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Journal of Mathematics
Volume 2013 (2013), Article ID 409604, 5 pages
http://dx.doi.org/10.1155/2013/409604
Research Article

Perturbations of Regularized Determinants of Operators in a Banach Space

Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel

Received 15 August 2012; Accepted 7 November 2012

Academic Editor: Mark J. Friedman

Copyright © 2013 Michael Gil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a separable Banach space with the approximation property. For an integer , let be a quasinormed ideal of compact operators in with a quasinorm , such that , where are the eigenvalues of and is a constant independent of . We suggest upper and lower bounds for the regularized determinants of operators from as well as bounds for the difference between determinants of two operators. Applications to the -summing operators, Hille-Tamarkin integral operators, Hille-Tamarkin matrices, Schatten-von Neumann operators, and Lorentz operator ideals are discussed.

1. Statement of the Main Result

Let be a separable Banach space with the approximation property and the unit operator . Let be the Weierstrass primary factor: For a Riesz operator whose eigenvalues counted with their algebraic multiplicities are denoted by   , introduce the -regularized determinant provided The classical theory of regularized determinants for Schatten-von Neumann operators has a long history, which is presented, in particular, in [1, 2]. König [3] developed the theory of regularized determinants for absolutely -summing operators in a Banach space. In [2, 4], following the classical pattern, regularized determinants are defined for operators of the form , in a Banach space where not necessarily itself but at least some power admits a trace. The idea is to replace in all formulas the undefined traces by zero.

Let be the von Neumann-Schatten ideal of compact operators in a separable Hilbert space with the finite norm , where is adjoint to . The following inequalities are well-known: with an unknown constant , see the books [1, page 1106] and [2, page 194]. In [5, 6] these inequalities were slightly improved. In [7] it was proved that one can take , where

In this paper we investigate a quasinormed ideal of compact operators in with a quasinorm . That is, satisfies all the usual properties of a norm, with the exception of the triangular inequality, which is replaced by with a constant independent of . Moreover, for an integer and any , the inequality holds, where is a constant independent of (but dependent on ). The aim of this paper is to generalize inequalities (4) to operators from . In addition, a lower bound for is established.

Now we are in a position to formulate our main result.

Theorem 1. Let for an integer . Then

This theorem is proved in the next section.

Note that if is a norm, then ; if , then .

2. Proof of Theorem 1

Let be an quasinormed space, that is, it is a linear space with a quasinorm . Namely, with a constant .

Lemma 2. For all , let be a scalar-valued entire function of and there be a monotone nondecreasing function , such that for all . Then

Proof. Put
Then is an entire function and Thanks to the Cauchy integral, Hence, In addition, by (11), Therefore according to (15), Taking , we get the required result.

We need also the following result, proved in [7, Lemma 2.3].

Lemma 3. For any integer and all , one has .

Proof of Theorem 1. By the previous lemma Now (8) follows from the latter inequality and (7).
Moreover, (8) and Lemma 2 imply (9).

3. Lower Bounds

Let and be a Jordan curve connecting   and , lying in the disc   and such that Let be the length of . For example, if does not have the eigenvalues on , then one can take . In this case and If the spectral radius of is less than one, then , .

Theorem 4. Let , , and condition (20) hold. Then

Proof . We have Clearly, But since So Hence, where Consequently, But for any , and thus by (7) Therefore, as claimed.

Since Theorems 1 and 4 imply the following result.

Corollary 5. Let for an integer , , and condition (20) hold. If, in addition, then is invertible.

4. Applications

Suppose and that a linear operator in . is said to be -summing, if there is a constant such that regardless of a natural number and regardless of the choice we have cf. [8]. The least for which this inequality holds is denoted by . The set of -summing operators in with the finite norm is an ideal, cf. [9], which is denoted by . By the well-known Theorem  3.7.2 in [9, page 159], (see also Theorem  17.4.3 in [10, page 298]). Since is a norm, Theorems 1 and 4 imply the following.

Corollary 6. Let for some integer . Then and If, in addition, (20) holds, then

Furthermore, let be the space of scalar functions defined on with a finite positive measure and the norm Let be the integral operator whose kernel defined on satisfies the condition where . Then is called a -Hille-Tamarkin operator. As it is well known [8, page 43], any -Hille-Tamarkin operator is a -summing operator and Since is a norm, by Theorems 1 and 4 we get.

Corollary 7. Let and be -Hille-Tamarkin operators in for an integer . Then and If, in addition, condition (20) holds for , then

Now let us consider a linear operator in    generated by an infinite matrix , satisfying Then is called a -Hille-Tamarkin matrix. As it is well known [8, page 43], any -Hille-Tamarkin matrix is a -summing operator with cf. [9, Sections and , page 230].

Since is a norm, Theorems 1 and 4 imply the following.

Corollary 8. Let and be -Hille-Tamarkin matrices for an integer . Then and If, in addition, condition (20) holds for , then

Now let be a separable Hilbert space and    the Lorentz ideal of compact operators with the finite quasinorm where are the singular numbers of taken with their multiplicities. So For the details, see [11, Section 1.1]. By [11, Lemma 1.4], For an integer , let and . Then simple calculations show that . By the Hölder inequality, for , we obtain with So we have Thus (51) implies the following result.

Lemma 9. For an integer and a , let with . Then

Now we can directly apply Theorems 1 and 4.

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