We prove the existence of a non-trivial hyperinvariant subspace for several sets polynomially compact operators. The main results paper are: (i) norm closed algebra $$\mathcal {A}\subseteq \mathcal {B}(\mathscr {X})$$ which consists quasinilpotent operators has subspace; (ii) if there exists non-zero operator in closure generated by an band {S}$$ , then subspace.

For a sequence {Tn} of continuous linear operators on a separable Fréchet space X, we discuss necessary conditions and sufficient conditions for {Tn} to be topologically mixing, and the relations between topological mixing and the Hypercyclicity Criterion. Among them are: 1) topological mixing is equivalent to being hereditarily densely hypercyclic; 2) the Hypercyclicity Criterion with respect ...

We show that each reflexive finite-dimensional subspace of operators is hyperreflexive. This gives a positive answer to a problem of Kraus and Larson. We also show that each n– dimensional subspace of Hilbert space operators is [ √ 2n]–hyperreflexive.

Let {Dn} be a sequence of bounded invertible operators on Hilbert space H. It is shown that the collection of operators T for which the norm-limit limDnTD n exists is an algebra. Furthermore, some sufficient conditions on this sequence are established for the corresponding algebra to have a nontrivial invariant subspace. By considering specific sequences of operators several invariant subspace ...

In this paper, we generalize to the context of algebras some recent results on existence common hypercyclic vectors for families products backward shift operators. We also give, in a multi-dimensional setting, positive answer question raised by F. Bayart, D. Papathanasiou and author about algebra ℓ1(N) with convolution product family shifts (Bw(λ))λ>0 induced weights wn(λ)=1+λ/n.