On utilizing the spectral representation of selfadjoint 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
Let be a selfadjoint 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 RiemannStieltjes integral:
which in terms of vectors can be written as
for any . The function is of 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 Ostrowskitype vector inequality [1].
Theorem 1.1.
Let be a selfadjoint 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
for any and for any .
Another result that compares the function of a selfadjoint 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
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
for any .
For various inequalities for functions of selfadjoint operators, see [4–8]. For recent results see [1, 9–12].
In this paper, we investigate the quantity
where are vectors in the Hilbert space and is a selfadjoint 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.
Let be a selfadjoint 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
Proof.
We observe
which is an equality of interest in itself.
Since are projections, we have for any and then we can write
Integrating by parts in the RiemannStieltjes integral and utilizing the spectral representation (1.1), we have
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
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 selfadjoint 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
for any .
Proof.
It is well known that if is a bounded function, is of bounded variation, and the RiemannStieltjes integral exists, then the following inequality holds:
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
for any .
Since are projections, in this case we have
then from (2.8), we deduce the first part of (2.6).
Now, by the same property (2.7) for vectorvalued functions with values in Hilbert spaces, we also have
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 selfadjoint 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
for any .
Proof.
Recall that if is a Riemann integrable function and is Lipschitzian with the constant , that is,
then the RiemannStieltjes integral exists and the following inequality holds:
Now, on applying this property of the RiemannStieltjes integral, then we have from the representation (2.5) that
for any and the first inequality in (2.11) is proved.
Further, observe that
for any .
If we use the vectorvalued version of the property (2.13), then we have
for any and the second part of (2.11) is proved.
Further on, by applying the doubleintegral version of the CauchyBuniakowskiSchwarz inequality, we have
for any .
Now, by utilizing the fact that are projections for each , then we have
for any .
If we integrate by parts and use the spectral representation (1.2), then we get
and by (2.18), we then obtain the following equality of interest:
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
we also have that
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 selfadjoint 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
for any .
Proof.
From the theory of RiemannStieltjes integral, it is also well known that if is of bounded variation and is continuous and monotonic nondecreasing, then the RiemannStieltjes integrals and exist and
Now, on applying this property of the RiemannStieltjes integral, we have from the representation (2.5) that
for any , which proves the first inequality in (2.23).
On utilizing the CauchyBuniakowskiSchwarztype inequality for the RiemannStieltjes integral of monotonic nondecreasing integrators, we have
for any .
Observe that
and, integrating by parts in the RiemannStieltjes integral, we have
for any .
On making use of the equalities (2.28), we have
for any .
Therefore, we obtain the following equality of interest in itself as well:
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 selfadjoint operator in the Hilbert space with the spectrum for some real numbers with .
If , then for any ,
where
where
The proof follows from Theorem 2.4 applied for the power function.
Proposition 3.2.
Let be a selfadjoint operator in the Hilbert space with the spectrum for some real numbers with .
If , then for any
The proof follows from Theorem 2.5.
Now, consider the logarithmic function . We have the following
Proposition 3.3.
Let be a selfadjoint operator in the Hilbert space with the spectrum for some real numbers with . Then one has the inequalities
The proof follows from Theorems 2.4 and 2.5 applied for the logarithmic function.
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