WEIGHTED COMPOSITION OPERATORS BETWEEN DIRICHLET SPACES

In this article, we study the boundedness of weighted composition operators between different vector-valued Dirichlet spaces. Some sufficient and necessary conditions for such operators to be bounded are obtained exactly, which are different completely from the scalar-valued case. As applications, we show that these vector-valued Dirichlet spaces are different counterparts of the classical scalar-valued Dirichlet space and characterize the boundedness of multiplication operators between these different spaces.

WEIGHTED COMPOSITION OPERATORS ON WEIGHTED DIRICHLET SPACES

We characterize the boundedness and compactness of weighted composition operators on weighted Dirichlet spaces in terms of Nevanlinna counting functions and Caxleson measure.

Toeplitz Algebras on Dirichlet Spaces

In the present paper, some properties of Toeplitz algebras on Dirichlet spaces for several complex variables are discussed; in particular, the automorphism group of the Toeplitz C * algebra, J (C 1) , generated by Toeplitz operators with C 1 symbols is discussed. In addition, the first cohomology group of J (C 1) is computed.

MULTIPLICATION OPERATORS ON WEIGHTED DIRICHLET SPACES

In this paper,we study multiplication operators on weighted Dirichlet spaces D_(β)(β∈R).Let n be a positive integer and β∈R,we show that the multiplication operator M_(z)^(n) on D_(β) is similar to the operator ⊕_(1)^(n)M_(z)on the space⊕_(1)^(n)D_(β).Moreover,we prove that M_(z)^(n)(≥2)on Dβis unitarily equivalent to ⊕_(1)^(n)M_(z) on⊕_(1)^(n)D_(β) if and only if β=0.In addition,we completely characterize the unitary equivalence of the restrictions of M_(z)^(n) to different invariant subspaces z^(k)D_(β)(k≥1),and the unitary equivalence of the restrictions of M_(z)^(n) to different invariant subspaces S_(j)(0≤j<n).Abkar,Cao and Zhu[Complex Anal Oper Theory,2020,14:Art 58]pointed out that it is an important,natural,and difficult question in operator theory to identify the commutant of a bounded linear operator.They characterized the commutant A′( M_(z)^(n))of M_(z)^(n)on a family of analytic function spaces A_(α)^(2)(α∈R)on D(in fact,the family of spaces A_(α)^(2)(α∈R)is the same with the family of spaces D_(β)(β∈R))in terms of the multiplier algebra of the underlying function spaces.In this paper,we give a new characterization of the commutant A′( M_(z)^(n))of M_(z)^(n)on D_(β),and characterize the self-adjoint operators and unitary operators in A'(M_(z)^(n)).We find that the class of self-adjoint operators(unitary operators)in A'(M_(z)^(n))when β≠0 is different from the class of self-adjoint operators(unitary operators)in A′( M_(z)^(n))when β=0.

Linear Combinations of Composition Operators on Weighted Dirichlet Spaces

In this paper, linear combinations of composition operators acting on weighted Dirichlet spaces are studied. By using the first derivative of the kernel function, we obtain a lower estimate for the essential norms of these operators acting on the Dirichlet space D and S2. For general weighted Dirichlet space, by using complex interpolation methods, we characterize the compactness of these operators induced by linear fractional self-maps of the disk.

Generalized Cesàro Operator on Dirichlet Spaces

In this paper, we study the boundedness of the generalized Cesàro operator on the weighted Dirichlet spaces Dα={f∈H（D）;‖f‖Dα^2=｜f（0）｜^2＋∫D｜f＇（z）｜^2（1-｜z｜^αdm（z）〈＋∞},where -1 〈 α 〈＋∞ and H（D） is the class of all holomorphic functions on the unit disc D.

TOEPLITZ OPERATORS AND ALGEBRAS ON DIRICHLET SPACES

The automorphism group of the Toeplitz C^*- algebra, J(C^1), generated by Toeplitz op-erators with C^1-symbols on Dirichlet space D is discussed; the K0, K1-groups and the first cohomology group of J(C^1) are computed. In addition, the author proves that the spectra of Toeplitz operators with C^1-symbols are always connected, and discusses the algebraic prop-erties of Toeplitz operators.In particular, it is proved that there is no nontrivial selfadjoint Toeplitz operator on D and Tψ^* = Tψ^- if and only if Tψ is a scalar operator.

Toeplitz Operators on Dirichlet Spaces

In this paper we study Toeplitz operators on Dirichlet spaces and describe the boundedness and compactness of Toeplitz operators on Dirichlet spaces. Meanwhile, we give density theorems for Toeplitz operators on Dirichlet spaces

Composition Operators Between the Weighted Dirichlet Spaces on Bounded Symmetric Domains of C^n

In this paper, we define the weighted Dirichlet space D p α(Ω) on bounded symmetric domains Ω of C n. Using η-α Carleson measure,we study the boundedness and compactmess of the composition operators between the weighted Dirichlet spaces.

