This is an announcement for the paper "A new method of constructing invariant subspaces" by George Androulakis.

Abstract: The method of compatible sequences is introduced in order to produce non-trivial (closed) invariant subspaces of (bounded linear) operators. Also a topological tool is used which is new in the search of invariant subspaces: the extraction of continuous selections of lower semicontinuous set valued functions. The advantage of this method over previously known methods is that if an operator acts on a reflexive Banach space then it has a non-trivial invariant subspace if and only if there exist compatible sequences (their definition refers to a fixed operator). Using compatible sequences a result of Aronszajn-Smith is proved for reflexive Banach spaces. Also it is shown that if $X$ be a separable reflexive Banach space, $T \in {\mathcal L} (X)$, and $A$ is any closed ball of $X$, then there exists $v \in A$ such that either $Tv=0$, or $\overline{\text{Span}}, \text{Orb}_T (Tv)$ is a non-trivial invariant subspace of $T$, or there exists a continuous function $f:A \to A$ where $A$ is endowed with the weak topology, such that $f(x) \in \overline{\text{Span}}, { T^k x : k \in {\mathbb N} } $ for all $x \in A$ and $f(v)=v$.

Archive classification: Functional Analysis

Mathematics Subject Classification: 47A15

The source file(s), ISP.tex: 60926 bytes, is(are) stored in gzipped form as 0506284.gz with size 17kb. The corresponding postcript file has gzipped size 78kb.

Submitted from: giorgis@math.sc.edu

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