Abstract <p> We show that for all <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1177_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(n,p>1\)</EquationSource> </InlineEquation>, there exists a unitary operator <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1177_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(U\)</EquationSource> </InlineEquation> such that the tensor product <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1177_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="156" /> </InlineMediaObject> <EquationSource Format="TEX">\(U\otimes U^p\otimes\dots\otimes U^{p^{n-1}}\)</EquationSource> </InlineEquation> is a unitary operator with simple Lebesgue spectrum. Moreover, there exists an ergodic automorphism <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1177_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(T\)</EquationSource> </InlineEquation> such that the spectrum of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1177_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(T\odot T\)</EquationSource> </InlineEquation> is simple, while the spectrum of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1177_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(T\otimes T\otimes T\)</EquationSource> </InlineEquation> is absolutely continuous. </p>

Tensor Factorizations of a Unitary Operator with Simple Lebesgue Spectrum

期刊论文
发表日期：

2025年6月1日

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Valerii Ryzhikov

摘要

Abstract

We show that for all \(n,p>1\) , there exists a unitary operator \(U\) such that the tensor product \(U\otimes U^p\otimes\dots\otimes U^{p^{n-1}}\) is a unitary operator with simple Lebesgue spectrum. Moreover, there exists an ergodic automorphism \(T\) such that the spectrum of \(T\odot T\) is simple, while the spectrum of \(T\otimes T\otimes T\) is absolutely continuous.