For banded Toeplitz operators, the resolvent grows inverse-linearly near spectral boundary points outside a finite exceptional set, within non-tangential domains.
Nikolski, Toeplitz Matrices and Operators, Cambridge Studies in Advanced Math- ematics, v.182, 2020
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On local non-tangential growth of the resolvent of a banded Toeplitz operator
For banded Toeplitz operators, the resolvent grows inverse-linearly near spectral boundary points outside a finite exceptional set, within non-tangential domains.