OPERATORS WHICH ARE POLYNOMIALLY ISOMETRIC TO A NORMAL OPERATOR

Abstract

Let H be a complex, separable Hilbert space and B(H) denote the algebra of all bounded linear operators acting on H. Given a unitarily-invariant norm k · ku on B(H) and two linear operators A and B in B(H), we shall say that A and B are polynomially isometric relative to k · ku if kp(A)ku = kp(B)ku for all polynomials p. In this paper, we examine to what extent an operator A being polynomially isometric to a normal operator N implies that A is itself normal. More explicitly, we first show that if k · ku is any unitarilyinvariant norm on Mn(C), if A, N ∈ Mn(C) are polynomially isometric and N is normal, then A is normal. We then extend this result to the infinite-dimensional setting by showing that if A, N ∈ B(H) are polynomially isometric relative to the operator norm and N is a normal operator whose spectrum neither disconnects the plane nor has interior, then A is normal, while if the spectrum of N is not of this form, then there always exists a non-normal operator B such that B and N are polynomially isometric. Finally, we show that if A and N are compact operators with N normal, and if A and N are polynomially isometric with respect to the (c, p)-norm studied by Chan, Li and Tu, then A is again normal.