The Finite Rank Theorem for Toeplitz Operators on the Fock Space

Abstract

We consider Toeplitz operators on the Fock space, under rather general conditions imposed on the symbols. It is proved that if the operator has finite rank then the operator and the symbol should be zero. The method of proof is different from the ones used previously for finite rank theorems, and it enables us to get rid of the compact support condition and even allow a certain growth of the symbol. LATP,

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

“…One can find a detailed historical overview in [4,12,14]. In particular, in [4] a finite rank theorem has been proved for operators in the Fock space of analytical functions on C 1 square integrable with weight function ω(z) = π −1 e −|z| 2 having symbols-functions subject to rather mild, almost sharp, growth restrictions. However, the reasoning in [4] does not apply to symbols-distributions.…”

“…In particular, in [4] a finite rank theorem has been proved for operators in the Fock space of analytical functions on C 1 square integrable with weight function ω(z) = π −1 e −|z| 2 having symbols-functions subject to rather mild, almost sharp, growth restrictions. However, the reasoning in [4] does not apply to symbols-distributions. The only presently known approach to deal with this latter case, developed for compactly supported symbols in [1], is based upon the result on the solvability of the∂-equation, namely on Lemma 1.2.…”