Upper Semi-Weyl and Upper Semi-Browder Spectra of Unbounded Upper Triangular Operator Matrices

Wurichaihu Bai,Qingmei Bai,Alatancang Chen

doi:10.1155/2018/7871352

Abstract

In this paper, we study the unbounded upper triangular operator matrix with diagonal domain. Some sufficient and necessary conditions are given under which upper semi-Weyl spectrum (resp. upper semi-Browder spectrum) of such operator matrix is equal to the union of the upper semi-Weyl spectra (resp. the upper semi-Browder spectra) of its diagonal entries. As an application, the corresponding spectral properties of Hamiltonian operator matrix are obtained.

Highlights

Let K, H be the infinite dimensional separable Hilbert spaces and C(H, K)(C+(H, K)) be the set of all closed linear operators from H into K
Applying different method—space decomposition technique—we present some sufficient and necessary conditions for (3) in this paper
With domain being D(A) ⊕ D(A∗) ⊂ H ⊕ H, where H = L2(0, 1) ⊕ L2(0, 1), A = AC[0, 1], and

Summary

Introduction

Let K, H be the infinite dimensional separable Hilbert spaces and C(H, K)(C+(H, K)) be the set of all closed (closable) linear operators from H into K. For a (linear) operator T between Hilbert spaces, we use D(T), R(T), and N(T) to denote the domain, the range, and the kernel of T and write α(T) and β(T) for the dimensions of the kernel N(T) and the quotient space H/R(T), respectively. An operator T ∈ C(H, K) with dense domain is said to be upper semi-Fredholm (resp., lower semi-Fredholm) if α(T) < ∞ (resp., β(T) < ∞) and R(T) is closed. If both α(T) and β(T) are finite, T is called Fredholm operator.

Some Properties of Upper Triangular Operator Matrix

Main Results

Applications