Numerical radius inequality for C-0 contractions

Abstract

We show that if A is a C-0 contraction with minimal function phi such that w(A) = w(S(phi)), where w(.) denotes the numerical radius of an operator and S(phi) is the compression of the shift on (HH2)-H-2 circle minus phi, and B commutes with A, then w(AB) <= w(A)parallel to B parallel to. This is in contrast to the known fact that if A = S(phi) (even on dimensional space) and B commutes with A, then w(AB) <= w parallel to A parallel to w(B) is not necessarily true. As a a finite consequence, we have w(AB) <= w(A)parallel to B parallel to for any quadratic operatorA and any B commuting with A. (c) 2007 Elsevier Inc. All rights reserved.