ANew Identity for Resolvents of Operators

Let XX be a complex Banach space with a norm ‖ ⋅ ‖ and the unit operator II. For a linear operator AA in XX, ‖AA‖ A supxxxXX‖AAxx‖A‖xx‖, σσσAAσ is the spectrum, AA −1 is the inverse operator, and RRλλσAAσ A σAA − λλIIσ σλλ λ σσσAAσσ is the resolvent. Everywhere in the following AA and 󵰒󵰒 AA are bounded operators in XX, and EE A 󵰒󵰒 AA − AA. Recall the Hilbert identity RRλλσ󵰒󵰒 AAσ − RRλλσAAσ A −RRλλσAAσEERRλλσ󵰒󵰒 AAσ [1]. In particular, it gives the following important result: if a λλ x λ is regular forAA and


Introduction and the Main Result
Let  be a complex Banach space with a norm ‖ ⋅ ‖ and the unit operator .For a linear operator  in , ‖‖  sup  ‖‖‖‖,  is the spectrum,  −1 is the inverse operator, and      −  −1    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. us, Taking   1, we obtain the required result.Furthermore, we have (  ) ≤  2 ( ) where ( )  inf ∈()  − , the distance between  and the spectrum of .
For the proof see [3, eorem 6.4.1].Now Corollary 3 implies the following.then  is regular for  .
For any  ∈ (), due to Corollary 7, we have Hence it follows that ( ) ≤  0 , where  0 is the unique positive root of But   (  )  sup ∈(  ) ( ).We thus arrive at our next result.eorem 8. Let  be a Hilbert-Schmidt operator and   be an arbitrary bounded operator in .en   (  ) ≤  0 , where  0 is the unique positive root of (17).
In Section 3 we obtain an estimate for  0 .If  is normal, then   0, and consequently  0   . Assume
eorem 1.Let a    be regular for  and  .en      −           2   −  2   .(2) It is clear that    and  is invertible.We have  −   0 and ( )  0. erefore Corollary 3 gives us the sharp result:   is invertible for all nonzero .At the same time (1) gives the invertibility condition   .Let    1 +  2 be a direct sum of two spaces  1 and  2 .Besides  1   2   is a Banach space with a norm  ⋅   .e norm in  is introduced by    1   +  2   , with an    1 +  2 ( 1 ∈  1   2 ∈  2 ).Let us consider the operator matrices Again     − .It is simple to check that  2  0,     , and  −   0. Corollary 3 implies   (  ).At the same time, due to (1), we can assert that   (  ) only if     ()  1.
Corollary 3. If a   () and ( )  ()  1, then   (  ).Example 4. Let    and    + with a nonzero number  and a nilpotent operator  in , such that  2  0. where  and  are commuting operators in .Let   ().and the Hausdorff distance between the spectra of  and   is hd    ∶ max          () .* )/2.In our reasonings in the following one can replace () by any of its upper bounds.In particular, one can replace () by √ 2 2 (  ).We need the following result.eorem 6.Let  be a Hilbert-Schmidt operator.en   () ≤ that both  and   are Hilbert-Schmidt operators.Note that    0 as   0 and   0. Similarly,    0 as   0 and   0.If, in addition, the condition  ≥  holds, then  0 ≥ ln  − ln ln .Put in (27)   2/ 2 .en we obtain (28) with   / 2 .Since   ≥  0   0 / we get inequality (22).Similarly, inequality (23) can be proved.Now eorem 8 and Corollary 9 imply the following.Let  be a Hilbert-Schmidt operator and   an arbitrary bounded operator in .en         .If both  and   are Hilbert-Schmidt operators, then hd        .