On a separable infinite dimensional complex Hilbert space, we show that the set of hypercyclic operators is dense in the strong operator topology, and moreover the linear span of hypercyclic operators is dense in the operator norm topology. Both results continue to hold if we restrict to only those hypercyclic operators with an infinite dimensional closed hypercyclic subspace. Our works make connections with the classical result on the nondenseness of cyclic operators in the operator norm topology, as well as the recent development on hypercyclic subspaces.
If X is a topological vector space and T is a cotinuous linear mapping from X into X, then T is said to be hypercyclic when there is a vector f in X such that the set {f, Tf, T^2f, ...} is dense in X. When X is a separable Frechet space, Gethner and Shapiro obtained a sufficient condition for the mapping T to be hypercyclic. In the present paper, we obtain an analogous sufficient condition when X is one nonmetrizable space, namely the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology. Using our result, we further provide a sufficient condition for a mapping T on H to have a closed infinite dimensional subspace of hypercyclic vectors. This condition was first found by Montes-Rodriguez for a general Banach space, but the approach that we take is entirely different and simpler.
Using the functional calculus for a normal operator, we provide a result for generalized Toeplitz operators, analogous to the theorem of Axler and Shields on harmonic extensions of the disc algebra. Besides that result, we prove that if T is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under T cannot be orthogonal to each other. Then we show that the C*-algebra generated by T and the identity operator contains all the compact operators as its commutator ideal, and give a characterization of that C*-algebra in terms of generalized Toeplitz operators. Motivated by these results, we further obtain their several-variable analogues, which generalize and unify Coburn's theorems for the Hardy Space and the Bergman space of the unit ball.
In this paper, we discuss necessary conditions and sufficient conditions for compression of an analytic Toeplitz opeator onto a shift coninvariant subspace to have nontrivial reducing subspaces. First we show that a nontrivial reducing subspace of the compression can be dervied from the kernel of a nonanalytic Toeplitz operator. This result enables us to study reducing subspaces when the symbol of the compression is an infinite Blaschke product, by using techniques in function theory. For other analytic symbols, we use techniques in operator theory.
We study integral operators on two subclasses of univalent functions defined on the unit disk, namely the subclass of starlike functions and the subclass of functions of bounded turning. In particular, we give different sufficient conditions for the operators to map from one subclass into itself, and from one subclass into the other.
We give sufficient conditions for analytic functions on the unit disk to be starlike functions. In addition, some sufficient conditions as given in terms of integral transformations on a certain class of analytic functions.
We show that if a subspace of the Dirichlet space for a finitely connected region is invariant under the algebra of multipllication operators, then it contains a dense set of bounded functions. This implies that it is possible to have a complete characterization of such subspaces similar to those in the Hardy space.
For the unit disk, no complete characterization for the universal interpolating sequences of the Dirichlet space is known. Nevertheless, we are able to show that such sequences of a finitely connected region are the finite unions of conformal images of those for the disk. Our method is very general and can be applied to extned many properties of the Dirichlet space for the disk to the setting of finitely connected regions. For example, we extend the corona theorem for the algebra of bounded functions in the Dirichlet space, which was proved by V. A. Tolokonnikov and independently by A. Nicolau.
We show that the translation operator on a Hilbert space of entire functions of slow growth is hypercyclic, which means that there is a function in the space whose translates are dense. This extends the work of G. D. Birkhoff who showed the corresponding result for the Frechet space of all entire functions. In this setting, we give an example to show that hypercyclic operators can arise a pertubations of the identity by "arbitrary compact" operators, despite the fact that no finite dimensional perturbation of teh identity can be hypercyclic.
For the Dirichlet space of a finitely connected region, we study the structure of its subspaces that are invariant under the algebra of multiplication operators. Motivated by our study, we also give conditions for a function to be in such a subsapce. Finally we continue the work of W. B. Arveson on transitive operator algebra in our setting.
We prove that there is a dense set of entire functions of N variables such that each one of them is cyclic for any nonscalar partial differntial operator with constant coefficients. Common cyclicity can also be studied in other settings, for example the Banach spaces of analytic functions; see the article below.
For a wide class of Banach spaces of analytic functions, we show that every space in the class contains a dense set of functions each of which is cyclic for the adjoint of any nonscalar multiplication operator. This extends the result of W. R. Wogen who showed that this is true for the Hardy space.