ON DIRICHLET TYPE SPACES AND α-BLOCH SPACES IN THE UNIT BALL OF C^n

In this paper, the authors study the inclusion relations between Dirichlet type spaces DΥT and α-Bloch spaces βα by means of higher radial derivative. The strictness and the best possibility of the inclusion relations are shown with constructive methods. Furthermore, they sharpen one of the results when Υ=n, which proves that a conjecture in [7] is true.

SPACES OF ANALYTIC FUNCTIONS REPRESENTED BY DIRICHLET SERIES OF TWO COMPLEX VARIABLES

We consider the space X of all analytic functionsof two complex variables s1 and s2, equipping it with the natural locally convex topology and using the growth parameter, the order of f as defined recently by the authors. Under this topology X becomes a Frechet space Apart from finding the characterization of continuous linear functionals, linear transformation on X, we have obtained the necessary and sufficient conditions for a double sequence in X to be a proper bases.

Volterra Type Operators on Weighted Dirichlet Spaces

The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Peláez,who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces.However,their characterizations for the boundedness are not complete.In this paper,the author completely characterizes the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces D^(p)_(α)to _(Dβ)^(q)(-1<α,βand 0<p<q<∞),which essentially complete their works.Furthermore,the author investigates the order boundedness of Volterra type operators between weighted Dirichlet spaces.

Dirichlet and Q_(p,0) Spaces on the Unit Ball of C^n

In this paper, we first give some new characterizations of Qp,0 spaces by means of vanishing p-Carleson measure. Using these results, we can prove that Dirichlet space is a subset of Qp,0 for(n-1)/n<p≤1.

ON F(p,q,s)SPACES

The family of spaces F(p,q,s)was introduced by the author in 1996.Since then,there has been great development in the theory of these spaces,due to the fact that these spaces include many classical function spaces,and have connections with many other areas of mathematics.In this survey we present some basic properties and recent results on F(p,q,s)spaces.

S_p-CLASS(0

In this paper,we construct a function u in L2,1（Bn,dA）,which is unbounded on any neighborhood of each boundary point of B n,such that Toeplitz operator Tu is compact on Dirichlet space D（Bn,dA）.Furthermore,Schatten p-class（0〈p〈∞） Toeplitz operators on Dirichlet space D（Bn,dA） with unbounded symbols are also obtained.

GALERKIN-PETROV METHODS OF TOEPLITZ OPERATORS ON DIRICHLET SPACE

The convergence of several Galerkin-Petrov methods, including polynomial collocation and analytic element collocation methods of Toeplitz operators on Dirichlet space, is established. In particular, it is shown that such methods converge if the basis and test function own certain circular symmetry.

BOUNDEDNESS OF COMPOSITION OPERATORS ON D_T SPACES FOR T > 1

In this note, some conditions of composition operators on DT spaces to be bounded are given by means of Carleson measures and pointwise multipliers, for some ranges of T. The authors prove that (i) Let 1 < T n + 2 and 2k < T 2k + 1 (or 2k - 1 < T 2k) for some positive integer k. Suppose = ( 1,… , n) be a univalent mapping from B into itself, denote dμj(l) (z) = R(l) j (z) 2(k-l+2) (1 - z 2)2k-t+1dv(z) for l= 1, 2,… , k + 1. If μj(l)-1 are (T - 2k + 2l-4)-Carleson measures for all l, then the composition operator C on DT is bounded; (ii) Let 1 < T n + 2, = ( 1,… , n) be univalent and the Frechet derivative of -1 be bounded on (B). If R j ∈ M(DT-2) for all j, then the composition operator C on DT is bounded; (iii) Let T > n + 2 and as in (ii). If j ∈ DT for all j, then the composition operator C on DT is bounded.

Linear Combination of Derivative Weighted Composition Operators

The relation between composition operators on the Dirichlet spaces in the open unit disk and derivative weighted composition operators on the Bergman spaces in the open unit disk is investigated firstly,and for a combination of several derivative weighted composition operators which acts on classic Bergman space,the lower bound of its essential norm is estimated in terms of the boundary data of the symbols of d-composition operators.Some similar results about composition operators on the Dirichlet space are also presented.A necessary condition is given to determine the compactness of the combination of several derivative weighted composition operators on Bergman spaces.

Small Hankel Operators on the Dirichlet-Type Spaces on the Unit Ball of C^n

In this paper the small Hankel operators on the Dirichlet-type spaces D p on the unit ball of C n are considered. A similar result to that of the one-dimensional setting is given, which characterizes the boundedness of the small Hankel operators on D p .

Commuting Dual Toeplitz Operators on the Orthogonal Complement of the Dirichlet Space

In this paper we characterize commuting dual Toeplitz operators with harmonic symbols on the orthogonal complement of the Dirichlet space in the Sobolev space. We also obtain the sufficient and necessary conditions for the product of two dual Toeplitz operators with harmonic symbols to be a finite rank perturbation of a dual Toeplitz operator.