Supercyclicity and weighted shifts

Hyponormal Operators, Weighted Shifts and Weak Forms of Supercyclicity

“…Question 1.3 seems interesting because N -supercyclic operators which appear in [6] or in [3] are constructed as direct sums of supercyclic operators, and it would be nice to obtain some other, less 'ad hoc' examples. Moreover, hypercyclic and supercyclic weighted shifts have already been characterized by Salas [15,16], so it is natural to consider N -supercyclic case.…”

“…Combined with the Berger-Shaw theorem (a basic fact in the theory of hyponormal operators), this result is used in § 3, where Questions 1.1 and 1.2 are solved. Question 1.3 is solved in § 4, the main tool here being the characterization of supercyclic weighted shifts given by Salas [16].…”

“…By a result of Salas [16], the fact that T is not supercyclic can be expressed as follows: there exist q ∈ N and α > 0 such that, for any n ∈ N, max γ j−nn γ hn ; |j|, |h| q α. By the definition of γ in , the ith coordinate of x n is given by e * i , x n = γ in (u n (1)a i+n (1) + · · · + u n (N )a i+n (N )) = γ inΘn (a i+n ), whereΘ n is a linear form on C N depending only on n. The key point in the proof is the following claim.…”

Common hypercyclic vectors for families of operators

“…In this respect, Abakumov and Gordon [1], answering a question posed by Salas [15], have shown the existence of common hypercyclic vectors in 2 for the family {λB : |λ| > 1}.…”

Asymptotic behaviour of Hilbert space operators with applications

“…But there is no characterization for cyclic bilateral shift operators which is quite surprising since for other cyclic type properties such characterizations exist (see e.g. [42] and [43]). Therefore it is a challenging problem to give this characterization for cyclicity.…”