Triangularizability of trace-class operators with increasing spectrum

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• 作者：Roman Drnov&scaron ; ek ^{roman.drnovsek@fmf.uni-lj.si" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Invariant subspace ; Standard subspace ; Trace-class operator ; Kernel operator ; Positive operator ; Triangularization
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：1102-1115
• 全文大小：360 K

文摘

For any measurable set E   of a measure space a1923ad83b69f8ec8fc0d297c0d9" title="Click to view the MathML source">(X,μ)$(X,\mu )$, let a16a7dd92cab81ce" title="Click to view the MathML source">P_E$P_E$ be the (orthogonal) projection on the Hilbert space L²(X,μ)$L^2(X,\mu )$ with the range 8b6c99f">$\mathrm{ran}{P}_E=\{f\in L^2(X,\mu ):f=0\text{a.e. on}{E}^c\}$ that is called a standard subspace of L²(X,μ)$L^2(X,\mu )$. Let T   be an operator on L²(X,μ)$L^2(X,\mu )$ having increasing spectrum relative to standard compressions, that is, for any measurable sets E and F   with a12d1ceecbd" title="Click to view the MathML source">E⊆F$E\subseteq F$, the spectrum of the operator a157a0c9dbb3e85ea58baf31954244">$P_{E}T{|}_{\mathrm{ran}{P}_E}$ is contained in the spectrum of the operator e53ed">$P_{F}T{|}_{\mathrm{ran}{P}_F}$. In 2009, Marcoux, Mastnak and Radjavi asked whether the operator T   has a non-trivial invariant standard subspace. They answered this question affirmatively when either the measure space a1923ad83b69f8ec8fc0d297c0d9" title="Click to view the MathML source">(X,μ)$(X,\mu )$ is discrete or the operator T has finite rank. We study this problem in the case of trace-class kernel operators. We also slightly strengthen the above-mentioned result for finite-rank operators.