<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {D}=\{z\in \mathbb {C}: |z|<1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">D</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo><</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {T}=\{z\in \mathbb {C}: |z|=1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$a\in \mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mi mathvariant="double-struck">D</mi> </mrow> </math></EquationSource> </InlineEquation>, consider <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varphi _a(z)=\displaystyle {\frac{a-z}{1-\bar{a}z}}$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <msub> <mi>φ</mi> <mi>a</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>a</mi> <mo>-</mo> <mi>z</mi> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mover accent="true"> <mrow> <mi>a</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$C_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> the composition operator in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^2(\mathbb {T})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> induced by <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\varphi _a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>φ</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation>: <Equation ID="Equ19"> <EquationSource Format="TEX">$\begin{aligned} C_a f=f\circ \varphi _a. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>C</mi> <mi>a</mi> </msub> <mi>f</mi> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <msub> <mi>φ</mi> <mi>a</mi> </msub> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Clearly <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$C_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$C_a^2=I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>C</mi> <mi>a</mi> <mn>2</mn> </msubsup> <mo>=</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation>, i.e., is a non-selfadjoint reflection. In this paper we study the operator algebras related to <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$C_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation>: the <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\text {C}^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>C</mtext> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra generated by <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$C_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation>, its commutant and its double commutant.</p>

Algebras of reflections in $L^2(\mathbb {T})$

期刊论文
发表日期：

2026年2月25日

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Esteban Andruchow

摘要

Let $\mathbb {D}=\{z\in \mathbb {C}: |z|<1\}$ $D = {z \in C : | z | < 1}$ and $\mathbb {T}=\{z\in \mathbb {C}: |z|=1\}$ $T = {z \in C : | z | = 1}$ . For $a\in \mathbb {D}$ $a \in D$ , consider $\varphi _a(z)=\displaystyle {\frac{a-z}{1-\bar{a}z}}$ $φ_{a} (z) = \frac{a - z}{1 - \bar{a} z}$ and $C_a$ $C_{a}$ the composition operator in $L^2(\mathbb {T})$ $L^{2} (T)$ induced by $\varphi _a$ $φ_{a}$ : $\begin{aligned} C_a f=f\circ \varphi _a. \end{aligned}$ $\begin{matrix} C_{a} f = f \circ φ_{a} . \end{matrix}$ Clearly $C_a$ $C_{a}$ satisfies $C_a^2=I$ $C_{a}^{2} = I$ , i.e., is a non-selfadjoint reflection. In this paper we study the operator algebras related to $C_a$ $C_{a}$ : the $\text {C}^*$ $C^{*}$ -algebra generated by $C_a$ $C_{a}$ , its commutant and its double commutant.

Algebras of reflections in \(L^2(\mathbb {T})\)

摘要