Spectrum perturbations of compact operators in a Banach space

Michael Gil’

doi:10.1515/math-2019-0085

Artikel Open Access

Spectrum perturbations of compact operators in a Banach space

Michael Gil’

Veröffentlicht/Copyright: 14. September 2019

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Aus der Zeitschrift Open Mathematics Band 17 Heft 1

Abstract

For an integer p ≥ 1, let Γ_p be an approximative quasi-normed ideal of compact operators in a Banach space with a quasi-norm N_{Γ_p}(.) and the property

∑k=1∞|λk(A)|p≤apNΓpp(A)(A∈Γp),

where λ_k(A) (k = 1, 2, …) are the eigenvalues of A and a_p is a constant independent of A. Let A, Ã ∈ Γ_p and

Δp(A,A~):=NΓp(A−A~)expapbpp1+12(NΓp(A+A~)+NΓp(A−A~))p,

where b_p is the quasi-triangle constant in Γ_p. It is proved the following result: let I be the unit operator, I – A^p be boundedly invertible and

Δp(A,A~)expapNΓpp(A)ψp(A)<1,

where ψ_p(A) = inf_k=1,2,… |1 – λkp (A)|. Then I – Ã^p is also boundedly invertible. Applications of that result to the spectrum perturbations of absolutely p-summing and absolutely (p, 2) summing operators are also discussed. As examples we consider the Hille-Tamarkin integral operators and matrices.

Keywords: Banach space; compact operators; perturbations; absolutely p-summing operators; absolutely (p, 2)-summing operators; integral operators; infinite matrices

MSC 2010: 47A10; 47A55; 47B10; 47A75

1 Introduction and statement of the main result

Roughly speaking, the spectrum perturbation theory for linear operators consists of two approaches. In the framework of the first one some structure on the error is imposed; for example, they may be analytic functions of a complex variable. The problem is then to determine how this structure affects the perturbed spectrum: e.g., when are they analytic functions of the variable, what kind of paths do they follow in the complex plane? That approach is well developed. For various results of this kind see for instance the book by Kato [1]. In the framework of the second approach the errors are unstructured and perturbations are bounded in terms of some norm of the errors. That approach in the case of operators in a Banach space to the best of our knowledge is at an early stage of development. Below we suggest perturbation results for compact operators in a Banach space, which are connected with the second approach.

Throughout this paper 𝓧 is a Banach space with the approximation property [2], the unit operator I and norm ∥.∥_𝓧 = ∥.∥, (𝓑(𝓧)) is the algebra of all bounded linear operators in 𝓧. For a compact operator A, λ_k(A) (k = 1, 2, …) are the eigenvalues counted with their algebraic multiplicities. A point λ ∈ ℂ is said to be Φ-regular for A if I – λA is boundedly invertible; σ_Φ(A) denotes the Fredholm spectrum (the complement of all Φ-regular points in the closed complex plane).

For an integer p ≥ 1 introduce the two-sided quasi-normed ideal Γ_p of compact operators in (𝓑(𝓧)) with a quasi-norm N_{Γ_p}(.) and the property

∑k=1∞|λk(A)|p≤apNΓpp(A)(A∈Γp), (1.1)

where a_p is a constant independent of A, and Γ_p is approximative (i.e. the set of all finite rank operators is dense in Γ_p). Below b_p denotes the quasi-triangle constant in Γ_p:

NΓp(A+A~)≤bp(NΓp(A)+NΓp(A~))(A,A~∈Γp). (1.2)

For the theory of the approximative normed and quasi-normed ideals see [2, 3] and references given therein. In the sequel constant a_p in (1.1) will be called the eigenvalue constant.

Put

Δp(A,A~):=NΓp(A−A~)expapbpp1+12(NΓp(A+A~)+NΓp(A−A~))p

and

ψp(A)=infk=1,2,...|1−λkp(A)|.

Now we are in a position to formulate the main result of the paper.

Theorem 1.1

For an integer p ≥ 1, let A, Ã ∈ Γ_p and I – A^p be boundedly invertible. If, in addition,

Δp(A,A~)expapNΓpp(A)ψp(A)<1,

then I – Ã^p is also boundedly invertible.

The proof of this theorem is presented in the next section. Replacing in Theorem 1.1 A and Ã by λA and λÃ, respectively, we get the following result.

Corollary 1.2

Let A, Ã ∈ Γ_p and λ^p ∉ σ_Φ(A^p). If, in addition,

Δp(λA,λA~)expapNΓpp(λA)ψp(λA)<1,

then λ^p is Φ-regular also for Ã^p.

