Some Spectral Properties of the Banach Algebra A ?d B with the Direct-Sum Product

Let A be a commutative Banach algebra and B be a closed subalgebra of A. Then A B is a commutative algebra with co-ordinatewise linear operations and the direct-sum product: (a; b) (c; d) = (ac + ad + bc; bd) (a; c 2 A; b; d 2 B). In fact, it is a Banach algebra with a suitable norm; it is denoted by A d B. Here, we study some important spectral properties of this algebra.