Abstract
We construct a functionuinL2Bn, dVwhich is unbounded on any neighborhood of each boundary point ofBnsuch that Toeplitz operatorTuis a Schattenp-class0<p<∞operator on Dirichlet-type spaceDBn, dV. Then, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Dirichlet-type spaceDBn, dV. We determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. We investigate the zero-product problem for several Toeplitz operators with radial symbols. Furthermore, the corresponding commuting problem of Toeplitz operators whose symbols are of the formξkuis studied, wherek ∈ Zn,ξ ∈ ∂Bn, anduis a radial function.
Highlights
Let Cn.Bn represent The Sobolev the open unit ball in several complex spaces space L2,1(Bn, dV) consists of the functions which satisfy f2,1 = n (∑ ∫ i=1 Bn (∂f ∂zi (z)2 + (z)2) dV)
Using Lemma 3, we can construct a compact Toeplitz operator with a symbol that is unbounded on any neighborhood of every point in unit surface
Is the product of two Toeplitz operators equal to a Toeplitz operator? In general, the answer is negative, but Brown and Halmos [14] showed that two bounded Toeplitz operators Tφ and Tψ commute on the Hardy space if and only if (I) both φ and ψ are analytic, (II) both φ and ψ are analytic, or (III) one is a linear function of the other
Summary
The Dirichlet-type space D is the subspace of all analytic functions g in L2,1(Bn) with g(0) = 0. Many mathematicians are interested in function theory and operator theory on the Dirichlet-type space (See [1, 2]). In [3, 4], for the Dirichlet-type space of one complex variable, that is, n = 1, Rochberg and Wu defined the Toeplitz operator with nonnegative measure μ on B1 as follows: Tμ : D → D by. Lu and Sun define Toeplitz operators on a Dirichlet-type space of several variables in [5]. Suppose that μ is a finite measure on Bn. Toeplitz operators on the Dirichlet-type space with symbol μ are defined as follows: Tμ (f) (w) = ∫ f (z) K(z, w)dμ (z) , f ∈ D.
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