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

Abstract

Theorems about characterization of finite rank Toeplitz operators in Fock-Segal-Bargmann spaces, known previously only for symbols with compact support, are carried over to symbols without that restriction, however with a rather rapid decay at infinity. The proof is based upon a new version of the Stone-Weierstrass approximation theorem.

Some weighted estimates for the ∂̅-equation and a finite rank theorem for Toeplitz operators in the Fock space

“…There are several discussions of this topic in the literature, see, for example, [2, 10-13] and references therein. These papers, however, consider the case of F being a function (or, as in [13], a measure) with certain growth limitations, or a distribution with compact support. We will gradually extend the set of admissible symbols, to reach, finally, a certain class of noncompactly supported distributions.…”

The Finite Rank Theorem for Toeplitz Operators on the Fock Space

“…Our aim now is to define the operator for a larger class of symbols. There are several discussions of this topic in the literature, see, e.g., [2], [9], [16], [17] and references therein. If we drop the boundedness condition for F , the Toeplitz operator is not necessarily bounded; it is defined on the set of functions u ∈ B satisfying T(F )u ∈ B.…”

“…Our aim now is to define the operator for a larger class of symbols. There are several discussions of this topic in the literature, see, e.g., [2], [9], [16], [17] and references therein.…”

“…The decay conditions imposed on the symbols in [17], look obviously excessive; on the other hand, the result in [17] does not apply to symbols-distributions, more singular than measures, which is an essential drawback for applications. Therefore the attack on the finite rank problem was continued.…”

“…It was shown in [2], [8] that if the compact support condition is dropped, then the finite rank theorem may become wrong even for rank zero, unless some restrictions on the behavior of the symbol at infinity are imposed. The compact support conditions were relaxed in [17], where for symbols-measures, very fast decaying at infinity, the finite rank theorem has been proved. Again, the approach by Luecking was used, and the result was established by introducing a new version of the Stone-Weierstrass theorem (proved in the same paper) where the uniform convergence of functions on a compact space is replaced by the weighted convergence on a locally compact space.…”