On Upper Triangular Operator 2×2 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 thedefinition of A-Fredholm and semi-A-Fredholm operators given in [3], [4], we obtain conditions relating semi-A-Fredholmness of these operators 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 elements in the C∗-algebra A.Considering these new spectra in A of bounded, adjointable operators on Hilbert C∗-modules over A related to the classes of A-Fredholm and semi-A-Fredholm operators, we prove ananalogue or a generalized version of the results in [1] concerning the relationship between the spectra of 2 by 2 upper triangular operator matrices and the spectra of their diagonal entries.