A New Identity for Resolvents of Operators

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 𝑋 be a complex Banach space with a norm ‖⋅‖ and the unit operator 𝐼. For a linear operator 𝐴 in 𝑋, ‖𝐴‖=sup_𝑥∈𝑋‖𝐴𝑥‖/‖𝑥‖, 𝜎(𝐴) is the spectrum, 𝐴⁻¹ is the inverse operator, and 𝑅_𝜆(𝐴)=(𝐴−𝜆𝐼)⁻¹(𝜆∉𝜎(𝐴)) is the resolvent.

Everywhere in the following 𝐴 and 𝐴 are bounded operators in 𝑋, and 𝐸=𝐴−𝐴. Recall the Hilbert identity 𝑅_𝜆(𝐴)−𝑅_𝜆(𝐴)=−𝑅_𝜆(𝐴)𝐸𝑅_𝜆(𝐴) [1]. In particular, it gives the following important result: if a 𝜆∈ℂ is regular for 𝐴 and ‖𝐸‖𝑅_𝜆(𝐴)<1, (1) then 𝜆 is also regular for 𝐴. 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 𝑍=𝐴𝐸−𝐸𝐴. Now we are in a position to formulate and prove our main result.

Theorem 1. Let a 𝜆∈ℂ be regular for 𝐴 and 𝐴. Then 𝑅_𝜆𝐴−𝑅_𝜆(𝐴)=𝑅_𝜆𝐴𝑍𝑅²_𝜆(𝐴)−𝐸𝑅²_𝜆(𝐴). (2)

Proof. We have 𝑅_𝜆𝐴𝐴𝐸−𝐸𝐴𝑅²_𝜆(𝐴)−𝐸𝑅²_𝜆(𝐴)=𝑅_𝜆𝐴𝐴𝐸−𝐸𝐴−𝐸𝑅²_𝜆(𝐴)=𝑅_𝜆𝐴𝐴𝐸−𝐸𝐴−𝐴−𝜆𝐼𝐸𝑅²_𝜆(𝐴)=𝑅_𝜆𝐴(𝐸𝜆−𝐸𝐴)𝑅²_𝜆(𝐴)=−𝑅_𝜆𝐴𝐸𝑅_𝜆(𝐴)=−𝑅_𝜆𝐴𝐴−𝜆𝐼−(𝐴−𝜆𝐼)𝑅_𝜆(𝐴)=−𝐼−𝑅_𝜆𝐴(𝐴−𝜆𝐼)𝑅_𝜆(𝐴)=𝑅_𝜆𝐴−𝑅_𝜆(𝐴), (3) as claimed.

Denote 𝜂(𝐴,𝐸,𝜆)=sup_0≤𝑡≤1𝑡‖(𝐴𝐸−𝐸𝐴+𝑡𝐸²)𝑅²_𝜆(𝐴)‖.

Corollary 2. Let a 𝜆∈ℂ be regular for 𝐴 and 𝜂(𝐴,𝐸,𝜆)<1. Then 𝜆 is regular also for 𝐴.

Indeed, put 𝐴_𝑡=𝐴+𝑡𝐸  (𝑡∈[0,1]). Since the regular sets of operators are open, 𝜆 is regular for 𝐴_𝑡, provided 𝑡 is small enough. By Theorem 1, we get 𝑅_𝜆𝐴_𝑡−𝑅_𝜆(𝐴)=𝑅_𝜆𝐴_𝑡(𝑡(𝐴+𝑡𝐸)𝐸−𝑡𝐸𝐴)𝑅²_𝜆(𝐴)−𝑡𝐸𝑅²_𝜆(𝐴). (4) Hence, 𝑅_𝜆𝐴_𝑡−𝑅_𝜆(𝐴)−𝑡𝐸𝑅²_𝜆(𝐴≤𝑅_𝜆)𝐴_𝑡𝑡(𝐸𝐴−𝐴𝐸)+𝑡²𝐸²𝑅²_𝜆(𝐴≤𝑅_𝜆)𝐴_𝑡𝜂(𝐴,𝐸,𝜆). (5) Thus, 𝑅_𝜆𝐴_𝑡≤𝑅_𝜆(𝐴)−𝑡𝐸𝑅²_𝜆(𝐴)().1−𝜂𝐴,𝐸,𝜆 (6) Taking 𝑡=1, we obtain the required result.

Furthermore, we have 𝜂(𝐴,𝐸,𝜆)≤𝜁²(𝐴,𝐸)‖𝑅²_𝜆(𝐴)‖, where 𝜁(𝐴,𝐸)∶=‖𝐴𝐸−𝐸𝐴‖+𝐸². (7) Now Corollary 2 implies.

Corollary 3. If a 𝜆∉𝜎(𝐴) and 𝜁(𝐴,𝐸)‖𝑅_𝜆(𝐴)‖<1, then 𝜆∉𝜎(𝐴).

