Invariant subspaces for non-normable Fréchet spaces

Abstract

A Fréchet space X satisfies the Hereditary Invariant Subspace (resp. Subset) Property if for every closed infinite-dimensional subspace M in X, each continuous operator on M possesses a non-trivial invariant subspace (resp. subset). In this paper, we exhibit a family of non-normable separable infinite-dimensional Fréchet spaces satisfying the Hereditary Invariant Subspace Property and we show that many non-normable Fréchet spaces do not satisfy this property. We also state sufficient conditions for the existence of a continuous operator without non-trivial invariant subset and deduce among other examples that there exists a continuous operator without non-trivial invariant subset on the space of entire functions HpCq.2010 Mathematics Subject Classification. 47A15; 47A16.

Linear Dynamical Systems

“…We note in the Fréchet space setting that the existence of hypercyclic subspaces has recently been investigated by Menet [106,108,110].…”

The invariant subspace problem for the space of smooth functions on the real line

“…• No such fundamental system exists. In this case it is not clear whether an operator without non-trivial invariant subspaces exists -this case is left open in [8]. * In this paper we construct an operator without non-trivial invariant subspaces on the space of smooth functions on the real line C ∞ (R) with the usual topology of uniform convergence of functions and their derivatives on compact sets.…”

“…In [8] Q. Menet was able to show that there is a big family of Fréchet spaces with a continuous norm that support an operator without non-trivial invariant subspaces (even subsets).…”

“…• There exists a fundamental increasing sequence of seminorms (p n ) n∈N for X such that ker p n+1 is of finite codimension in ker p n for all n (e.g., the space of all sequences ω). Then every operator on X has a non-trivial invariant subspace (see [8,Theorem 2.1]).…”

“…• No such fundamental system exists. In this case it is not clear whether an operator without non-trivial invariant subspaces exists -this case is left open in [8].…”