Quasihomogeneous Toeplitz operators on the harmonic Bergman space

Abstract

In this paper we study the product of Toeplitz operators on the harmonic Bergman space of the unit disk of the complex plane C. Mainly, we discuss when the product of two quasihomogeneous Toeplitz operators is also a Toeplitz operator, and when such operators commute.Mathematics Subject Classification (2010). Primary 47B35; Secondary 47B38.

Quasihomogeneous Toeplitz Operators With Integrable Symbols on the Harmonic Bergman Space

“…(3) e ik 1 θ r m and e ik 2 θ ϕ are linearly dependent; This has been partially proved in [11,Theorem 3.8], in the case when |k 1 | ≤ |k 2 |. Also, with the additional hypothesis that the symbols are bounded, Louhichi and Zakariasy [18] proved some special cases of the above theorem, in the case when 0 < k 1 ≤ k 2 . The remaining case, that is, when |k 1 | > |k 2 |, was left open.…”

“…The corresponding problem has been well studied for many years on the classical Hardy space and the analytic Bergman space; for example, see [2-4, 7, 10, 13, 14, 17, 20]. Recently, there has been an increasing interest in the present harmonic Bergman space case; see [5,6,8,18,19] and the references therein.…”

Algebraic Properties of Quasihomogeneous and Separately Quasihomogeneous Toeplitz Operators on the Pluriharmonic Bergman Space

“…As for = 1, a separately radial function is a radial and it follows that Toeplitz operators with separately radial symbols commute, there is no contradiction with an extension of a result in [29] to the case of ≥ 2. It is given in [29] that a Toeplitz operator on 2 ℎ (D) with radial symbol commutes with another Toeplitz operator if that operator also has a radial symbol. The following theorem will show that this result is not true on 2 ℎ (B ) ( ≥ 2).…”

“…Zhou and Dong [9] studied the commuting problem for quasihomogeneous Toeplitz operators. In 2012, Dong and Zhou [28] and Louhichi and Zakariasy [29] characterized the commuting Toeplitz operators with radial or quasihomogeneous symbols on the harmonic Bergman space of the unit disk. In papers [19,[30][31][32][33], the authors studied the wide classes of (nongeometrically defined) commutative Banach algebras generated by Toeplitz operators of the Bergman spaces on the unit ball.…”

Algebraic properties of small Hankel operators on the harmonic Bergman space

“…On the harmonic Bergman space, [14] has studied the algebra of Toeplitz operators and small Hankel operators. Some studies focusing on the algebraic properties of Toepltz operators with harmonic symbols [5], [7] or quasihomogeneous symbols [8], [9], [10], [18] showed that the results obtained are also quite different from the case on the Hardy or Bergman space.…”