# A New Identity for Resolvents of Operators

Authors: Michael Gil'
Journal: International Journal of Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction and the Main Result

LetX be a complex Banach space with a norm ∥·∥ and the unit operator I. For a linear operator A in X, ∥A∥=supx∈X∥Ax∥/∥x∥, σ(A) is the spectrum, A-1 is the inverse operator, and Rλ(A)=(A-λI)-1(λ∉σ(A)) is the resolvent.Everywhere in the followingA and A~ are bounded operators in X, and E=A~-A. Recall the Hilbert identity Rλ(A~)-Rλ(A)=-Rλ(A)ERλ(A~) [1]. In particular, it gives the following important result: if a λ∈ℂ is regular for A and
(1)∥E∥∥Rλ(A)∥<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.PutZ=A~E-EA. Now we are in a position to formulate and prove our main result.Theorem 1.
Let aλ∈ℂ be regular for A and A~. Then
(2)Rλ(A~)-Rλ(A)=Rλ(A~)ZRλ2(A)-ERλ2(A).Proof.
We have(3)Rλ(A~)(A~E-EA)Rλ2(A)-ERλ2(A)=[Rλ(A~)(A~E-EA)-E]Rλ2(A)=Rλ(A~)[A~E-EA-(A~-λI)E]Rλ2(A)=Rλ(A~)(Eλ-EA)Rλ2(A)=-Rλ(A~)ERλ(A)=-Rλ(A~)(A~-λI-(A-λI))Rλ(A)=-(I-Rλ(A~)(A-λI))Rλ(A)=Rλ(A~)-Rλ(A),
as claimed.Denoteη(A,E,λ)=sup0≤t≤1t∥(AE-EA+tE2)Rλ2(A)∥.Corollary 2.
Let aλ∈ℂ be regular for A and η(A,E,λ)<1. Then λ is regular also for A~.Indeed, putAt=A+tE  (t∈[0,1]). Since the regular sets of operators are open, λ is regular for At, provided t is small enough. By Theorem 1, we get
(4)Rλ(At)-Rλ(A)=Rλ(At)(t(A+tE)E-tEA)Rλ2(A)-tERλ2(A).
Hence,
(5)∥Rλ(At)∥-∥Rλ(A)-tERλ2(A)∥≤∥Rλ(At)∥∥[t(EA-AE)+t2E2]Rλ2(A)∥≤∥Rλ(At)∥η(A,E,λ).
Thus,
(6)∥Rλ(At)∥≤∥Rλ(A)-tERλ2(A)∥1-η(A,E,λ).
Taking t=1, we obtain the required result.Furthermore, we haveη(A,E,λ)≤ζ2(A,E)∥Rλ2(A)∥, where
(7)ζ(A,E):=∥AE-EA∥+∥E2∥.
Now Corollary 2 implies.Corollary 3.
If aλ∉σ(A) and ζ(A,E)∥Rλ(A)∥<1, then λ∉σ(A~).Example 4.
LetA=cI and A~=cI+V with a nonzero number c and a nilpotent operator V in X, such that V2=0.It is clear thatE=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.
LetX=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
(8)A=(B00B),A~=(BC0B),
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∥Rλ(A)∥<1.Furthermore, following the notions of the matrix perturbation theory, cf. [2], we will say that the spectral variation ofA~with respect toAis(9)svA(A~)∶=supμ∈σ(A~)infs∈σ(A)|μ-s|,and the Hausdorff distance between the spectra ofAandA~is(10)hd(A,A~)∶=max{svA(A~),svA~(A)}.


## 2. Perturbations of Hilbert-Schmidt Operators

In this sectionX=H is a separable Hilbert space. Let
(11)N2(A)∶=[Trace(AA*)]1/2<∞.
That is, A is a Hilbert-Schmidt operator. Introduce the quantity
(12)g(A)∶=[N22(A)-∑k=1∞‍|λk(A)|2]1/2.
The following relations are checked in [3, Section 6.4]:
(13)g2(A)≤N22(A)-|TraceA2|,g2(A)≤N22(A-A*)2=2N22(AI),
where AI=(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(AI).We need the following result.Theorem 6.
LetA be a Hilbert-Schmidt operator. Then
(14)∥Rλ(A)∥≤∑k=0∞‍gk(A)ρk+1(A,λ)k!(λ∉σ(A)),
where ρ(A,λ)=infs∈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.
Ifλ is regular for A, condition (11) holds and
(15)ζ(A,E)∑k=0∞‍gk(A)ρk+1(A,λ)k!<1,
then λ is regular for A~.For anyμ∈σ(A), due to Corollary 7, we have
(16)ζ(A,E)∑k=0∞‍gk(A)ρk+1(A,μ)k!≥1.
Hence it follows that ρ(A,μ)≤x0, where x0 is the unique positive root of
(17)ζ(A,E)∑k=0∞‍gk(A)xk+1k!=1.
But svA(A~)=supμ∈σ(A~)ρ(A,μ). We thus arrive at our next result.Theorem 8.
LetA be a Hilbert-Schmidt operator and A~ be an arbitrary bounded operator in H. Then svA(A~)≤x0, where x0 is the unique positive root of (17).In Section3 we obtain an estimate for x0.IfA is normal, then g(A)=0, and consequently x0=ζ(A,E).Assume that bothA and A~ are Hilbert-Schmidt operators. Set
(18)g^=max{g(A),g(A~)}.
Now Theorem 8 implies the following.Corollary 9.
Let bothA and A~ be Hilbert-Schmidt operators. Then hd(A,A~)≤z(g^), where z(g^) is the unique positive root of the equation
(19)ζ(A,E)∑k=0∞‍g^kxk+1k!=1.
In the following, we suggest an estimate also for z(g^).Note that in [3, Theorem 8.5.1], the inequality svA(A~)≤x^(∥E∥,A) is proved, where x^(∥E∥,A) is the unique positive root
(20)∥E∥xexp[12+g2(A)x2]=1.
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 svA(A~)=hd(A,A~)=0.Theorem6 supplements the recent perturbation results for operators see the interesting papers [4–9] and references given therein.


## 3. Estimates forx0 and z(g^)

Denote(21)γ(b,c)∶=b2ln1/2[1/2+1/4+b2/c2].
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:(22)x0≤γ(g(A),ζ(A,E)),(23)z(g^)≤γ(g^,ζ(A,E)).Proof.
Substitutingx=g(A)y into (17), with the notation q=ζ(A,E)/g(A), we get
(24)1=q∑k=0∞‍1yk+1k!.
By the Schwarz inequality
(25)(∑k=0∞‍1ykk!)2=(∑k=0∞‍(2)k(2)kykk!)2≤∑j=0∞‍2-j∑k=0∞‍2ky2kk!=2e2/y2.
Let y0=x0/g(A) be the unique positive root of (24). Then
(26)1≤q2y0e1/y02or1≤2q2y02e2/y02
and therefore, y0≤y^, where y^ is the unique positive root of
(27)1=2q2y2e2/y2.
We need the following simple result proved in [10, Lemma 1.6.5].Lemma 11.
The unique positive rootz0 of the equation
(28)zez=a(a=const>0)
satisfies the estimate
(29)z0≥ln[12+14+a].
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
(30)y^≤2ln1/2[1/2+1/4+(1/q2)].
Since y^≥y0=x0/g(A) we get inequality (22). Similarly, inequality (23) can be proved.Now Theorem8 and Corollary 9 imply the following.Corollary 12.
LetA be a Hilbert-Schmidt operator and A~ an arbitrary bounded operator in H. Then svA(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)).



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