Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> be a separable complex Hilbert space. A conjugate-linear map <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C:\mathcal {H}\rightarrow \mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo>:</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">H</mi> </mrow> </math></EquationSource> </InlineEquation> is called a conjugation if it is an involutive isometry. In this paper, we focus on the following interpolation problems: Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{x_i\}_{i\in I}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\{y_i\}_{i\in I}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be orthonormal sets of vectors in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\{N_k\}_{k\in K}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>N</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be a set of mutually commuting normal operators. We seek to determine under which conditions there exists a conjugation C on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> such that <OrderedList> <ListItem> <ItemNumber>(a)</ItemNumber> <ItemContent> <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$Cx_i=y_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$CN_kC=N_k^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <msub> <mi>N</mi> <mi>k</mi> </msub> <mi>C</mi> <mo>=</mo> <msubsup> <mi>N</mi> <mi>k</mi> <mo>∗</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$i\in I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$k\in K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation>; or </ItemContent> </ListItem> <ListItem> <ItemNumber>(b)</ItemNumber> <ItemContent> <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$Cx_i=y_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$CN_kC=-N_k^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <msub> <mi>N</mi> <mi>k</mi> </msub> <mi>C</mi> <mo>=</mo> <mo>-</mo> <msubsup> <mi>N</mi> <mi>k</mi> <mo>∗</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$i\in I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$k\in K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation>. </ItemContent> </ListItem> </OrderedList> We provide complete answers to problems (a) and (b) using the spectral projections of normal operators. Our results are then applied to the study of complex-symmetric and skew-symmetric operators, as well as to the characterization of hyperinvariant subspaces of normal operators through conjugations.

Interpolation theorems for conjugations

期刊论文
发表日期：

2026年2月24日

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Zouheir Amara

摘要

Let $\mathcal {H}$ $H$ be a separable complex Hilbert space. A conjugate-linear map $C:\mathcal {H}\rightarrow \mathcal {H}$ $C : H \to H$ is called a conjugation if it is an involutive isometry. In this paper, we focus on the following interpolation problems: Let $\{x_i\}_{i\in I}$ ${x_{i}}_{i \in I}$ and $\{y_i\}_{i\in I}$ ${y_{i}}_{i \in I}$ be orthonormal sets of vectors in $\mathcal {H}$ $H$ , and let $\{N_k\}_{k\in K}$ ${N_{k}}_{k \in K}$ be a set of mutually commuting normal operators. We seek to determine under which conditions there exists a conjugation C on $\mathcal {H}$ $H$ such that (a)

$Cx_i=y_i$ $C x_{i} = y_{i}$ and $CN_kC=N_k^*$ $C N_{k} C = N_{k}^{*}$ for all $i\in I$ $i \in I$ and $k\in K$ $k \in K$ ; or

(b)

$Cx_i=y_i$ $C x_{i} = y_{i}$ and $CN_kC=-N_k^*$ $C N_{k} C = - N_{k}^{*}$ for all $i\in I$ $i \in I$ and $k\in K$ $k \in K$ .

We provide complete answers to problems (a) and (b) using the spectral projections of normal operators. Our results are then applied to the study of complex-symmetric and skew-symmetric operators, as well as to the characterization of hyperinvariant subspaces of normal operators through conjugations.