<p>We study the growth of the resolvent of a Hardy–Toeplitz operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$T_b$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>b</mi> </msub> </math></EquationSource> </InlineEquation> with a Laurent polynomial symbol <i>b</i>. The matrix of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T_b$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>b</mi> </msub> </math></EquationSource> </InlineEquation> is banded in this case. First, we obtain a local upper bound on the norm of the resolvent of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T_b$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>b</mi> </msub> </math></EquationSource> </InlineEquation> in a neighborhood of a point <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$w_0\in \partial \sigma (T_b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>∈</mo> <mi>∂</mi> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>b</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on the boundary of its spectrum. Second, the local growth bound on the resolvent is extended to a global one on the whole resolvent set of the operator <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T_b$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>b</mi> </msub> </math></EquationSource> </InlineEquation>, under additional assumption on the spectrum.</p>

On the Local Growth of Resolvent of Banded Toeplitz Operators

期刊论文
发表日期：

2025年12月15日

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L. Golinskii,
S. Kupin

摘要

We study the growth of the resolvent of a Hardy–Toeplitz operator $T_b$ $T_{b}$ with a Laurent polynomial symbol b. The matrix of $T_b$ $T_{b}$ is banded in this case. First, we obtain a local upper bound on the norm of the resolvent of $T_b$ $T_{b}$ in a neighborhood of a point $w_0\in \partial \sigma (T_b)$ $w_{0} \in \partial σ (T_{b})$ on the boundary of its spectrum. Second, the local growth bound on the resolvent is extended to a global one on the whole resolvent set of the operator $T_b$ $T_{b}$ , under additional assumption on the spectrum.