Topological Transitivity of Shift Similar Operators on Nonseparable Hilbert Spaces

Abstract

In this paper, we investigate topological transitivity of operators on nonseparable Hilbert spaces which are similar to backward weighted shifts. In particular, we show that abstract differential operators and dual operators to operators of multiplication in graded Hilbert spaces are similar to backward weighted shift operators.

On Direct Sum of Topologically Transitive Operators

“…in the affirmative. Thus, we will modify Theorem 1.3 of Zagorodnyuk [23] to prove our main results of this study on topologically transitive operators.…”

Topological transitivity of translation operators in a non-separable Hilbert space

“…In 1929, G.D. Birkhoff proved that the translation operator f (x) → f (x + a) is hypercyclic on the Fréchet space of entire functions on the complex plane C for any nonzero complex number a (see [7]). This result was generalized by many authors for composition operators in spaces of analytic functions in finite-dimensional spaces [8,9,13], and in infinite-dimensional Banach spaces [2,11,16,[18][19][20]22].…”

The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces

“…Note that condition (ii) in Theorem 2 is evidently true if F k are isomorphisms. Such topologically transitive operators for the case of Hilbert spaces were considered in [14].…”

“…In [6] G. Godefroy and J. Shapiro suggested how to extend the notion of hypercyclicity to non-separable spaces using the concept of topological transitivity (see definition below). The property of the topological transitivity of some analogues of the weighted backward shift on non-separable Hilbert spaces were studied in [12][13][14]. Note that not every non-separable Banach space admits a topological transitive operator [15].…”