关于Halmos问题的说明

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-09-24 DOI:10.1112/blms.13159

Maximiliano Contino, Eva A. Gallardo-Gutiérrez

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引用次数: 0

摘要

我们讨论了算子T $T$在Banach空间上的非平凡闭不变子空间的存在性，当它们的平方T 2 $T^2$具有，或者更一般地说，是否存在一个多项式p $p$与deg (p)小于2 $\mbox{deg}(p)\geqslant 2$使得p (T)的不变子空间的格$p(T)$是非平凡的。在希尔伯特空间设置中，t2 $T^2$ -问题是由Halmos在70年代提出的，2007年，Foias, Jung， Ko和Pearcy推测它可以等同于不变子空间问题。

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A note on a Halmos problem

We address the existence of non-trivial closed invariant subspaces of operators $T$ on Banach spaces whenever their square $T^{2}$ have or, more generally, whether there exists a polynomial $p$ with $deg (p) ⩾ 2$ such that the lattice of invariant subspaces of $p (T)$ is non-trivial. In the Hilbert space setting, the $T^{2}$ -problem was posed by Halmos in the seventies and in 2007, Foias, Jung, Ko and Pearcy conjectured it could be equivalent to the Invariant Subspace Problem.