Joint spectra and commutativity of systems of (2 × 2) selfadjoint matrices

Abstract

McIntosh and Pryde [4, 5] introduced a notion of joint spectrum γ(A), for commuting n-tuples A = (A 1,…,An ) of bounded linear operators on a Banach space, which has proved to be very useful in certain applications. In this note we investigate the joint spectrum γ(A) for the non-commutative setting, not in full generality, but rather for the particular case of selfadjoint operators Aj ,1≤jL≤n, in a Hilbert space H. For finite dimensional spaces H quite a lot can be said about such sets γ(A). For instance, γ(A) is characterized as precisely those points which are joint eigenvalues of A and hence, γ(A)≠O if and only if there exists a common joint eigenvector for A. As an application, the general results are applied to the special case of dim(H) = 2 to show that γ(A)≠O if and only if the selfadjoint operators Aj ,1≤jL≤n mutually commute. This fact is combined with a recent result from [3] to give various other characterizations of commutativity of systems of (2 × 2) selfadjoint matrices.