Example 4. Let 𝐴=𝑐𝐼 and 𝐴=𝑐𝐼+𝑉 with a nonzero number 𝑐 and a nilpotent operator 𝑉 in 𝑋, such that 𝑉²=0.

It is clear that 𝐸=𝑉 and 𝐴 is invertible. We have 𝐴𝐸−𝐸𝐴=0 and 𝜁(𝐴,𝐸)=0. Therefore Corollary 3 gives us the sharp result: 𝐴 is invertible for all nonzero 𝑐.

At the same time (1) gives the invertibility condition ‖𝑉‖<|𝑐|.

Example 5. Let 𝑋=𝑌₁+𝑌₂ be a direct sum of two spaces 𝑌₁ and 𝑌₂. Besides 𝑌₁=𝑌₂=𝑌 is a Banach space with a norm ‖⋅‖_𝑌. The norm in 𝑋 is introduced by ‖𝑥‖=‖𝑦₁‖_𝑌+‖𝑦₂‖_𝑌, with an 𝑥=𝑦₁+𝑦₂ (𝑦₁∈𝑌₁,𝑦₂∈𝑌₂). Let us consider the operator matrices 𝐴=,𝐵00𝐵𝐴=𝐵𝐶0𝐵, (8) where 𝐶 and 𝐵 are commuting operators in 𝑌. Let 𝜆∉𝜎(𝐴). Again 𝐸=𝐴−𝐴. It is simple to check that 𝐸²=0, ‖𝐸‖=‖𝐶‖_𝑌, and 𝐴𝐸−𝐸𝐴=0. Corollary 3 implies 𝜆∉𝜎(𝐴). At the same time, due to (1), we can assert that 𝜆∉𝜎(𝐴) only if ‖𝐶‖_𝑌‖𝑅_𝜆(𝐴)‖<1.

Furthermore, following the notions of the matrix perturbation theory, cf. [2], we will say that the spectral variation of 𝐴 with respect to 𝐴 is 𝑠𝑣_𝐴𝐴∶=sup_𝜇∈𝜎𝐴inf_{(𝐴)𝑠∈𝜎}𝜇−𝑠, (9)and the Hausdorff distance between the spectra of𝐴 and 𝐴 ishd𝐴,𝐴∶=max𝑠𝑣_𝐴𝐴,𝑠𝑣_𝐴(𝐴). (10)

2. Perturbations of Hilbert-Schmidt Operators

In this section 𝑋=𝐻 is a separable Hilbert space. Let 𝑁₂(𝐴)∶=Trace𝐴𝐴^∗1/2<∞. (11) That is, 𝐴 is a Hilbert-Schmidt operator. Introduce the quantity 𝑔(𝐴)∶=𝑁²₂(𝐴)−^∞_𝑘=1𝜆_𝑘(𝐴)^21/2. (12) The following relations are checked in [3, Section 6.4]: 𝑔²(𝐴)≤𝑁²₂(𝐴)−Trace𝐴²,𝑔²(𝐴)≤𝑁²₂𝐴−𝐴^∗2=2𝑁²₂𝐴_𝐼, (13) where 𝐴_𝐼=(𝐴−𝐴^∗)/2𝑖. In our reasonings in the following one can replace 𝑔(𝐴) by any of its upper bounds. In particular, one can replace 𝑔(𝐴) by √2𝑁₂(𝐴_𝐼).

We need the following result.

Theorem 6. Let 𝐴 be a Hilbert-Schmidt operator. Then 𝑅_𝜆(𝐴)≤^∞_𝑘=0𝑔^𝑘(𝐴)𝜌^𝑘+1(𝐴,𝜆)√((𝐴,𝑘!𝜆∉𝜎)) (14) where 𝜌(𝐴,𝜆)=inf_{𝑠∈𝜎(𝐴)}|𝑠−𝜆|, the distance between 𝜆 and the spectrum of 𝐴.

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

Corollary 7. If 𝜆 is regular for 𝐴, condition (11) holds and 𝜁(𝐴,𝐸)^∞_𝑘=0𝑔^𝑘(𝐴)𝜌^𝑘+1(𝐴,𝜆)√𝑘!<1, (15) then 𝜆 is regular for 𝐴.

For any 𝜇∈𝜎(𝐴), due to Corollary 7, we have 𝜁(𝐴,𝐸)^∞_𝑘=0𝑔^𝑘(𝐴)𝜌^𝑘+1𝐴,𝜇√𝑘!≥1. (16) Hence it follows that 𝜌(𝐴,𝜇)≤𝑥₀, where 𝑥₀ is the unique positive root of 𝜁(𝐴,𝐸)^∞_𝑘=0𝑔^𝑘(𝐴)𝑥^𝑘+1√𝑘!=1. (17) But 𝑠𝑣_𝐴(𝐴)=sup_𝜇∈𝜎(𝐴₎𝜌(𝐴,𝜇). We thus arrive at our next result.

