C*-module operators which satisfy the generalized Cauchy–Schwarz type inequality

Abstract

Let L ( H ) denote the C ∗ -algebra of adjointable operators on a Hilbert C ∗ -module H . In this paper, we introduce the generalized Cauchy–Schwarz inequality for operators in L ( H ) . More precisely, an operator A ∈ L ( H ) is said to satisfy the generalized Cauchy–Schwarz inequality if there exists ν ∈ ( 0 , 1 ) such that ‖ ⟨ A x , y ⟩ ‖ ≤ ( ‖ A x ‖ ‖ y ‖ ) ν ( ‖ A y ‖ ‖ x ‖ ) 1 − ν ( x , y ∈ H ) . We investigate various properties of operators which satisfy the generalized Cauchy–Schwarz inequality. In particular, we prove that if A satisfies the generalized Cauchy–Schwarz inequality such that A has the polar decomposition, then A is paranormal. In addition, we show that if for A the equality holds in the generalized Cauchy–Schwarz inequality, then A is cohyponormal. Among other things, when A has the polar decomposition, we prove that A is semi-hyponormal if and only if ‖ ⟨ A x , y ⟩ ‖ ≤ ‖ | A | 1 / 2 x ‖ ‖ | A | 1 / 2 y ‖ for all x , y ∈ H .