Some Research On Convex-cyclicity And Disjoint Diskcyclicity For Weighted Shifts

This thesis is concerned with dynamics of linear operators and we will give some results. It is divided into four chapters:Firstly, we characterize subspace-diskcyclic operators on separable Hilbert space over the real or complex scalar field, We provide some properties of projection, kernel about subspace-diskcyclic operators. In addition, we obtain some brief reviews of work-s on subspace-diskcyclic operators and show an example to prove M-diskcyclic does not imply M-disk transitive. We introduce some properties of subspace-supercyclic semigroup and develop the basic theory of M-supercyclic semigroup. Additionally,two criterions to yield a subspace-supercyclic semigroup related to a subspace M are obtained.Secondly, we introduce that how to prove a given operator is super convex-cyclic.In particular, Super Convex-cyclicity Criterion is described. Additionally, we charac-terize convex-cyclicity of (invertible) bilateral weighted shifts in terms of their weight sequences. Meanwhile, we give super convex-cyclicity of (invertible) weighted shifts in the bilateral case.Thirdly, We introduce a new notion called disjoint diskcyclicity . More precisely,we provide disjoint disk blow-up/collapse property and Disjoint Diskcyclicity Criterion.In addition, we characterize the disjoint diskcyclicity for weighted shifts both in the bilateral and unilateral cases.Finally, We characterize disjointness of supercyclic operators who map a holomor-phic function to a partial sum of the Taylor expansion. In particular, we show that disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type Oper-ators. Moreover, we give a sufficient condition to yield the disjoint supercyclicity for families of Taylor-type operators.