From this corollary it follows

Corollary 1.3

Let A, Ã ∈ Γ_p and μ^p ∈ σ_Φ(Ã^p). Then either μ^p ∈ σ_Φ(A^p), or

Δp(μA,μA~)expapNΓpp(μA)ψp(μA)≥1. (1.3)

Note that (1.3) can be rewritten as

|μ|NΓp(A−A~)expap|μ|pNΓpp(A)ψp(μA)+apbpp1+|μ|2(NΓp(A+A~)+NΓp(A−A~))p≥1. (1.4)

2 Proof of Theorem 1.1

For an A ∈ Γ₁ introduce the determinant by

det(I−A)=∏k=1∞(1−λk(A)).

Obviously,

|det(I−A)|≤∏k=1∞(1+|λk(A)|)≤exp∑k=0∞|λk(A)|.

So from (1.1) we have

|det(I−A)|≤expa1NΓ1(A).

Hence, the convergence of the product follows. Since Γ₁ is approximative, we get

limn→∞det(I−An)=det(I−A)

for a sequence {A_n} of n-dimensional operators (n < ∞) converging to A in N_Γ₁(.).

Various approaches to the determinants of operators in a Banach space can be found, in particular, in the well-known publications [2, 4, 5].

Similarly, if A ∈ Γ_p for p > 1, we can write

det(I−Ap)=∏k=1∞(1−λkp(A))

and according to (1.1)

|det(I−Ap)|≤exp[apNΓpp(A)]. (2.1)

Lemma 2.1

Let A, Ã ∈ Γ_p. Then

|det(I−Ap)−det(I−A~p)|≤Δp(A,A~). (2.2)

Proof

Let A and Ã be n-dimensional (n < ∞). Consider the function

f(λ)=det(I−[12(A+A~)+λ(A−A~)]).

First assume that I – 12 (A + Ã) is invertible. Then

f(λ)=detI−12(A+A~))(I−λ(I−12(A+A~))−1(A−A~))=det(I−12(A+A~))det(I−λC),

where C = (I – 12 (A + Ã))^–1 (A – Ã). But

det(I−λC)=∏k=1n(1−λλk(C))

is a polynomial. So f(λ) is a polynomial. Similarly, we can prove that

det(I−[12(Ap+A~p)+λ(Ap−A~p)])

is a polynomial, if I – 12 (A^p + Ã^p) is invertible. Making use of Lemma 1.4.1 [6] (see also [7]), according to (2.1) we get (2.2). If I – 12 (A^p + Ã^p) is not invertible, then (2.2) can be proved by a small perturbation of the considered operators and continuity of determinants. So for finite dimensional operators the lemma is proved. The approximativity of Γ_p implies the required result.□

Corollary 2.2

Let A, Ã ∈ Γ_p and

|det(I−Ap)|>Δp(A,A~).

Then

|det(I−A~p)|≥|det(I−Ap)|−Δp(A,A~)>0.

Lemma 2.3

If the condition

∑k=1∞|λk(A)|<∞ (2.3)

holds and 1 ∉ σ_Φ(A), then

|det(I−A)|≥exp[−1ψ1(A)∑k=1∞|λk(A)|].

Proof

By the usual procedure for the calculations of an extremum we find that max_x≥0 e^–xx = 1/e. Hence

1|1−z|e−1|1−z|≤1/e(z∈C)

and

|1−z|≥e1−1|1−z|=e(|1−z|−1)/|1−z|≥e(1−|z|−1)/|1−z|=e−|z||1−z|.

So if 1 ∉ σ_Φ(A), then

|1−λk(A)|≥exp[−|λk(A)||1−λk(A)|]≥exp[−|λk(A)|ψ1(A)].

Hence

|det(I−A)|=∏k=1∞|1−λk(A)|≥exp[−1ψ1(A)∑k=1∞|λk(A)|],

as claimed.□

Now let A ∈ Γ_p. From the previous lemma, due to (1.1)

|det(I−Ap)|≥exp−apNΓpp(A)ψp(A)(1∉σΦ(Ap)). (2.4)

Corollary 2.2 implies

Corollary 2.4

Let A, Ã ∈ Γ_p for an integer p ≥ 1. If, 1 ∉ σ_Φ (A^p) and

exp−apNΓpp(A)ψp(A)>Δp(A,A~), (2.5)

then

|det(I−A~p)|≥exp−appNΓpp(A)ψp(A)−Δp(A,A~)>0.

