具有非三维封闭不变仿射子空间的算子

IF 0.9 3区数学 Q2 MATHEMATICS

Aequationes Mathematicae Pub Date : 2024-06-20 DOI:10.1007/s00010-024-01090-0

Janko Bračič

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引用次数: 0

摘要

我们关注的问题是复巴纳赫空间上的算子 A 是否存在不变的适当仿射子空间。事实证明，在 A 的谱或其邻接算子 \(A^*\)的谱中是否存在数字 1 至关重要。例如，当且仅当 1 是一个代数算子的特征值时，它才有一个不变的适当仿射子空间。对于任意算子 A，我们证明，只有当 1 是 \(A^*\) 的特征值时，它才有一个不变的适当超平面。如果 A 是一个幂有界算子，那么每个不变的适当仿射子空间都包含在一个不变的适当超平面中，此外，A 还有一个非三维不变锥。

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Operators with a non-trivial closed invariant affine subspace

We are concerned with the question of the existence of an invariant proper affine subspace for an operator A on a complex Banach space. It turns out that the presence of the number 1 in the spectrum of A or in the spectrum of its adjoint operator \(A^*\) is crucial. For instance, an algebraic operator has an invariant proper affine subspace if and only if 1 is its eigenvalue. For an arbitrary operator A, we show that it has an invariant proper hyperplane if and only if 1 is an eigenvalue of \(A^*\). If A is a power bounded operator, then every invariant proper affine subspace is contained in an invariant proper hyperplane, moreover, A has a non-trivial invariant cone.

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来源期刊

Aequationes Mathematicae MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.70

自引率

12.50%

发文量

审稿时长

>12 weeks

期刊介绍： aequationes mathematicae is an international journal of pure and applied mathematics, which emphasizes functional equations, dynamical systems, iteration theory, combinatorics, and geometry. The journal publishes research papers, reports of meetings, and bibliographies. High quality survey articles are an especially welcome feature. In addition, summaries of recent developments and research in the field are published rapidly.