof.

It is well known that if is a bounded function, is of bounded variation, and the Riemann-Stieltjes integral exists, then the following inequality holds:

(2.7)

where denotes the total variation of on .

Utilising this property and the representation (2.5), we have by the Schwarz inequality in Hilbert space that

(2.8)

for any .

Since are projections, in this case we have

(2.9)

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

Now, by the same property (2.7) for vector-valued functions with values in Hilbert spaces, we also have

(2.10)

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.

Let be a self-adjoint operator in the Hilbert space with the spectrum for some real numbers , and let be its spectral family. If is a Lipschitzian function with the constant on and with , then one has the inequality

(2.11)

for any .

Proof.

Recall that if is a Riemann integrable function and is Lipschitzian with the constant , that is,

(2.12)

then the Riemann-Stieltjes integral exists and the following inequality holds:

(2.13)

Now, on applying this property of the Riemann-Stieltjes integral, then we have from the representation (2.5) that

(2.14)

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

Further, observe that

(2.15)

for any .

If we use the vector-valued version of the property (2.13), then we have

(2.16)

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

Further on, by applying the double-integral version of the Cauchy-Buniakowski-Schwarz inequality, we have

(2.17)

for any .

Now, by utilizing the fact that   are projections for each , then we have

(2.18)

for any .

If we integrate by parts and use the spectral representation (1.2), then we get

(2.19)

and by (2.18), we then obtain the following equality of interest:

(2.20)

for any .

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

Finally, by utilizing the elementary inequality in inner product spaces

(2.21)

we also have that

(2.22)

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

The case of nondecreasing monotonic functions is as follows.

Theorem 2.5.

Let be a self-adjoint operator in the Hilbert space with the spectrum for some real numbers , and let be its spectral family. If is a monotonic nondecreasing function on , then one has the inequality

(2.23)

for any .

Proof.

From the theory of Riemann-Stieltjes integral, it is also well known that if is of bounded variation and is continuous and monotonic nondecreasing, then the Riemann-Stieltjes integrals and exist and

(2.24)

Now, on applying this property of the Riemann-Stieltjes integral, we have from the representation (2.5) that

(2.25)

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

On utilizing the Cauchy-Buniakowski-Schwarz-type inequality for the Riemann-Stieltjes integral of monotonic nondecreasing integrators, we have

(2.26)

for any .

Observe that

(2.27)

and, integrating by parts in the Riemann-Stieltjes integral, we have

(2.28)

for any .

On making use of the equalities (2.28), we have

(2.29)

for any .

Therefore, we obtain the following equality of interest in itself as well:

(2.30)

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.

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 .

If , then for any ,

(3.1)

where

(3.2)

where

(3.3)

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 .

If , then for any

(3.4)

The proof follows from Theorem 2.5.

Now, consider the logarithmic function . We have the following

Proposition 3.3.

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

(3.5)

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