The assertion of Theorem 1.1 directly follows from Corollary 2.4.□

3 Particular cases

3.1 Absolutely p-summing operators

An operator A ∈ (𝓑(𝓧)) is said to be absolutely p-summing (1 ≤ p < ∞), if there is a constant ν, such that regardless of a natural number m and regardless of the choice x₁, …, x_m ∈ 𝓧 we have

[∑k=1m∥Axk∥p]1/p≤νsup{[∑k=1m|〈x∗,xk〉|p]1/p:x∗∈X∗,∥x∗∥=1}.

Here 〈 ., .〉 means the functional on 𝓧, 𝓧^* means the space adjoint to 𝓧[2, 3, 8, 9]. The least ν for which this inequality holds is a norm and is denoted by π_p(A). The set of absolutely p-summing operators in 𝓧 with the finite norm π_p is a normed ideal in the set of bounded linear operators, which is denoted by Π_p, cf. [2].

As is well-known,

∑k=1∞|λk(A)|p≤πpp(A)(A∈Πp;2≤p<∞), (3.1)

cf. Theorem 17.4.3 from [9] (see also Theorem 3.7.2 from [2, p. 159]). Thus, Π_p (p ≥ 2) has the properties of ideal Γ_p. Besides, N_{Γ_p}(A) = π_p(A), b_p = 1 and a_p = 1.

3.2 Ideal 𝓔_p and absolutely (p, 2)-summing operators

Recall [2, p. 79] that s_n(T) (n = 1.2, …) is called the n-th s-number (n-th singular number) of T ∈ (𝓑(𝓧)), if the following conditions are satisfied:

(S1)∥T∥=s1(T)≥s2(T)≥...≥0;(S2)sn+m−1(S+T)≤sm(T)+sn(S)(S∈(B(X)));(S3)sn(A1TA2)≤∥A1∥sn(T)∥A2∥(A1,A2∈(B(X)));(S4) If rank(T)<n, then sn(T)=0;(S5)sn(Iln2)=1.

Here Iln2 is the unit operator in the n-dimensional Hilbert space ln2 with the traditional scalar product.

Let L(l², 𝓧) denote the space of linear operators acting from the Hilbert space l² with the traditional scalar product into 𝓧. The n-th Weyl number of T ∈ (𝓑(𝓧))) is defined by

xn(T):=sup{an(TZ):Z∈L(l2,X),∥Z∥=1},

where a_n(T) is the n-th approximation number defined by

an(T):=inf{∥T−Tn∥:Tn∈(B(X)),rankTn<n}.

x_n(T) is an s-number with the sub-multiplicative property

(S6)xn+m−1(TS)≤xn(T)xm(S)(S,T∈(B(X))),

cf. [2, Theorem 2.4.14] and [2, Proposition 2.4.17]. For an integer p ≥ 1, let 𝓔_p. be the set of compact operators A acting in 𝓧 and satisfying

NEp(A):=(∑k=1∞xkp(A))1/p<∞.

Since x_k(A) ≤ x_k–1(A) and x_2k–1(A + Ã) ≤ x_k(A) + x_k(Ã), we have

∑k=1∞xkp(A+A~)=∑j=1∞x2j−1p(A+A~)+x2jp(A+A~)≤2∑j=1∞x2j−1p(A+A~)≤2∑j=1∞(xj(A)+xj(A~))p.

By the Minkovsky inequality

(∑j=1∞(xj(A)+xj(A~))p)1/p≤(∑j=1∞xjp(A))1/p+(∑j=1∞xjp(A~))1/p.

Then

NEp(A+A~)≤21/p(NEp(A)+NEp(A~)). (3.2)

So 𝓔_p is a quasinormed ideal with the quasi-triangular constant b_p = 2^1/p. It is approximative, cf. [2, 3]. We need the following Weyl type inequality:

∑k=1∞|λk(A)|p≤cpp∑k=1∞xkp(A)=cppNEpp(A) (3.3)

with

cp=21/p2e.

cf. [3, Theorem 2.a.6, p. 85].

So 𝓔_p is an example of ideal Γ_p with N_{Γ_p}(A) = N_{𝓔_p}(A), a_p = cpp and b_p = 2^1/p.

About the recent investigations of the singular numbers and Weyl type inequalities see [10]-[16].

Let us point an estimate for N_{𝓔_p}(A). To this end recall that an A ∈ (𝓑(𝓧)) is said to be absolutely (p, q)-summing (p ≥ q), if there is a constant ν such that regardless a natural number m and regardless of the choice x₁, …, x_m ∈ 𝓧 we have

[∑k=1m∥Axk∥p]1/p≤νsup{[∑k=1m|〈x∗,xk〉|q]1/q:x∗∈X∗,∥x∗∥=1}

cf. [2, 3, 8, 9]. The least ν for which this inequality holds is denoted by π_p,q(A). The set of absolutely (p, q)-summing operators is denoted by Π_p,q.

