Cyclic Phenomena for Composition Operators

Let H(U) be the set of all holomorphic functions on the unit ball U of the complex plane. The Hardy space H² is the set of all functions f(z)= that belongs to H(U) such that  < ¥. Let j be a holomorphic self map of U. The composition operator C_j on H² is defined as follows:

C_jf = foj, for all f Î H²

Littlewood's principle shows that C_j is a bounded operator on H². Recall that, an operator T on a Hilbert space H is said to be cyclic operator if there exists a vector x in H, such that span {Tⁿx : n =0, 1, …} is dense in H, the operator T is supercyclic if there is a vector x in H, such that the set {l_nTⁿx : l_n Î , n =0, 1, …} is dense in H. It may happen that orb(T, x) = {Tⁿx : n = 0, 1, …} is dense in H, in this case T is called a hypercyclic operator.One of our main concerns in this thesis was to give conditions that are necessary and (or) sufficient for the composition operator to be a cyclic (hypercyclic, supercyclic) operator.We give some known results with details of the proofs, specially when j is a linear fractional transformation, i.e.

j(z) = , z Î U

Where a, b, c and d are complex numbers.

This thesis contains some new results (to the best of our knowledge) for the cyclicity of the operator, where  is the adjoint of the composition operator C_j. .

Let H(U) be the set of all holomorphic functions on the unit ball U of the complex plane. The Hardy space H² is the set of all functions f(z)= that belongs to H(U) such that  < ¥. Let j be a holomorphic self map of U. The composition operator C_j on H² is defined as follows:

C_jf = foj, for all f Î H²

Littlewood's principle shows that C_j is a bounded operator on H². Recall that, an operator T on a Hilbert space H is said to be cyclic operator if there exists a vector x in H, such that span {Tⁿx : n =0, 1, …} is dense in H, the operator T is supercyclic if there is a vector x in H, such that the set {l_nTⁿx : l_n Î , n =0, 1, …} is dense in H. It may happen that orb(T, x) = {Tⁿx : n = 0, 1, …} is dense in H, in this case T is called a hypercyclic operator.

One of our main concerns in this thesis was to give conditions that are necessary and (or) sufficient for the composition operator to be a cyclic (hypercyclic, supercyclic) operator.We give some known results with details of the proofs, specially when j is a linear fractional transformation, i.e.

j(z) = , z Î U

Where a, b, c and d are complex numbers.

This thesis contains some new results (to the best of our knowledge) for the cyclicity of the operator where is the adjoint of the composition operator C_j. .