COMPLEX SYMMETRY OF TOEPLITZ OPERATORS OVER THE BIDISK

In this paper,we investigate the complex symmetric structure of Toeplitz operators T_(φ)on the Hardy space over the bidisk.We first characterize the weighted composition operators,W_(u,v)which are J-symmetric and unitary.As a consequence,we characterize conjugations of the form A_(u,v).In addition,a class of conjugations of the form C_(λ,a)is introduced.We show that the class of conjugations C_(λ,a)coincides with the class of conjugations A_(u,v);we then characterize the complex symmetry of the Toeplitz operators T_(φ)with respect to the conjugation C_(λ,a).

COMPLEX SYMMETRIC C_(0)-SEMIGROUPS ON THE WEIGHTED HARDY SPACES H_(γ)(D)

In this paper,we study the complex symmetric C_(0)-semigroups of weighted composition operators W_(ψ,φ)on the weighted Hardy spaces H_(γ) of the unit disk D.It is well-known that there are only two classes of weighted composition conjugations A_(u,v) on H_(γ)(D):either C_(1) or C_(2).We completely characterize C_(1)-symmetric(C_(2)-symmetric)C_(0)-semigroups of weighted composition operators W_(ψ,φ) on H_(γ)(D).

UNBOUNDED COMPLEX SYMMETRIC TOEPLITZ OPERATORS

In this paper,we study unbounded complex symmetric Toeplitz operators on the Hardy space H^(2)(D) and the Fock space g^(2).The technique used to investigate the complex symmetry of unbounded Toeplitz operators is different from that used to investigate the complex symmetry of bounded Toeplitz operators.

Complex Symmetric C0-semigroups on A^2(C+)

In this paper,we study complex symmetric C0-semigroups on the Bergman space A^2(C+) of the right half-plane C+.In contrast to the classical case,we prove that the only involutive composition operator on A^2(C+) is the identity operator,and the class of J-symmetric composition operators does not coincide with the class of normal composition operators.In addition,we divide semigroups{φt}of linear fractional self-maps of C+into two classes.We show that the associated composition operator semigroup{Tt}is strongly continuous and identify its infinitesimal generator.As an application,we characterize Jσ-symmetric C0-semigroups of composition operators on A^2(C+).

Composition Operators with Universal Translates on S^(2)(D)

It is known that the Invariant Subspace Problem for Hilbert spaces is equivalent to the statement that all minimal non-trivial invariant subspaces for a universal operator are one dimensional.In this paper,we characterize all linear fractional composition operators and their adjoints that have universal translates on the space S^(2)(D).Moreover,we characterize all adjoints of linear fractional composition operators that have universal translates on the Hardy space H^(2)(D).In addition,we consider the minimal invariant subspaces of the composition operator Cφa on S^(2)(D),where φa(z)=az+1-a,a ∈(O,1).Finally,some relationships between complex symmetry and universality for bounded linear operators and commuting pairs of operators on a complex separable,infinite dimensional Hilbert space are explored.

Weighted composition operators on the Fock space

In this paper,we study weighted composition operators on theFock space F^(2).We prove that each bounded composition operator on F^(2) is complex symmetric.This is in sharp contrast with the phenomenon on the Hardy space H^(2)(D).We characterize Hermitian weighted composition operators and algebraic weighted composition operators with degree less than or equal to two on F^(2).In addition,we investigate cyclicity and hypercyclicity of complex symmetric weighted composition operators.We also characterize those weighted compositionoperators that preserveframes,tight frames or normalized tight frames on F^(2).Finally,we study mean ergodicity and uniformly mean ergodicity of weighted composition operators.