The notion of an invariant subspace is fundamental to the subject of operator theory. Given a linear operator T on a Banach space X, a closed subspace M of X is said to be a non-trivial invariant subspace for T if T (M) ⊆M and M 6= {0}, X. This generalizes the idea of eigenspaces of n×n matrices. A famous unsolved problem, called the “invariant subspace problem,” asks whether every bounded line...

The question whether every operator on H has an hyperinvariant subspace is one of the most difficult problems in operator theory. The purpose of this paper is to make a beginning on the hyperinvariant subspace problems for another class of operators closely related to the normal operators namely, the class of k -quasi-class A operators. A necessary and sufficient condition for the hypercyclicit...

By the linearization property of Lipschitz-free spaces, any Lipschitz map $$f : M \rightarrow N$$ between two pointed metric spaces may be extended uniquely to a bounded linear operator $${\widehat{f}} {\mathcal {F}}(M) {F}}(N)$$ their corresponding spaces. In this note, we explore connections dynamics self-maps M$$ and extensions {F}}(M)$$ . This not only allows us relate topological dynamical...

The famous Lomonosov’s invariant subspace theorem states that if a continuous linear operator T on an infinite-dimensional normed space E “commutes” with a compact operator K 6= 0, i.e., TK = KT, then T has a non-trivial closed invariant subspace. We generalize this theorem for multivalued linear operators. We also provide some applications to singlevalued linear operators.