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Dive into the research topics where Kit C. Chan is active.

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Featured researches published by Kit C. Chan.


Integral Equations and Operator Theory | 2001

Hypercyclic subspaces of a Banach space

Kit C. Chan; D Ronald TaylorJr.

Recently a lot of research has been done on hypercyclicity of a bounded linear operator on a Banach space, based on the hypercyclicity criterion obtained by Kitai in 1982, and independently by Gethner and Shapiro in 1987. By combining this criterion with one extra condition, Montes-Rodríguez obtained in 1996 a sufficient condition for the operator to have a closed infinite dimensional hypercyclic subspace, with a very technical proof. Since then, this result has been used extensively to generate new results on hypercyclic subspaces. In the present paper, we give a simple proof of the result of Montes-Rodríguez, by first establishing a few elementary results about the algebra of operators on a Banach space.


Integral Equations and Operator Theory | 2001

Chaotic unbounded differentiation operators

Juan Bès; Kit C. Chan; Steven M. Seubert

We construct dense sets of hypercyclic vectors for unbounded differention operators, including differentiation operators on the Hardy spaceH2, and the Laplacian operator onL2((Ω), for any bounded open subset Ω of ℝ2. Furthermore, we show that these operators are chaotic, in the sense of Devaney.


Journal of Mathematical Analysis and Applications | 2003

Approximation by chaotic operators and by conjugate classes

Juan Bès; Kit C. Chan

We show that every bounded linear operator on a separable, infinite-dimensional Hilbert space H is the sum of two operators in the norm-closure of the set of operators on H that are chaotic in the sense of Devaney. We also observe that the density of several classes of cyclic operators, with respect to the strong operator topology, may be derived from a result by Hadwin et al. (Proc Amer. Math. Soc. 76 (1979) 250–252).


Complex Variables and Elliptic Equations | 2001

Universal meromorphic functions

Kit C. Chan

Using the techniques of the hypercyclicity criterion, we prove that there is a meromorphic function f(z) on the complex plane whose translates f(z + n) for all n ≥ 1 are dense in the metric space of meromorphic functions on any region in the plane. In additions, we prove the analogue of the result for non-Euclidean translation on the unit disk.


Proceedings of the American Mathematical Society | 2012

Prescribed compressions of dual hypercyclic operators

Kit C. Chan

On a separable, infinite dimensional Banach space X, a bounded linear operator T : X → X is said to be hypercyclic on X if there is a vector x ∈ X whose orbit orb (T, x) = {x, Tx, T x, T x, . . .} is dense in X. Such a vector x is called a hypercyclic vector for T . When the operator T : X → X is hypercyclic on X and its adjoint operator T ∗ : X∗ → X∗ is hypercyclic on the dual space X∗ of X, then we say that T is dual hypercyclic. Since an orbit is a countable set, dual hypercyclicity can only take place when both X and X∗ are separable. However, X is separable whenever its dual X∗ is, but the converse is not always true. Separability does not present an issue when the Banach space X is indeed a Hilbert space H, because the adjoint T ∗ is a bounded linear operator on H itself. In fact, it was the Hilbert space setting that the concept of dual hypercyclicity started to develop. A fundamental question is whether dual hypercyclic operators on H can ever exist. This question was originally raised by Herrero [4]. The first example of such an operator was found by Salas [6]. Later he [7] provided another example using a general result for hypercyclic bilateral weighted shift operators. Recently generalizations of dual hypercyclic operators to a Banach space X were studied. For instance, Petersson [5] showed that any infinite dimensional Banach space X with a shrinking symmetric basis, such as c0 and p with 1 < p < ∞, admits a dual hypercyclic operator T : X → X. Then Salas [8] showed that any Banach space X with a separable dual spaceX∗ admits a dual hypercyclic operator. More recently, Shkarin [10] studied dual hypercyclic tuples of operators on Banach spaces, and Salas [9] studied dual disjoint hypercyclic operators. In the present paper we return to the setting of a separable, infinite dimensional Hilbert space H and study compressions of dual hypercyclic operators T : H → H, making use of unique Hilbert space properties. Our main result is Theorem 2 below, which states that the compression of a dual hypercyclic operator T onto a closed subspace M of infinite codimension in H can coincide with any prescribed operator A on M . In other words, if P : H → H is the orthogonal projection onto a closed subspace M with dim (H/M) = ∞, then for any bounded linear


Integral Equations and Operator Theory | 1997

Reducing subspaces of compressed analytic Toeplitz operators on the Hardy space

Kit C. Chan; Steven M. Seubert

In this paper we discuss necessary conditions and sufficient conditions for the compression of an analytic Toeplitz operator onto a shift coinvariant subspace to have nontrivial reducing subspaces. We give necessary and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial and obtain examples of reducing subspaces from these kernels. Motivated by this result we give necessary conditions and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial in terms of the supports of the two inner functions. By studying the commutant of a compression, we are able to give a necessary condition for the existence of reducing subspaces on certain shift coinvariant subspaces.


Transactions of the American Mathematical Society | 1990

On the Dirichlet space for finitely connected regions

Kit C. Chan

This paper is devoted to the study of the Dirichlet space Dir(G) for finitely connected regions G; we are particularly interested in the algebra of bounded multiplication operators on this space. Results in different directions are obtained. One direction deals with the structure of closed subspaces invariant under all bounded multiplication operators. In particular, we show that each such subspace contains a bounded function. For regions with circular boundaries we prove that a finite codimensional closed subspace invariant under multiplication by z must be invariant under all bounded multiplication operators, and furthermore it is of the form p Dir(G) , where p is a polynomial with all its roots lying in G. Another direction is to study cyclic and noncyclic vectors for the algebra of all bounded multiplication operators. Typical results are: if f E Dir(G) and f is bounded away from zero then f is cyclic; on the other hand, if the zero set of the radial limit function of f on the boundary has positive logarithmic capacity, then f is not cyclic. Also, some other sufficient conditions for a function to be cyclic are given. Lastly, we study transitive operator algebras containing all bounded multiplication operators; we prove that they are dense in the algebra of all bounded operators in the strong operator topology.


Integral Equations and Operator Theory | 1990

Common cyclic entire functions for partial differential operators

Kit C. Chan

Let H(ℂN) denote the Fréchet space of all entire functions of N variables (N≥1). The purpose of this paper is to prove the existence of a dense set of functions f in H(ℂN) such that if L is any nonscalar linear differential operator with constant coefficients, then the set {p(L)f∶p(·) is a polynomial} is dense in H(ℂN).


Archive | 2004

A WEAKLY HYPERCYCLIC OPERATOR THAT IS NOT NORM HYPERCYCLIC

Kit C. Chan; Rebecca Sanders


Archive | 1999

HYPERCYCLICITY OF THE OPERATOR ALGEBRA FOR A SEPARABLE HILBERT SPACE

Kit C. Chan

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Juan Bès

Bowling Green State University

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Irina Seceleanu

Bridgewater State University

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Steven M. Seubert

Bowling Green State University

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D Ronald TaylorJr.

Bowling Green State University

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G. Turcu

Bowling Green State University

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