多圆盘Bergman空间上Toeplitz算子的乘积和交换性

该文给出了多圆盘Bergman空间上两个带有某种符号的Toeplitz算子的乘积等于另一个Toeplitz算子的充分必要条件,并且给出了乘积算子所带符号的公式.接下来,相应的研究了它的交换性.这些研究结果都是根据符号函数的Mellin变换.     

Positivity of Toeplitz operators via Berezin transform

In this paper, we study positive Toeplitz operators on the Bergman space via their Berezin transforms. Surprisingly we show that the positivity of a Toeplitz operator on the Bergman space is not completely determined by the positivity of the Berezin transform of its symbol. In fact, we show that even if the minimal value of the Berezin transform of a quadratic polynomial of |z|$|z|$ on the unit disk is positive, the Toeplitz operator with the function as the symbol may not be positive.     

Positivity of Toeplitz operators on harmonic Bergman space

In this paper, we study positive Toeplitz operators on the harmonic Bergman space via their Berezin transforms. We consider the Toeplitz operators with continuous harmonic symbols on the closed disk and show that the Toeplitz operator is positive if and only if its Berezin transform is nonnegative on the disk. On the other hand, we construct a function such that the Toeplitz operator with this function as the symbol is not positive but its Berezin transform is positive on the disk. We also consider the harmonic Bergman space on the upper half plane and prove that in this case the positive Toeplitz operators with continuous integrable harmonic symbols must be the zero operator.     

A question of Toeplitz operators on the harmonic Bergman space

Choe and Lee [B.R. Choe, Y.J. Lee, Commuting Toeplitz operators on the harmonic Bergman space, Michigan Math. J. 46 (1999) 163-174] put the question: If an analytic Toeplitz operator and a co-analytic Toeplitz operator on the harmonic Bergman space commute, then is one of their symbols constant? If one of their symbols is bounded, then we will show that the answer is yes.     

Weighted reproducing kernels and Toeplitz operators on harmonic Bergman spaces on the real ball

For the standard weighted Bergman spaces on the complex unit ball, the Berezin transform of a bounded continuous function tends to this function pointwise as the weight parameter tends to infinity. We show that this remains valid also in the context of harmonic Bergman spaces on the real unit ball of any dimension. This generalizes the recent result of C. Liu for the unit disc, as well as the original assertion concerning the holomorphic case. Along the way, we also obtain a formula for the corresponding weighted harmonic Bergman kernels.

     

Algebraic properties of Toeplitz and small Hankel operators on the harmonic Bergman space

In this paper,we discuss some algebraic properties of Toeplitz operators and small Hankel operators with radial and quasihomogeneous symbols on the harmonic Bergman space of the unit disk in the complex plane C.We solve the product problem of quasihomogeneous Toeplitz operator and quasihomogeneous small Hankel operator.Meanwhile,we characterize the commutativity of quasihomogeneous Toeplitz operator and quasihomogeneous small Hankel operator.     

Toeplitz operators with BMO symbols of several complex variables

In this note we prove that the boundedness and compactness of the Toeplitz operator on the Bergman space L_a² (\mathbbB_n )L_a^2 (\mathbb{B}_n ) for several complex variables with a BMO¹ symbol is completely determined by the boundary behavior of its Berezin transform.     

Isometries and Toeplitz operators of Bergman space of bounded symmetric domains

In this paper we completely characterize the isometries of Bergman space (0?p<∞, p≠2) of bounded symmetric domains. We also prove that a pair of Toeplitz operators Tf and Tg on (0<p<∞, p≠2) is isometric equivalence if and only if there is a τ∈Aut(Ω), such that g=f○τ, where Aut(Ω) is the automorphism group of Ω.     

Bergman空间上拟齐次Toeplitz算子的乘积

使用Mellin变换作为工具,讨论了Bergman空间上以拟齐次函数为符号的Toeplitz算子的乘积问题,得出了当拟齐次函数的度处于三种不同情况时两个Toeplitz算子乘积仍是Toeplitz算子的充分必要条件.     