Due to [9, Theorem 16.3.1] π_p,q is a norm and Π_p,q with that norm is a Banach space. If A ∈ Π_p,q, then ∥A∥ ≤ π_p,q(A), since

∥Ax∥=[∥Ax∥p]1/p≤πp,q(A)sup{[|〈x∗,x〉|q]1/q:x∗∈X∗,∥x∗∥=1}≤πp,q(A)∥x∥

for any x ∈ 𝓧. If, in addition R and S are bounded operators acting in 𝓧, then π_p,q(SAR) ≤ ∥R∥_𝓧∥S∥_𝓧 π_p,q(A).

We need Corollary 2.a.3 from [3, p. 81] (see also Corollary 17.2.2 from [9, p. 293]), which asserts the following: if A ∈ Π_p₀,2 (2 ≤ p₀ < ∞), then

xn(A)≤πp0,2(A)n1/p0(n=1,2,...). (3.4)

Hence, for any p > p₀ we have

NEp(A)=(∑k=1∞xnp(A))1/p≤πp0,2(A)(∑k=1∞1kp/p0)1/p=ζ1/p(p/p0)πp0,2(A)(A∈Πp0,2), (3.5)

where

ζ(z)=∑k=1∞1kz(ℜz>1)

is the Riemann zeta-function.

4 Additional upper bounds for determinants

Lemma 4.1

For an integer p ≥ 1 and A ∈ Γ_p one has

|det(I−Ap)|≤ψp(A)exp[apNΓpp(A)].

Proof

Evidently,

|det(I−Ap)|=|1−λmp(A)|∏k=1,k≠m∞|1−λkp(A)|≤|1−λmp(A)|exp[∑k=1∞|λk(A)|p]

for any m ≥ 1. Taking into account (1.1) and choosing m in such a way that |1 – λmp (A)| = ψ_p(A), we prove the lemma.□

Furthermore, let E_p(z) be the Weierstrass primary factor:

E1(z)=(1−z);Ep(z)=(1−z)exp[∑m=1p−1zmm](p=2,3,...;z∈C).

Put

γp:=p−1p(p≠1;p≠3) and γ1=γ3=1.

According to Theorem 1.5.3 [6],

|Ep(z)|≤exp⁡[γp|z|p](z∈C). (4.1)

For an A ∈ Γ_p, p ≥ 2, introduce the p-regularized determinant by

detp(I−A):=∏k=1∞Ep(λk(A)).

Due to (1.1) and (4.1)

|detp(I−A)|≤exp[γp∑k=1∞|λk(A)|p]≤exp[a0γpNΓpp(A)](p≥2), (4.2)

and therefore the product converges.

Lemma 4.2

For an integer p ≥ 2 and any A ∈ Γ_p one has

|detp(I−A)|≤ψ1(A)exp[∑k=1p−1rsk(A)k]exp[apγpNΓpp(A)],

where r_s(A) is the spectral radius of A.

Proof

By (4.1) and (1.1),

|detp(I−A)|=|E(λm(A))|∏k=1,k≠m∞|E(λk(A))|≤|E(λm(A))|exp[γp∑k=1,k≠m∞|λk(A)|p]≤|E(λm(A))|exp[apγpNΓpp(A)]

for any m ≥ 1. But

|Ep(λm(A))|=|1−λm(A)||exp[∑k=1p−1λm(A)kk]|≤|1−λm(A)|exp[∑k=1p−1rsk(A)k].

|detp(I−A)|≤|1−λm(A)|exp[∑k=1p−1rsk(A)k]exp[apγpNΓpp(A)].

Hence, choosing m in such a way that |1 – λ_m(A)| = ψ₁(A), we prove the lemma.□

5 Hille-Tamarkin integral operators “close” to Volterra ones

In this section and in the next one, we consider some concrete integral and matrix operators. We need the following result.

Corollary 5.1

Let W ∈ Γ_p be a quasi-nilpotent operator (i.e. its spectrum is {0}). Then for an arbitrary Ã ∈ Γ_p one has

|det(I−A~p)−1|≤Δp(W,A~).

Indeed, this result is due to Lemma 2.1 and the equality det(I – W^p) = 1.

Let L^p = L^p(0, 1) (2 ≤ p < ∞) be the space of scalar functions f defined on [0, 1] and endowed the norm

∥f∥=[∫01|f(t)|pdt]1/p.

