<p>In the paper we fully describe the Taylor spectrum of pairs of isometries defined by diagrams. In most cases both isometries in such pairs have nontrivial shift part, and its Taylor spectrum is a proper subset (of Lebesgue measure in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(0,\pi ^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mi>π</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>) of the closed bidisk. In particular, there are pairs of isometries which spectrum is equal to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{(\mu ,\lambda )\in \overline{\mathbb {D}}^2: |\mu |^a\le |\lambda |\le |\mu |^b\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mover> <mi mathvariant="double-struck">D</mi> <mo>¯</mo> </mover> <mn>2</mn> </msup> <mo>:</mo> <mo stretchy="false">|</mo> <mi>μ</mi> <msup> <mo stretchy="false">|</mo> <mi>a</mi> </msup> <mo>≤</mo> <mo stretchy="false">|</mo> <mi>λ</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mo stretchy="false">|</mo> <mi>μ</mi> <msup> <mo stretchy="false">|</mo> <mi>b</mi> </msup> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$0<a\le b<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo>≤</mo> <mi>b</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

The Taylor Spectrum of Pairs of Isometries

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2026年8月4日

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Zbigniew Burdak,
Patryk Pagacz

摘要

In the paper we fully describe the Taylor spectrum of pairs of isometries defined by diagrams. In most cases both isometries in such pairs have nontrivial shift part, and its Taylor spectrum is a proper subset (of Lebesgue measure in $(0,\pi ^2)$ $(0, π^{2})$ ) of the closed bidisk. In particular, there are pairs of isometries which spectrum is equal to $\{(\mu ,\lambda )\in \overline{\mathbb {D}}^2: |\mu |^a\le |\lambda |\le |\mu |^b\}$ ${(μ, λ) \in {\bar{D}}^{2} : | μ |^{a} \leq | λ | \leq | μ |^{b}}$ for any $0<a\le b<\infty $ $0 < a \leq b < \infty$ .