On tensor products with C-normal operators

Abstract

A bounded linear operator A on H is said to be conjugate normal if CA⁎AC=AA⁎ for some conjugation C on H (in this case, A is said to be C-normal). In this paper, we investigate when the conjugate normality of an operator can be preserved under the operation of tensor product. Given an operator A, we show that the tensor product A⊗B is conjugate normal for any Hilbert space operator B if and only if A2=0. Also we obtain similar results concerning tensor products with skew symmetric operators, g-normal operators and Z-normal operators.