Theorem 8. Let 𝐴 be a Hilbert-Schmidt operator and 𝐴 be an arbitrary bounded operator in 𝐻. Then 𝑠𝑣_𝐴(𝐴)≤𝑥₀, where 𝑥₀ is the unique positive root of (17).

In Section 3 we obtain an estimate for 𝑥₀.

If 𝐴 is normal, then 𝑔(𝐴)=0, and consequently 𝑥₀=𝜁(𝐴,𝐸).

Assume that both 𝐴 and 𝐴 are Hilbert-Schmidt operators. Set 𝑔=max𝑔(𝐴),𝑔𝐴. (18) Now Theorem 8 implies the following.

Corollary 9. Let both 𝐴 and 𝐴 be Hilbert-Schmidt operators. Then hd(𝐴,𝐴)≤𝑧(𝑔), where 𝑧(𝑔) is the unique positive root of the equation 𝜁(𝐴,𝐸)^∞_𝑘=0𝑔^𝑘𝑥^𝑘+1√𝑘!=1. (19) In the following, we suggest an estimate also for 𝑧(𝑔).

Note that in [3, Theorem 8.5.1], the inequality 𝑠𝑣_𝐴(𝐴)≤𝑥(‖𝐸‖,𝐴) is proved, where 𝑥(‖𝐸‖,𝐴) is the unique positive root ‖𝐸‖𝑥1exp2+𝑔²(𝐴)𝑥²=1. (20) If 𝐸≠0 this inequality gives us a nonzero result. At the same time, if 𝑍=0 (as in the above given examples), then Theorem 8 and Corollary 9 give us the sharp result 𝑠𝑣_𝐴(𝐴)=hd(𝐴,𝐴)=0.

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

3. Estimates for 𝑥₀ and 𝑧(𝑔)

Denote 𝛾()𝑏√𝑏,𝑐∶=2ln^1/21/2+1/4+𝑏²/𝑐². (21) Note that 𝛾(𝑏,𝑐)→0 as 𝑏→0 and 𝑐>0. Similarly, 𝛾(𝑏,𝑐)→0 as 𝑐→0 and 𝑏>0.

Lemma 10. The following inequalities are true: 𝑥₀𝑔(𝐴)(),𝑧𝑔().≤𝛾,𝜁𝐴,𝐸≤𝛾𝑔,𝜁𝐴,𝐸 (22)(23)

Proof. Substituting 𝑥=𝑔(𝐴)𝑦 into (17), with the notation 𝑞=𝜁(𝐴,𝐸)/𝑔(𝐴), we get 1=𝑞^∞_𝑘=01𝑦^𝑘+1√.𝑘! (24) By the Schwarz inequality ^∞_𝑘=01𝑦^𝑘√𝑘!²=^∞_𝑘=0(√2)^𝑘(√2)^𝑘𝑦^𝑘√𝑘!²≤^∞_𝑗=02^∞−𝑗_𝑘=02^𝑘𝑦^2𝑘𝑘!=2𝑒^2/𝑦2. (25) Let 𝑦₀=𝑥₀/𝑔(𝐴) be the unique positive root of (24). Then 𝑞√1≤2𝑦₀𝑒^1/𝑦20or1≤2𝑞²𝑦²₀𝑒^2/𝑦20 (26) and therefore, 𝑦₀≤𝑦, where 𝑦 is the unique positive root of 1=2𝑞²𝑦²𝑒^2/𝑦2. (27) We need the following simple result proved in [10, Lemma 1.6.5].

Lemma 11. The unique positive root 𝑧₀ of the equation 𝑧𝑒^𝑧=𝑎(𝑎=const>0) (28) satisfies the estimate 𝑧₀1≥ln2+14+𝑎. (29) If, in addition, the condition 𝑎≥𝑒 holds, then 𝑧₀≥ln𝑎−lnln𝑎.
Put in (27) 𝑧=2/𝑦². Then we obtain (28) with 𝑎=1/𝑞². Now (29) implies √𝑦≤2ln^1/21/2+1/4+1/𝑞². (30) Since 𝑦≥𝑦₀=𝑥₀/𝑔(𝐴) we get inequality (22). Similarly, inequality (23) can be proved.

Now Theorem 8 and Corollary 9 imply the following.

Corollary 12. Let 𝐴 be a Hilbert-Schmidt operator and 𝐴 an arbitrary bounded operator in 𝐻. Then 𝑠𝑣_𝐴(𝐴)≤𝛾(𝑔(𝐴),𝜁(𝐴,𝐸)). If both 𝐴 and 𝐴 are Hilbert-Schmidt operators, then hd(𝐴,𝐴)≤𝛾(𝑔,𝜁(𝐴,𝐸)).