A commuting pair of Hilbert space operators having the closed symmetrized bidisc <Equation ID="Equ11"> <EquationSource Format="TEX">$ \Gamma =\{(z_1+z_2, z_1z_2) \ : \ |z_1| \le 1, |z_2| \le 1\} $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="4pt" /> <mo>:</mo> <mspace width="4pt" /> <mo stretchy="false">|</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> <mo>,</mo> <mo stretchy="false">|</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </Equation>as a spectral set is called a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-contraction. A <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-unitary is a commuting pair of normal operators with its Taylor joint spectrum inside the distinguished boundary <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$b\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mi mathvariant="normal">Γ</mi> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> and a <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-isometry is the restriction of a <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-unitary to a joint invariant subspace of its components. Also, a pure <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-contraction is a <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-contraction (S, P) for which P is a pure contraction, i.e., <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$P^{n} \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>P</mi> <mrow> <mrow /> <mo>∗</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> strongly as <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$n \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. A <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-contraction (S, P) is called <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-distinguished if (S, P) is annihilated by a polynomial <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$q \in \mathbb {C}[z_1,z_2]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">[</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> whose zero set Z(q) defines a distinguished variety in the symmetrized bidisc <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\mathbb {G}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. There is a Schaffer-type minimal <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-isometric dilation of a <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-contraction (S, P) in the literature. In this article, we study when such a minimal <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-isometric dilation is <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-distinguished provided that (S, P) is a <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-distinguished <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-contraction. We show that a pure <InlineEquation ID="IEq24"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-isometry (T, V) with defect space <InlineEquation ID="IEq25"> <EquationSource Format="TEX">$\dim \mathscr {D}_{V^}< \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>dim</mo> <msub> <mi mathvariant="script">D</mi> <msup> <mi>V</mi> <mo>∗</mo> </msup> </msub> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, is <InlineEquation ID="IEq26"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-distinguished if and only if the fundamental operator of <InlineEquation ID="IEq27"> <EquationSource Format="TEX">$(T^,V^)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mo>,</mo> <msup> <mi>V</mi> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has spectral radius less than 1. Further, it is proved that a <InlineEquation ID="IEq28"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-contraction acting on a finite-dimensional Hilbert space dilates to a <InlineEquation ID="IEq29"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-distinguished <InlineEquation ID="IEq30"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-isometry if its fundamental operator has numerical radius less than 1. We also provide sufficient conditions for a pure <InlineEquation ID="IEq31"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-contraction to be <InlineEquation ID="IEq32"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-distinguished. Wold decomposition splits an isometry into two orthogonal parts of which one is a unitary and the other is a completely non-unitary contraction. In this direction, we find a few decomposition results for the <InlineEquation ID="IEq33"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-distinguished <InlineEquation ID="IEq34"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-unitaries and <InlineEquation ID="IEq35"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-distinguished pure <InlineEquation ID="IEq36"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-isometries.

Constrained Dilation and $\Gamma $-contractions

期刊论文
发表日期：

2026年6月1日

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Sourav Pal,
Nitin Tomar

摘要

A commuting pair of Hilbert space operators having the closed symmetrized bidisc $ \Gamma =\{(z_1+z_2, z_1z_2) \ : \ |z_1| \le 1, |z_2| \le 1\} $ $Γ = {(z_{1} + z_{2}, z_{1} z_{2}) : | z_{1} | \leq 1, | z_{2} | \leq 1}$ as a spectral set is called a $\Gamma $ $Γ$ -contraction. A $\Gamma $ $Γ$ -unitary is a commuting pair of normal operators with its Taylor joint spectrum inside the distinguished boundary $b\Gamma $ $b Γ$ of $\Gamma $ $Γ$ and a $\Gamma $ $Γ$ -isometry is the restriction of a $\Gamma $ $Γ$ -unitary to a joint invariant subspace of its components. Also, a pure $\Gamma $ $Γ$ -contraction is a $\Gamma $ $Γ$ -contraction (S, P) for which P is a pure contraction, i.e., $P^{*n} \rightarrow 0$ $P^{* n} \to 0$ strongly as $n \rightarrow \infty $ $n \to \infty$ . A $\Gamma $ $Γ$ -contraction (S, P) is called $\Gamma $ $Γ$ -distinguished if (S, P) is annihilated by a polynomial $q \in \mathbb {C}[z_1,z_2]$ $q \in C [z_{1}, z_{2}]$ whose zero set Z(q) defines a distinguished variety in the symmetrized bidisc $\mathbb {G}_2$ $G_{2}$ . There is a Schaffer-type minimal $\Gamma $ $Γ$ -isometric dilation of a $\Gamma $ $Γ$ -contraction (S, P) in the literature. In this article, we study when such a minimal $\Gamma $ $Γ$ -isometric dilation is $\Gamma $ $Γ$ -distinguished provided that (S, P) is a $\Gamma $ $Γ$ -distinguished $\Gamma $ $Γ$ -contraction. We show that a pure $\Gamma $ $Γ$ -isometry (T, V) with defect space $\dim \mathscr {D}_{V^*}< \infty $ $dim D_{V^{*}} < \infty$ , is $\Gamma $ $Γ$ -distinguished if and only if the fundamental operator of $(T^*,V^*)$ $(T^{*}, V^{*})$ has spectral radius less than 1. Further, it is proved that a $\Gamma $ $Γ$ -contraction acting on a finite-dimensional Hilbert space dilates to a $\Gamma $ $Γ$ -distinguished $\Gamma $ $Γ$ -isometry if its fundamental operator has numerical radius less than 1. We also provide sufficient conditions for a pure $\Gamma $ $Γ$ -contraction to be $\Gamma $ $Γ$ -distinguished. Wold decomposition splits an isometry into two orthogonal parts of which one is a unitary and the other is a completely non-unitary contraction. In this direction, we find a few decomposition results for the $\Gamma $ $Γ$ -distinguished $\Gamma $ $Γ$ -unitaries and $\Gamma $ $Γ$ -distinguished pure $\Gamma $ $Γ$ -isometries.

Constrained Dilation and \(\Gamma \)-contractions

摘要