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 .