Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices

Pembe Ipek Al,Zameddin Ismailov

doi:10.31801/cfsuasmas.474512

Abstract

In this paper, the boundedness and compactness properties of infinite tridiagonal block operator matrices in the direct sum of Hilbert spaces are studied. The necessary and sufficient conditions for these operators belong to Schatten-von Neumann class are given. Then, the results are supported by applications.

Highlights

The general theory of singular or characteristic numbers for linear compact operators in Hilbert spaces has been investigated by Gohberg and Krein [11], Pietsch [18], [19]
Throughout this paper, we use the following notations: ( : ; : )H := ( : ; : ); k : kH := k : k and ( : ; : )Hn := ( : ; : )n; k : kHn := k : kn; n 1: There are numerous physical problems arising in the modelling of processes of multiparticle quantum mechanics, quantum ...eld theory and the physics of rigid bodies. These problems support to study the theory of linear operators in the direct sum of Hilbert spaces
We study the compactness properties of in...nite tridiagonal block operator matrices in the in...nite direct sum of Hilbert spaces

Summary

Introduction

The general theory of singular or characteristic numbers for linear compact operators in Hilbert spaces has been investigated by Gohberg and Krein [11], Pietsch [18], [19]. Direct sum of Hilbert spaces, in...nite tridiagonal block operator matrices, compact operator, Schatten-von Neumann classes. These problems support to study the theory of linear operators in the direct sum of Hilbert spaces (see [14],[25] and references in them) They have been widely studied in view of spectral analysis of ...nite or in...nite dimensional real and complex entries special matrices (upper and lower triangular double-band or third-band or Toeplitz types) in sequences spaces !; c; c0; bs; b!p; lp (see [1],[2],[4],[5],[6],[9],[23]). The compactness property and membership to Schatten-von Neumann classes of diagonal operator matrices in the direct sum of Hilbert spaces have been examined in [12]. We study the compactness properties of in...nite tridiagonal block operator matrices in the in...nite direct sum of Hilbert spaces. It can be proved that B 2 C1(H) if and only if Bn 2 C1(Hn+1; Hn); n 1 and lim n!1

Membership of T to Schatten-von Neumann classes

1: According to above derived arguments

Examples