In this paper we give necessary and sufficient conditions for a symbol that produces a Toeplitz operator on the Bergman space that commutes with another such operator whose symbol is a monomial. The result is stated in terms of the Mellin transform of the symbol. As a corollary, we show that a Toeplitz operator with a radial symbol may only commute with another such operator with a radial symbol. 1998 Academic Press 195
Let A be a closed, point-separating sub-algebra of C 0 (X), where X is a locally compact Hausdorff space. Assume that X is the maximal ideal space of A.
Abstract. In this note we show that if two Toeplitz operators on a Bergman space commute and the symbol of one of them is analytic and nonconstant, then the other one is also analytic.Let Ω be a bounded open domain in the complex plane and let dA denote area measure on Ω. The Bergman space L 2 a (Ω) is the subspace of L 2 (Ω, dA) consisting of the square-integrable functions that are analytic on Ω. For a bounded measurable function ϕ on Ω, the Toeplitz operator T ϕ with symbol ϕ is the operator on L 2 a (Ω) defined by More general results concerning which operators, not necessarily Toeplitz, commute with an analytic Hardy space Toeplitz operator are due to Thompson ([10] and [11]) and Cowen [6]. On the Bergman space, the situation is more complicated. The Brown-Halmos result mentioned above fails. For example, if Ω is the unit disk, then any two Toeplitz operators whose symbols are radial functions commute (proof: an easy calculation shows that every Toeplitz operator with radial symbol has a diagonal matrix with respect to the usual orthonormal basis; any two diagonal matrices commute).
In this paper we discuss an unusual phenomenon in the context of Toeplitz operators in the Bergman space on the unit disc: If two Toeplitz operators commute with a quasihomogeneous Toeplitz operator, then they commute with each other. In the Bourbaki terminology, this result can be stated as follows: The commutant of a quasihomogeneous Toeplitz operator is equal to its bicommutant.
Mathematics Subject Classification (2000). Primary 47B35; Secondary 47L80.Let D denote the unit disc in the complex plane C, and dA = rdr dθ π , where (r, θ) are polar coordinates, is the normalized Lebesgue measure. So that D has area 1. The Bergman space L 2 a is the Hilbert space of analytic functions on D that are square integrable with respect to the measure dA. It is well known that L 2 a is a closed subspace of the Hilbert space L 2 (D, dA) and has { √ n + 1z n | n ≥ 0} as an orthonormal basis. Let P be the orthogonal projection from L 2 (D, dA) onto L 2 a . For any function φ ∈ L ∞ (D, dA), the Toeplitz operator T φ with symbol φ is the operator on L 2 a defined by T φ f = P (φf ), for any f ∈ L 2 a . The question when two Toeplitz operators in the Bergman space commute, was worked on by many people [1,4,5,2,6,7,9] since Brown and Halmos solved the analogous problem on the Hardy space of the unit circle T of C [3]. In fact they prove that T φ T ψ = T ψ T φ for some φ and ψ in L ∞ (T) if and only if (a) φ and ψ are both analytic, or (b)φ andψ are both analytic, or (c) one of the two symbols is a linear function of the other.
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