On higher dimensional Luecking's theorem

Finite-Rank Products of Toeplitz Operators in Several Complex Variables

“…That the converse is also true is the content of the following theorem, which had been an open conjecture for about twenty years. See [1,6,7]. Theorem 1.1.…”

“…In [7], G. Rozenblum and N. Shirokov give a proof by induction on the dimension n. In the base case (n = 1), they use the above Luecking's result. In [1], B. Choe follows Luecking's scheme with modifications (to the setting of several variables) to prove Theorem 1.1 for all n ≥ 1. In this note, we modify Choe's proof to obtain a refined version of Theorem 1.1.…”

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.…”

“…• The extension to the case of operators in the harmonic Bargmann space in [1] and to operators in d-harmonic Bargmann space in [5] in R d .…”

“…The alternative to [5] proof of the multi-dimensional extension of Luecking's theorem in [18] does not carry over directly to non-compactly supported measures. This proof uses the induction on dimension and the relation of the finite rank property for the Toeplitz operator in the Bargmann space and this property for the operator with the same symbol, but acting in the Bergman space of analytical functions in a bounded domain containing the support of the symbol.…”