Abstract <p> We study the block projection operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal{P}_n\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n \geq 2\)</EquationSource> </InlineEquation>, in the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\)</EquationSource> </InlineEquation>-algebra <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(S(\mathcal{M}, \tau)\)</EquationSource> </InlineEquation> of all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation>-measurable operators, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation> is a faithful normal semi-finite trace on the von Neumann algebra <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathcal{M}\)</EquationSource> </InlineEquation>. For a state <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation>, we study the variance <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\mathbb D\)</EquationSource> </InlineEquation> of an operator from <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(L_2(\mathcal{M},\tau)\)</EquationSource> </InlineEquation>. These results are new, even for the <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\)</EquationSource> </InlineEquation>-algebra <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(\mathcal{M}=\mathcal{B}(\mathcal{H})\)</EquationSource> </InlineEquation> of all bounded linear operators in <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(\mathcal{H}\)</EquationSource> </InlineEquation> equipped with canonical trace <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(\tau =\mathrm{tr}\)</EquationSource> </InlineEquation>. For a random bounded linear operator <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(\mathbf{A}\)</EquationSource> </InlineEquation> in a Hilbert space <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\(\mathcal{H}\)</EquationSource> </InlineEquation>, we consider the operator-valued random variance <InlineEquation ID="IEq17"> <EquationSource Format="TEX">\(\mathbf{D}(\mathbf{A})\)</EquationSource> </InlineEquation>, which is used to investigate the validity of the law of large numbers for the product of independent identically distributed semigroups of bounded linear operators. We present conditions on a bounded self-adjoint operator <InlineEquation ID="IEq18"> <EquationSource Format="TEX">\(X\in \mathcal{B}(\mathcal{H})\)</EquationSource> </InlineEquation> and on a random operator <InlineEquation ID="IEq19"> <EquationSource Format="TEX">\(\mathbf{A}\)</EquationSource> </InlineEquation>, whose variances are related as <InlineEquation ID="IEq20"> <EquationSource Format="TEX">\(\mathbb{D}(X)=\tau (\mathbf{D}(\mathbf{A}))\)</EquationSource> </InlineEquation>. </p>

On the variance of measurable operators

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发表日期：

2026年5月26日

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A. M. Bikchentaev,
V. Zh. Sakbaev

摘要

Abstract

We study the block projection operator \(\mathcal{P}_n\) , \(n \geq 2\) , in the \(*\) -algebra \(S(\mathcal{M}, \tau)\) of all \(\tau\) -measurable operators, where \(\tau\) is a faithful normal semi-finite trace on the von Neumann algebra \(\mathcal{M}\) . For a state \(\tau\) , we study the variance \(\mathbb D\) of an operator from \(L_2(\mathcal{M},\tau)\) . These results are new, even for the \(*\) -algebra \(\mathcal{M}=\mathcal{B}(\mathcal{H})\) of all bounded linear operators in \(\mathcal{H}\) equipped with canonical trace \(\tau =\mathrm{tr}\) . For a random bounded linear operator \(\mathbf{A}\) in a Hilbert space \(\mathcal{H}\) , we consider the operator-valued random variance \(\mathbf{D}(\mathbf{A})\) , which is used to investigate the validity of the law of large numbers for the product of independent identically distributed semigroups of bounded linear operators. We present conditions on a bounded self-adjoint operator \(X\in \mathcal{B}(\mathcal{H})\) and on a random operator \(\mathbf{A}\) , whose variances are related as \(\mathbb{D}(X)=\tau (\mathbf{D}(\mathbf{A}))\) .