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

2022/05/10 by Ali Zamani, Zamani, Ali
Mathematics · #46L08 #47A63 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.2205.05164

openalex publication_date 2022/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L(\mathscrH) denote the C^*-algebra of adjointable operators on a Hilbert C^*-module \mathscrH. We introduce the generalized Cauchy-Schwarz inequality for operators in L(\mathscrH) and investigate various properties of operators which satisfy the generalized Cauchy--Schwarz inequality. In particular, we prove that if an operator A\inL(\mathscrH) 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 ‖⟨ Ax, y⟩‖ ≤ ‖|A|^1/2x‖‖|A|^1/2y‖ for all x, y ∈\mathscrH.

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