Research Article | Open Access
Michael Gil, "Perturbations of Regularized Determinants of Operators in a Banach Space", Journal of Mathematics, vol. 2013, Article ID 409604, 5 pages, 2013. https://doi.org/10.1155/2013/409604
Perturbations of Regularized Determinants of Operators in a Banach Space
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  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  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
We need also the following result, proved in [7, Lemma 2.3].
Lemma 3. For any integer and all , one has .
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.
Corollary 5. Let for an integer , , and condition (20) hold. If, in addition, then is invertible.
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. . The least for which this inequality holds is denoted by . The set of -summing operators in with the finite norm is an ideal, cf. , 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].
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
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