Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2828_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{T}=\left\{ T\left( t\right) \right\} _{t\ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">T</mi> <mo>=</mo> <msub> <mfenced close="}" open="{"> <mi>T</mi> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mfenced> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> be a <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2828_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$C_{0} $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-semigroup of contractions on a Hilbert space H with generator A and assume that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2828_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">T</mi> </math></EquationSource> </InlineEquation> is not a multiple of the identity. In addition, assume that <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2828_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">T</mi> </math></EquationSource> </InlineEquation> is of class <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2828_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$C_{*\cdot }$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>C</mi> <mrow> <mrow /> <mo>∗</mo> <mo>·</mo> </mrow> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>, that is, there exists a vector <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2828_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$x\in H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2828_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">$\inf _{t\ge 0}\left\| T\left( t\right) x\right\| >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo movablelimits="true">inf</mo> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mfenced close="∥" open="∥"> <mi>T</mi> <mfenced close=")" open="("> <mi>t</mi> </mfenced> <mi>x</mi> </mfenced> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. If <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2828_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _{A}\left( x\right) \cap i \mathbb {R} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>A</mi> </msub> <mfenced close=")" open="("> <mi>x</mi> </mfenced> <mo>∩</mo> <mi>i</mi> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is at most countable set, then <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2828_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">T</mi> </math></EquationSource> </InlineEquation> has a non-trivial hyperinvariant subspace, where <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2828_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _{A}\left( x\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>A</mi> </msub> <mfenced close=")" open="("> <mi>x</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> is the local spectrum of A at x.

Invariant Subspace of $C_{0}$-Semigroup of Contractions on Hilbert Space

期刊论文
发表日期：

2025年3月21日

查看全文

Heybetkulu Mustafayev

摘要

Let $\textbf{T}=\left\{ T\left( t\right) \right\} _{t\ge 0}$ $T = {\{T, (t)\}}_{t \geq 0}$ be a $C_{0} $ $C_{0}$ -semigroup of contractions on a Hilbert space H with generator A and assume that $\textbf{T}$ $T$ is not a multiple of the identity. In addition, assume that $\textbf{T}$ $T$ is of class $C_{*\cdot }$ $C_{* \cdot}$ , that is, there exists a vector $x\in H$ $x \in H$ such that $\inf _{t\ge 0}\left\| T\left( t\right) x\right\| >0$ ${inf}_{t \geq 0} ∥T, (t), x∥ > 0$ . If $\sigma _{A}\left( x\right) \cap i \mathbb {R} $ $σ_{A} (x) \cap i R$ is at most countable set, then $\textbf{T}$ $T$ has a non-trivial hyperinvariant subspace, where $\sigma _{A}\left( x\right) $ $σ_{A} (x)$ is the local spectrum of A at x.

Invariant Subspace of \(C_{0}\)-Semigroup of Contractions on Hilbert Space

摘要