A BAUER-FIKE THEOREM FOR THE JOINT SPECTRUM OF COMMUTING MATRICES

Let A = (A1,...,A(m)) and B = (B1,...,B(m)) be m-tuples of commuting n by n self-adjoint matrices. We obtain a number epsilon = \\Cliff(A - B)\\ such that within a distance epsilon of each joint eigenvalue of A there is a joint eigenvalue of B. The Clifford operator Cliff(A - B) of A - B can be represented by a square matrix of size 2(m)n and is defined using Clifford algebras. When m = 1, \\Cliff(A - B)\\ = \\A - B\\, the operator bound norm of A - B. Similar results are obtained for arbitrary commuting matrices A(j) and simultaneously diagonalizable matrices B(j).