A Refined Luecking’s Theorem and Finite-Rank Products of Toeplitz Operators

Abstract

Abstract. For any function f in L ∞ (D), let T f denote the corresponding Toeplitz operator the Bergman space A 2 (D). A recent result of D. Luecking shows that if T f has finite rank then f must be the zero function. Using a refined version of this result, we show that if all except possibly one of the functions f1, . . . , fm are radial and T f 1 · · · T fm has finite rank, then one of these functions must be zero.

Finite-Rank Products of Toeplitz Operators in Several Complex Variables

“…Thatμ is a linear combination of point masses implies that µ is also a linear combination of point masses. Thus, we recover Theorem 3.1 in [4].…”

“…We then apply the refined theorem to solve the problem about finite-rank products of Toeplitz operators in all dimensions, when all but possibly one of the operators are (weighted) shifts. This result is the content of Theorem 3.2, which is a generalization of the main result in [4].…”

“…, f r are bounded measurable functions on the disk, all except possibly one of them are radial functions and T f1 · · · T fr has finite rank, then one of these functions is the zero function. See [4] for more detail.…”

Finite rank Bargmann–Toeplitz operators with non-compactly supported symbols

“…After the proof in [12] appeared, a number of generalizations of Luecking's finite rank theorem have been obtained, see [1,5,11,17,18]. Some of them do not use the compactness of the support of the measure ν but rather build upon the theorem itself, and thus carry over to the noncompact case automatically (of course, with the condition of the type (6.1)) imposed.…”

“…• A generalization of the finite rank theorem to operators in the subspace in the Bargmann space B(C d ), spanned by monomials Z α , with a certain 'sparse' but infinite set of monomials removed [11,17].…”