Product of Toeplitz operators on the harmonic Dirichlet space

Abstract

In this paper, we study Toeplitz operators with harmonic symbols on the harmonic Dirichlet space, and show that the product of two Toeplitz operators is another Toeplitz operator only if one factor is constant.

Algebraic Properties of Toeplitz Operators on the Harmonic Dirichlet Space

“…As consequences, we characterize semi-commuting Toeplitz operators and zero Toeplitz operators. As special cases of when the symbols are harmonic, our results recover several known results in [14,15] mentioned above. As a corresponding problem, we also study the compact product problem of when The corresponding problems for Toeplitz operators acting on the Hardy space, (harmonic) Bergman space or Dirichlet space D have been well studied as in [1][2][3][4]7] or [8] and references therein.…”

“…First, taking U = V = 0 in Theorem 4.3, we characterize the zero Toeplitz operator. IEOT In a special case h = 0 in Corollary 4.5, we show that the zero product has only a trivial solution as shown in the following which recovers Corollary 1.2 of [15]. Specially, if we take U, V as harmonic functions in Corollary 4.7, we have the following simple corollary which recovers Theorem 1.2 of [14].…”

“…Then clearly H ∈ Δ 0 and by Proposition 2.1 T H (z) = D (1 − r)r dA = 0. Thus, (b) in Corollary 4.4 cannot be relaxed.As a special case of Theorem 4.3 when the symbols are all harmnoic, we have the following corollary which extends Theorem 1.1 of[15].…”

“…In recent papers [14,15], Zhao considered certain harmonic symbols inducing bounded Toeplitz operators, and then studied (semi-)commuting problem and product problem for Toeplitz operators.…”