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 Cj on H2 is defined as follows:
Cjf = foj, for all f Î H2
Littlewood's principle shows that Cj 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 {lnTnx : ln 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 Cj. .
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 Cj on H2 is defined as follows:
Cjf = foj, for all f Î H2
Littlewood's principle shows that Cj 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 {lnTnx : ln 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 Cj. .
