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International Journal of Analysis
Volume 2013 (2013), Article ID 243891, 4 pages
A New Identity for Resolvents of Operators
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.
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 . 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.
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. , 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 .
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 .
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.
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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