Dynamics Of Toeplitz Operators And Composition Operators

Toeplitz operators and composition operators are two important operators in function spaces,which have a wide range of applications in modern analysis.Dynamics of linear operators is a young and rapidly evolving branch of functional analysis,which has a familiar connection with ergodic theory,differential equation,the geometry of Banach space,matrix theory and several other fields.In this thesis,we study the dynamical properties of Hardy-Toeplitz operators and composition operators based on the operator theory in function spaces,such as the(frequent)hypercyclicity,mixing property,chaoticity and so on.The dissertation will be divided into six chapters:In Chapter 1,we introduce the background,research status at home and abroad of the dynamics of linear operators and its connections with other disciplines of mathematics.We also state the main results of this thesis.In Chapter 2,we list the prerequisite knowledge including the basic notions of the(frequent)hypercyclicity,mixing property,weakly mixing property,chaoticity of the linear operators on the infinite-dimensional topological vector spaces and their respective criterions.In addition,we also introduce the definitions of Hardy-Toeplitz operators and composition operators and their basic properties.In Chapter 3,based on the research of the hypercyclicity of Toeplitz operators TΦ with the symbol Φ(z)=p(1/z)+φ(z)due to A.Baranov and A.Lishanskii,we further characterize the frequent hypercyclicity of TΦ.In particular,we give a concrete example to verify our conditions.Meanwhile,we also study the hypercyclicity and frequent hypercyclicity of the tensor products TΦ1(?)TΦ2.Furthermore,we construct an example such that neither TΦ1 nor TΦ2 is hypercyclic,but TΦ1(?)TΦ2 is frequently hypercyclic!This strengthens the conclusion of F.Martinez-Gimenez and A.Peris:The tensor products of two non-hypercyclic operators can be hypercyclic.Finally,we also prove that TΦ and TΦ1(?)TΦ2 are mixing and chaotic under the given conditions.In Chapter 4,we study the hypercyclicity of weighted composition operator Cφψ*on the vector-valued analytic reproducing kernel Hilbert space Hε(K),which generalize the results about the hypercyclicity of Cφψ*on the scalar-valued reproducing kernel Hilbert spaces by Z.Kamali,B.K.Robati and K.Hedayatian.Meanwhile,we also show that Hardy spaces over the polydisk H2(Dn)is isomorphic to a special quasi-scalar reproducing kernel Hilbert space.As a supplement,we characterize the hypercyclicity of tuple of operators Cφ(1),ψ(1)*,Cφ(2),ψ(2)*)on Hε(K).In Chapter 5,we partially answer the open question raised by F.Colonna and R.A.Martinez-Avendano:Whether or not the weighted Dirichlet spaces Dαp can support hypercyclic composition operators when p-2<α<p?They have shown that when-1<α≤p-2,any composition operator can not be hypercyclic on Dαp,but hypercyclic composition operators exist on Dαp when α≥p.However,they left open the case p-2<α<p.In fact,as early as 2004,E.A.Gallardo-Gutierrez and A.Montes-Rodriguez have already thoroughly studied the case p=2,i.e.,the weighted Dirichlet-Hilbert spaces.By their results,we know that there exist hypercyclic composition operators on Dα2 for all α>0.Inspired by this,we study the hypercyclicity of linear fractional composition operators on Dαp by using the classification of linear fractional transformations according to their fixed points.We get the following main results:Assume that p-2<α<p,φ is a parabolic automorphism or a hyperbolic automorphism,then Cφ is hypercyclic on Dαp wheneverp>3;suppose that p-1<α<p,(p is a hyperbolic non automorphism,then Cφ is hypercyclic on Dαp for all p>1;assume that-1<α<p,φ is a parabolic non automorphism,then Cφ is not hypercyclic on Dαp when p>2.In the final chapter,we summarize the main research results of the thesis.Moreover,we point out the difficulties that have not been overcome and some questions that we will study in the further.