On subscalarity of some 2 × 2 class A operator matrices

Abstract

In this paper we provide some conditions for 2×2 operator matrices whose diagonal entries are class A operators to be subscalar. As a corollary, we get that such operators with rich spectra have nontrivial invariant subspaces. In addition, we show that the tensor product of a 2×2 upper triangular class A operator matrix and a class A operator has a scalar extension. Finally, we find some subscalar 2×2 operator matrices satisfying the operator equations ABA=A2 and BAB=B2. © 2012 Elsevier Inc. All rights reserved.