International Journal of Analysis

Volume 2013 (2013), Article ID 243891, 4 pages

http://dx.doi.org/10.1155/2013/243891

Research Article

A New Identity for Resolvents of Operators

Michael Gil'

Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel

Received 10 December 2012; Accepted 21 December 2012

Academic Editor: Jens Lorenz

Copyright © 2013 Michael Gil'. 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.

Abstract.

A new identity for resolvents of operators is suggested. We show that in appropriate situations it is more convenient than the Hilbert identity. In particular, we establish a new invertibility condition for perturbed operators as well as new bounds for the spectrum of perturbed operators. As a particular case we consider perturbations of Hilbert-Schmidt operators.

1. Introduction and the Main Result

Let be a complex Banach space with a norm and the unit operator . For a linear operator in , , is the spectrum, is the inverse operator, and is the resolvent.

Everywhere in the following and are bounded operators in , and . Recall the Hilbert identity [1]. In particular, it gives the following important result: if a is regular for and

then is also regular for . In the present paper we suggest a new identity for resolvents of operators, by which we derive a new invertibility condition for perturbed operators as well as new bounds for the spectrum of perturbed operators. It is shown that in appropriate situations our results improve condition (1). As a particular case we consider perturbations of Hilbert-Schmidt operators.

Put . Now we are in a position to formulate and prove our main result.

Theorem 1. Let a be regular for and . Then

Proof. We have

as claimed.

Denote .

Corollary 2. Let a be regular for and . Then is regular also for .

Indeed, put   . Since the regular sets of operators are open, is regular for , provided is small enough. By Theorem 1, we get

Hence,

Thus,

Taking , we obtain the required result.

Furthermore, we have , where

Now Corollary 2 implies.

Corollary 3. If a and , then .

Example 4. Let and with a nonzero number and a nilpotent operator in , such that .

It is clear that and is invertible. We have and . Therefore Corollary 3 gives us the sharp result: is invertible for all nonzero .

At the same time (1) gives the invertibility condition .

Example 5. Let be a direct sum of two spaces and . Besides is a Banach space with a norm . The norm in is introduced by , with an (). Let us consider the operator matrices

where and are commuting operators in . Let . Again . It is simple to check that , , and . Corollary 3 implies . At the same time, due to (1), we can assert that only if .

Furthermore, following the notions of the matrix perturbation theory, cf. [2], we will say that the spectral variation of with respect to is

and the Hausdorff distance between the spectra of and is

2. Perturbations of Hilbert-Schmidt Operators

In this section is a separable Hilbert space. Let

That is, is a Hilbert-Schmidt operator. Introduce the quantity

The following relations are checked in [3, Section 6.4]:

where . In our reasonings in the following one can replace by any of its upper bounds. In particular, one can replace by .

We need the following result.

Theorem 6. Let be a Hilbert-Schmidt operator. Then

where , the distance between and the spectrum of .

For the proof see [3, Theorem 6.4.1]. Now Corollary 3 implies the following.

Corollary 7. If is regular for , condition (11) holds and

then is regular for .

For any , due to Corollary 7, we have

Hence it follows that , where is the unique positive root of

But . We thus arrive at our next result.

Theorem 8. Let be a Hilbert-Schmidt operator and be an arbitrary bounded operator in . Then , where is the unique positive root of (17).

In Section 3 we obtain an estimate for .

If is normal, then , and consequently .

Assume that both and are Hilbert-Schmidt operators. Set

Now Theorem 8 implies the following.

Corollary 9. Let both and be Hilbert-Schmidt operators. Then , where is the unique positive root of the equation

In the following, we suggest an estimate also for .

Note that in [3, Theorem 8.5.1], the inequality is proved, where is the unique positive root

If this inequality gives us a nonzero result. At the same time, if (as in the above given examples), then Theorem 8 and Corollary 9 give us the sharp result .

Theorem 6 supplements the recent perturbation results for operators see the interesting papers [4–9] and references given therein.

3. Estimates for and

Denote

Note that as and . Similarly, as and .

Lemma 10. The following inequalities are true:

Proof. Substituting into (17), with the notation , we get

By the Schwarz inequality

Let be the unique positive root of (24). Then

and therefore, , where is the unique positive root of

We need the following simple result proved in [10, Lemma ].

Lemma 11. The unique positive root of the equation

satisfies the estimate

If, in addition, the condition holds, then .

Put in (27) . Then we obtain (28) with . Now (29) implies

Since we get inequality (22). Similarly, inequality (23) can be proved.

Now Theorem 8 and Corollary 9 imply the following.

Corollary 12. Let be a Hilbert-Schmidt operator and an arbitrary bounded operator in . Then . If both and are Hilbert-Schmidt operators, then .

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