LRG condition for Toeplitz operators with arbitrary Laurent polynomial symbols

Determine whether the Linear Resolvent Growth condition ||(Tb − w)^{-1}|| ≤ C(b)/dist(w,σ(Tb)) holds for Toeplitz operators Tb whose symbols b are arbitrary Laurent polynomials (without assuming regularity), for all w in the resolvent set ρ(Tb).

Background

The paper studies bounds on the resolvent norm of Toeplitz operators Tb in terms of the distance from w to the spectrum σ(Tb). Two growth regimes are considered: Linear Resolvent Growth (LRG), where ||(Tb − w)^{-1}|| ≤ C(b)/dist(w,σ(Tb)), and Quadratic Resolvent Growth (QRG), where ||(Tb − w)^{-1}|| ≤ C(b)/dist(w,σ(Tb)) * (1 + 1/dist(w,σ(Tb))).

Theorem 2.1 establishes the LRG bound for Toeplitz operators whose symbols b are regular Laurent polynomials, a class broader than Laurent polynomials with the Jordan property. In Section 3 the authors note that, while QRG can be shown for any polynomial symbol using general techniques, it remains unknown whether the stronger LRG bound extends from regular to all Laurent polynomial symbols. This uncertainty is explicitly stated and motivates the open problem.