On super-recurrent operators

Abstract

In this paper, we introduce and study the notion of super-recurrence of operators. We investigate some properties of this class of operators and show that it shares some characteristics with supercyclic and recurrent operators. In particular, we show that if T is super-recurrent, then σ(T ) and σp(T * ), the spectrum of T and the point spectrum of T * respectively, have some noteworthy properties.

On super-rigid and uniformly super-rigid operators

“…Recently in [2], recurrent operators have been generalized to a large class of operators called superrecurrent operators. We say that T ∈ B(X) is super-recurrent if for each open subset U of X, there exist λ ∈ K and n ∈ N such that λT n (U ) ∩ U = ∅.…”

“…However, if ω n = 1, for all n ≥ 1, and B w = B is the backward shift operator, thenσ p (B) = B(0, 1).Let T be a super-rigid operator acting on a Banach space X. Since any super-rigid operator is super-recurrent, it follows by[2, Theorem 4.2] that that if T is super-rigid, then the eigenvalues of T * ; the Banach adjoint operator of T , are of the same argument. This means that there existsR 1 > such that σ p (T * ) ⊂ {z ∈ C : |z| = R 1 }.…”