Products of composition and analytic Toeplitz operators

We will consider the problem of which the products of composition and analytic Toeplitz operators would be bounded or compact on the Hardy space H² and the Bergman space L_a².     

Compact operators and Toeplitz algebras on multiply-connected domains

If Ω is a smoothly bounded multiply-connected domain in the complex plane and S belongs to the Toeplitz algebra τ of the Bergman space of Ω, we show that S is compact if and only if its Berezin transform vanishes at the boundary of Ω. We also show that every element S in T, the C^?-subalgebra of τ generated by Toeplitz operators with symbols in H^∞(Ω), has a canonical decomposition for some R in the commutator ideal CT; and S is in CT iff the Berezin transform vanishes identically on the set M₁ of trivial Gleason parts.     

Toeplitz operators on the polydisk

In this paper it is shown that two analytic Toeplitz operators essentially doubly commute if and only if they doubly commute on the Bergman space of the polydisk.

     

Berezin transform and Toeplitz operators on harmonic Bergman spaces

For an operator which is a finite sum of products of finitely many Toeplitz operators on the harmonic Bergman space over the half-space, we study the problem: Does the boundary vanishing property of the Berezin transform imply compactness? This is motivated by the Axler-Zheng theorem for analytic Bergman spaces, but the answer would not be yes in general, because the Berezin transform annihilates the commutator of any pair of Toeplitz operators. Nevertheless we show that the answer is yes for certain subclasses of operators. In order to do so, we first find a sufficient condition on general operators and use it to reduce the problem to whether the Berezin transform is one-to-one on related subclasses.     

Finite rank Toeplitz operators on the Bergman space

Given a complex Borel measure with compact support in the complex plane the sesquilinear form defined on analytic polynomials and by , determines an operator from the space of such polynomials to the space of linear functionals on . This operator is called the Toeplitz operator with symbol . We show that has finite rank if and only if is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.

     

Commuting Toeplitz Operators on the Pluriharmonic Bergman Space

We prove that two Toeplitz operators acting on the pluriharmonic Bergman space with radial symbol and pluriharmonic symbol respectively commute only in an obvious case.     

Commuting Toeplitz operators on the Segal-Bargmann space

Consider two Toeplitz operators Tg, Tf on the Segal-Bargmann space over the complex plane. Let us assume that g is a radial function and both operators commute. Under certain growth condition at infinity of f and g we show that f must be radial, as well. We give a counterexample of this fact in case of bounded Toeplitz operators but a fast growing radial symbol g. In this case the vanishing commutator [Tg,Tf]=0 does not imply the radial dependence of f. Finally, we consider Toeplitz operators on the Segal-Bargmann space over Cn and n>1, where the commuting property of Toeplitz operators can be realized more easily.     

Toeplitz operators on the Dirichlet space

We study algebraic properties of Toeplitz operators acting on the Dirichlet space. We first characterize two harmonic symbols of commuting Toeplitz operators. Also, we give characterizations of the harmonic symbol for which the corresponding Toeplitz operator is self-adjoint or an isometry.     

Finite sums of Toeplitz products on the Dirichlet space

On the Dirichlet space of the unit disk, we consider a class of operators which contain finite sums of products of two Toeplitz operators with harmonic symbols. We give characterizations of when an operator in that class is zero or compact. Also, we solve the zero product problem for products of finitely many Toeplitz operators with harmonic symbols.     

Commuting Toeplitz operators on the polydisk

We obtain characterizations of (essentially) commuting Toeplitz operators with pluriharmonic symbols on the Bergman space of the polydisk. We show that commuting and essential commuting properties are the same for dimensions bigger than 2, while they are not for dimensions less than or equal to 2. Also, the corresponding results for semi-commutators are obtained.

     

Bergman空间上可交换的Hankel算子和Toeplitz算子

证明了Bergman空间上的两个小Hankel算子如果是可交换的且其中一个是拟齐次的小Hankel算子,则另一个也是拟齐次的.还研究了Toeplitz算子和小Hankel算子的交换性.