The quantum annulus of type r is the class of invertible operators with singular values in (1/r, r). Given an analytic function on the classical annulus of type r,  we may evaluate it on operators in the quantum annulus via the holomorphic functional calculus. The spectral constant gives the maximum ratio betweeen the supremum over the norm of evalutions at operators in the quantum annulus to the supremum over classical evaluations. We show that the limit of the spectral constant as r goes to infinity is 2. Via the correspondence between annuli and hyperbolae, our study degenerates the problem to one on the quantum cross, pairs of contractions with product zero, where the spectral constant is exactly 2. As a consequence, we see the following elementary sharp Cartan extension theorem with bounds in the limit case: Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f,g: \mathbb {D}\rightarrow \overline{\mathbb {D}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">→</mo> <mover> <mi mathvariant="double-struck">D</mi> <mo>¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> be analytic functions such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f(0)=g(0).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> There is an <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$h: \mathbb {D}^2\rightarrow 2\overline{\mathbb {D}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">→</mo> <mn>2</mn> <mover> <mi mathvariant="double-struck">D</mi> <mo>¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> analytic such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$h(z,0)=f(z), h(0,w)=g(w).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The essential technique is to rationally dilate Z to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\hat{Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation> which has <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$U =(\hat{Z}+(\hat{Z}^{-1})^)/(r+1/r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mover accent="true"> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> unitary and estimate <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$Uf(\hat{Z})U^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> <msup> <mi>U</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> directly.

The spectral constant for the quantum cross and asymptotically sharp bounds for annuli

期刊论文
发表日期：

2026年1月12日

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J. E. Pascoe

摘要

The quantum annulus of type r is the class of invertible operators with singular values in (1/r, r). Given an analytic function on the classical annulus of type r, we may evaluate it on operators in the quantum annulus via the holomorphic functional calculus. The spectral constant gives the maximum ratio betweeen the supremum over the norm of evalutions at operators in the quantum annulus to the supremum over classical evaluations. We show that the limit of the spectral constant as r goes to infinity is 2. Via the correspondence between annuli and hyperbolae, our study degenerates the problem to one on the quantum cross, pairs of contractions with product zero, where the spectral constant is exactly 2. As a consequence, we see the following elementary sharp Cartan extension theorem with bounds in the limit case: Let $f,g: \mathbb {D}\rightarrow \overline{\mathbb {D}}$ $f, g : D \to \bar{D}$ be analytic functions such that $f(0)=g(0).$ $f (0) = g (0) .$ There is an $h: \mathbb {D}^2\rightarrow 2\overline{\mathbb {D}}$ $h : D^{2} \to 2 \bar{D}$ analytic such that $h(z,0)=f(z), h(0,w)=g(w).$ $h (z, 0) = f (z), h (0, w) = g (w) .$ The essential technique is to rationally dilate Z to $\hat{Z}$ $\hat{Z}$ which has $U =(\hat{Z}+(\hat{Z}^{-1})^*)/(r+1/r)$ $U = (\hat{Z} + {({\hat{Z}}^{- 1})}^{*}) / (r + 1 / r)$ unitary and estimate $Uf(\hat{Z})U^*$ $U f (\hat{Z}) U^{*}$ directly.