Scholarlink

<p>The set of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathfrak {L}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">L</mi> </math></EquationSource> </InlineEquation>-resolvents of a densely defined symmetric operator in a Hilbert space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathfrak {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">H</mi> </math></EquationSource> </InlineEquation> with a proper gauge <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathfrak {L}(\subset \mathfrak {H})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">L</mi> <mo stretchy="false">(</mo> <mo>⊂</mo> <mi mathvariant="fraktur">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> was described by Kreĭn and Saakyan. The Kreĭn–Saakyan theory of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathfrak {L}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">L</mi> </math></EquationSource> </InlineEquation>-resolvent matrices was extended by Shmul’yan and Tsekanovskii to the case of improper gauge <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\mathfrak {L}(\not \subset \mathfrak {H})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">L</mi> <mo stretchy="false">(</mo> <mo>⊄</mo> <mi mathvariant="fraktur">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and by Langer and Textorius to the case of symmetric linear relations in Hilbert spaces. In the present paper we find connections between the theory of boundary triples and the Kreĭn–Saakyan theory of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\mathfrak {L}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">L</mi> </math></EquationSource> </InlineEquation>-resolvent matrices for symmetric linear relations with improper gauges in Hilbert spaces and extend the known formula for the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\mathfrak {L}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">L</mi> </math></EquationSource> </InlineEquation>-resolvent matrix in terms of boundary operators to this class of relations. Descriptions of spectral and pseudo-spectral functions of symmetric linear relations with improper gauges are given. The results are applied to linear relations generated by a canonical system.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

\({\mathfrak {L}}\)-Resolvents and Pseudo-Spectral Functions of Symmetric Linear Relations in Hilbert Spaces

  • 期刊论文

  • 发表日期：

    2026年3月28日

查看全文
  • Volodymyr Derkach

摘要

The set of \(\mathfrak {L}\) L -resolvents of a densely defined symmetric operator in a Hilbert space \(\mathfrak {H}\) H with a proper gauge \(\mathfrak {L}(\subset \mathfrak {H})\) L ( ⊂ H ) was described by Kreĭn and Saakyan. The Kreĭn–Saakyan theory of \(\mathfrak {L}\) L -resolvent matrices was extended by Shmul’yan and Tsekanovskii to the case of improper gauge \(\mathfrak {L}(\not \subset \mathfrak {H})\) L ( ⊄ H ) and by Langer and Textorius to the case of symmetric linear relations in Hilbert spaces. In the present paper we find connections between the theory of boundary triples and the Kreĭn–Saakyan theory of \(\mathfrak {L}\) L -resolvent matrices for symmetric linear relations with improper gauges in Hilbert spaces and extend the known formula for the \(\mathfrak {L}\) L -resolvent matrix in terms of boundary operators to this class of relations. Descriptions of spectral and pseudo-spectral functions of symmetric linear relations with improper gauges are given. The results are applied to linear relations generated by a canonical system.