Let K : L^p → L^p be the operator defined by

(Kf)(t)=∫01k(t,s)f(s)ds(f∈Lp,0≤t≤1),

whose kernel k defined on [0, 1]² satisfies the condition

k^p(K):=[∫01(∫01|k(t,s)|p′ds)p/p′dt]1/p<∞, (5.1)

where 1/p + 1/p′ = 1. Then K is called a (p, p′)-Hille-Tamarkin integral operator.

As is well known, [8, p. 43], any (p, p′)-Hille-Tamarkin operator K is an absolutely p-summing operator with π_p(K) ≤ k̂_p(K). Let the operator V be defined by

(Vf)(t)=∫0tk(t,s)f(s)ds(f∈Lp).

This operator is quasi-nilpotent. With Γ_p = Π_p we have

Δp(K,V)=πp(K−V)exp1+12(πp(K+V)+πp(K−V))p≤Δ^p(K,V),

where

Δ^p(K,V):=k^p(K−V)exp1+12(k^p(K+V)+k^p(K−V))p.

Note that

((K−V)f)(t)=∫x1k(t,s)f(s)ds.

Corollary 5.1 implies

Corollary 5.2

Let K be a (p, p′)-Hille-Tamarkin integral operator in L^p(0, 1) for an integer p ≥ 2 and 1/p + 1/p′ = 1. If Δ̂_p(K, V) < 1, then

|det(I−Kp)−1|≤Δ^p(K,V)

and therefore 1p∉σΦ(K), provided Δ̂_p(K, V) < 1.

6 Hille-Tamarkin infinite matrices “close” to triangular ones

Let us consider the linear operator T in l^p (2 ≤ p < ∞) generated by an infinite matrix (tjk)j,k=1∞, satisfying the condition

τp(T):=[∑j=1∞(∑k=1∞|tjk|p′)p/p′]1/p<∞, (6.1)

where 1/p + 1/p′ = 1.

Then T is called a (p, p′)-Hille-Tamarkin matrix. As is well known, any (p, p′)-Hille-Tamarkin matrix T is an absolutely p-summing operator with π_p(T) ≤ τ_p(T), cf. [8, p. 43], [2, Sections 5.3.2 and 5.3.3, p. 230]). So according to (3.1),

∑k=1∞|λk(T)|p≤τpp(T)(2≤p<∞). (6.2)

Let T+=(τjk)j,k=1∞ be the upper-triangular part of T : τ_jk = t_jk for 1 ≤ j ≤ k ≤ ∞ and τ_jk = 0 otherwise. Since p > p′, we obtain

(∑k=1∞|tjk|p′)p/p′≥∑k=1∞|tjk|p

and thus

∑j=1∞∑k=1∞|tjk|p<∞.

Since T₊ is triangular, its eigenvalues are the diagonal entries and

det(I−T+p)=d+,p:=∏k=1∞(1−tkkp).

Under consideration,

Δp(T,T+)=πp(T−T+)exp1+12(πp(T+T+)+πp(T−T+))p≤Δ^p(T,T+),

where

Δ^p(T,T+):=τp(T−T+)exp1+12(τp(T+T+)+τp(T−T+))p.

Note that T – T₊ is the strictly lower part of T.

Making use of Lemma 2.1, we arrive at

Corollary 6.1

Let T be a (p, p′)-Hille-Tamarkin matrix for an integer p ≥ 2 and 1/p + 1/p′ = 1. Then |det(I – T^p) – d_+,p| ≤ Δ̂_p(T, T₊), and therefore 1p∉σΦ(T), provided |d_+,p| > Δ̂_p(T, T₊).

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Received: 2018-09-18

Accepted: 2019-08-12

Published Online: 2019-09-14

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

https://doi.org/10.1515/math-2019-0085

Schlagwörter für diesen Artikel

Banach space; compact operators; perturbations; absolutely p-summing operators; absolutely (p, 2)-summing operators; integral operators; infinite matrices

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Spectrum perturbations of compact operators in a Banach space

Artikel

Abstract

1 Introduction and statement of the main result

Theorem 1.1

Corollary 1.2

Corollary 1.3

2 Proof of Theorem 1.1

Lemma 2.1

Proof

Corollary 2.2

Lemma 2.3

Proof

Corollary 2.4

3 Particular cases

3.1 Absolutely p-summing operators

3.2 Ideal 𝓔p and absolutely (p, 2)-summing operators

4 Additional upper bounds for determinants

Lemma 4.1

Proof

Lemma 4.2

Proof

5 Hille-Tamarkin integral operators “close” to Volterra ones

Corollary 5.1

Corollary 5.2

6 Hille-Tamarkin infinite matrices “close” to triangular ones

Corollary 6.1

References

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3.2 Ideal 𝓔_p and absolutely (p, 2)-summing operators