Cauchy--Schwarz norm inequalities for weak*-integrals of operator valued functions
type: Item
instance of: scholarly article
title: Cauchy--Schwarz norm inequalities for weak*-integrals of operator valued functions
zbMATH Open document ID: 1083.46024
author: Danko R. Jocić
published in: Journal of Functional Analysis
publication date: 2005-02-22T00:00:00Z
review text: In this paper, \( \mathcal{H}\) is a separable, infinite-dimensional Hilbert space, \(\mathcal{B}( \mathcal{H})\) is the space of all bounded linear operators on \( \mathcal{H}\), and \( \mathcal{C}_{p} (\mathcal{H})\) (\(p \geq 1\)) are the Schatten \(p\)-ideals. Each symmetric gauge function \( \varPhi \) on sequences determines a symmetric norm on \(\mathcal{B}( \mathcal{H})\) defined by \( \| X \| _{ \varPhi } = \varPhi ( \{ s_{n}(X) \}_{n=1}^{ \infty})\) with \( s_{1}(X) \geq s_{2}(X) \geq \dots \) being the singular values of \(X\); the symbol \(| | | . | | | \) is used for any such norm on a naturally associated norm ideal \( \mathcal{C}_{ | | | . | | | } (\mathcal{H}) \) of \( \mathcal{C}_{ \infty} (\mathcal{H}) \). For a measure space \( (\Omega, \mathfrak{M}, \mu) \) and for a weak*-integrable mapping \( \mathcal{A}: \Omega \to (\mathcal{B}( \mathcal{H}), \sigma(\mathcal{B}( \mathcal{H}), \mathcal{C}_{1}( \mathcal{H})))\) (note that \(\mathcal{B}( \mathcal{H})\) is the dual space of \(\mathcal{C}_{1}( \mathcal{H})\)), for any \( E \in \mathfrak{M}\), one gets a unique \(\mathcal{I}_{E} \in \mathcal{B}( \mathcal{H})\) such that \[ \text{tr}( \mathcal{I}_{E} Y) = \int_{E}\text{tr}( \mathcal{A}_{t} Y) d \mu (t), \; \text{ for all }Y \in \mathcal{C}_{1}( \mathcal{H}). \] This is denoted by \( \int_{E} \mathcal{A}_{t} d \mu (t) \) or \( \int_{E} \mathcal{A} d \mu \). The main result is: Suppose that \((\Omega, \mathfrak{M}, \mu)\) is a \(\sigma\)-finite measure space and that \(\mathcal{A}: \Omega \to (\mathcal{B}( \mathcal{H}), \sigma(\mathcal{B}(\mathcal{H}), \mathcal{C}_{1}( \mathcal{H})))\) and \( \mathcal{B}: \Omega \to (\mathcal{B}( \mathcal{H}), \sigma(\mathcal{B}( \mathcal{H}), \mathcal{C}_{1}( \mathcal{H})))\) are \(\mu\)-weak* measurable. Then for every \(p, q, r \geq 1\) such that \( \frac{1}{q} + \frac{1}{r}=\frac{2}{p} \), and \(X \in \mathcal{C}_{p} (\mathcal{H})\), \[ \left\| \int_{\Omega} \mathcal{A} X \mathcal{B} \right\| _{p} \leq \left\| \sqr…
reviewed by: Surjit Singh Khurana
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47A64
zbMATH DE Number: 2136762
zbMATH Keywords: Schatten ideals
zbMATH Keywords: unitarily invariant norms
zbMATH Keywords: weak* measurability
zbMATH Keywords: operator mean inequalities
MaRDI profile type: Publication
full work available at URL: j.jfa.2004.06.003
OpenAlex ID: W2082315005
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DOI: 10.1016/J.JFA.2004.06.003
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description: scientific article; zbMATH DE number 2136762