Spectrum perturbations of compact operators in a Banach space

Abstract For an integer p ≥ 1, let Γp be an approximative quasi-normed ideal of compact operators in a Banach space with a quasi-norm NΓp(.) and the property ∑k=1∞|λk(A)|p≤apNΓpp(A)(A∈Γp), $$\begin{array}{} \displaystyle \sum_{k=1}^{\infty} |\lambda_k(A)|^p\le a_p N_{{\it\Gamma}_p}^p(A) \;\;(A\in {\it\Gamma}_p), \end{array}$$ where λk(A) (k = 1, 2, …) are the eigenvalues of A and ap is a constant independent of A. Let A, Ã ∈ Γp and Δp(A,A~):=NΓp(A−A~)expapbpp1+12(NΓp(A+A~)+NΓp(A−A~))p, $$\begin{array}{} \displaystyle {\it\Delta}_p(A, \tilde A):= N_{{\it\Gamma}_p}(A-\tilde A) \;\exp\;\left[a_p b_p^p \;\left(1+\frac{1}2 (N_{{\it\Gamma}_p}(A+\tilde A) + N_{{\it\Gamma}_p}(A-\tilde A))\right)^p\right], \end{array}$$ where bp is the quasi-triangle constant in Γp. It is proved the following result: let I be the unit operator, I – Ap be boundedly invertible and Δp(A,A~)expapNΓpp(A)ψp(A)<1, $$\begin{array}{} \displaystyle {\it\Delta}_p(A, \tilde A)\exp\;\left[\frac{a_pN^p_{{\it\Gamma}_p}(A) } {\psi_p(A)}\right] \lt 1, \end{array}$$ where ψp(A) = infk=1,2,… |1 – λkp $\begin{array}{} \displaystyle \lambda_k^{p} \end{array}$(A)|. Then I – Ãp 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.


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 rst 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 a ects 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 = , , ...) are the eigenvalues counted with their algebraic multiplicities. A point λ ∈ C is said to be Φregular 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 ≥ introduce the two-sided quasi-normed ideal Γp of compact operators in (B(X)) with a quasi-norm N Γp (.) and the property where ap is a constant independent of A, and Γp is approximative (i.e. the set of all nite rank operators is dense in Γp). Below bp denotes the quasi-triangle constant in Γp: For the theory of the approximative normed and quasi-normed ideals see [2,3] and references given therein.
In the sequel constant ap in (1.1) will be called the eigenvalue constant. The proof of this theorem is presented in the next section. Replacing in Theorem 1.1 A andÃ by λA and λÃ, respectively, we get the following result. From this corollary it follows Note that (1.3) can be rewritten as

Proof of Theorem 1.1
For an A ∈ Γ introduce the determinant by Obviously, Hence, the convergence of the product follows. Since Γ is approximative, we get for a sequence {An} of n-dimensional operators (n < ∞) converging to A in N Γ (.). 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 ∈ Γp for p > , we can write and according to ( Proof. Let A andÃ be n-dimensional (n < ∞). Consider the function First assume that I − (A +Ã) is invertible. Then is a polynomial. So f (λ) is a polynomial. Similarly, we can prove that is a polynomial, if I − (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 − (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 nite dimensional operators the lemma is proved. The approximativity of Γp implies the required result. Proof. By the usual procedure for the calculations of an extremum we nd that max x≥ e −x x = /e. Hence and as claimed. 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 πp(A). The set of absolutely p-summing operators in X with the nite norm πp is a normed ideal in the set of bounded linear operators, which is denoted by Πp, cf. [2]. As is well-known, Here I l n is the unit operator in the n-dimensional Hilbert space l n with the traditional scalar product. Let L(l , X) denote the space of linear operators acting from the Hilbert space l with the traditional scalar product into X. The n-th Weyl number of T ∈ (B(X))) is de ned by

Now let
So Ep is a quasinormed ideal with the quasi-triangular constant bp = /p . It is approximative, cf. [2,3]. We need the following Weyl type inequality: cf. [3, Theorem 2.a.6, p. 85]. So Ep is an example of ideal Γp with N Γp (A) = N Ep (A), ap = c p p and bp = /p . About the recent investigations of the singular numbers and Weyl type inequalities see [10]- [16]. Let us point an estimate for N Ep (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 x , ..., xm ∈ X we have cf. [2,3,8,9]. The least ν for which this inequality holds is denoted by πp,q(A).

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.
As is well known, [8, p. 43], any (p, p )-Hille-Tamarkin operator K is an absolutely p-summing operator with πp(K) ≤kp(K). Let the operator V be de ned by