Finite Rank Bergman–Toeplitz and Bargmann–Toeplitz Operators in Many Dimensions

Abstract

Abstract. The recent theorem by D. Luecking that finite rank Toeplitz-Bergman operators must be generated by a measure consisting of finitely many point masses is carried over to the manydimensional case.

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.…”

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

“…This circumstance can be taken care of by a slight change in the proof. In fact, it is noticed [18] that the finite rank property for the measure ν implies the same property for the measure ν g = |g(Z )| 2 ν, where g is a function analytical in the neighborhood of the support of the measure ν. This reasoning does not hold water for the measure with a noncompact support.…”

“…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 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.…”