On upper triangular operator matrices over C*-algebras

Abstract

We study adjointable, bounded operators on the direct sum of
two copies of the standard Hilbert C-module over a unital C-algebra
A that are given by upper triangular 2 by 2 operator matrices. Using
the denition of A-Fredholm and semi-A-Fredholm operators given in
[3], [4], we obtain conditions relating semi-A-Fredholmness of these op-
erator and that of their diagonal entries, thus generalizing the results in
[1], [2]. Moreover, we generalize the notion of the spectra of operators
by replacing scalars by the center of the C-algebra A denoted by Z(A):
Considering these new spectra in Z(A) of bounded, adjointable opera-
tors on Hilbert C-modules over A related to the classes of A-Fredholm
and semi-A-Fredholm operators, we prove an analogue or a generalized
version of the results in [1] concerning the relationship between the spec-
tra of 2 by 2 upper triangular operator matrices and the spectra of their
diagonal entries.