Let H be a separable, complex Hilbert space and A be a bounded linear operator on H. For a closed subspace <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathscr {M}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation> of H, the local commutant of A at <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathscr {M}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation> is defined by <Equation ID="Equ2"> <EquationSource Format="TEX">$\begin{aligned} \mathscr {C}(A;\mathscr {M}):={\{T\in \mathscr {B}(H): TAx=ATx,\; \text {for all}\; x\in \mathscr {M}}\}. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi mathvariant="script">C</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>;</mo> <mi mathvariant="script">M</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>T</mi> <mi>A</mi> <mi>x</mi> <mo>=</mo> <mi>A</mi> <mi>T</mi> <mi>x</mi> <mo>,</mo> <mspace width="0.277778em" /> <mtext>for all</mtext> <mspace width="0.277778em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="script">M</mi> </mrow> <mo stretchy="false">}</mo> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>The subspace <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathscr {M}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation> is called ultrainvariant subspace for A if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr {M}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation> is invariant for every <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T\in \mathscr {C}(A;\mathscr {M})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>;</mo> <mi mathvariant="script">M</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Every ultrainvariant subspace is necessarily hyperinvariant, though the converse does not always hold. In this article, first, we discuss some important properties of ultrainvariant subspaces of Hilbert space operators. Later, we describe ultrainvariant subspaces of unilateral right and left shift operators on the Hardy Hilbert space. Finally, we completely determine the ultrainvariant subspaces of an isometry on a Hilbert space.

On ultrainvariant subspaces of shift operators and isometries

期刊论文
发表日期：

2025年12月21日

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G. Ramesh,
T.K. Shabeeba

摘要

Let H be a separable, complex Hilbert space and A be a bounded linear operator on H. For a closed subspace $\mathscr {M}$ $M$ of H, the local commutant of A at $\mathscr {M}$ $M$ is defined by $\begin{aligned} \mathscr {C}(A;\mathscr {M}):={\{T\in \mathscr {B}(H): TAx=ATx,\; \text {for all}\; x\in \mathscr {M}}\}. \end{aligned}$ $\begin{matrix} C (A; M) : = {T \in B (H) : T A x = A T x, for all x \in M} . \end{matrix}$ The subspace $\mathscr {M}$ $M$ is called ultrainvariant subspace for A if $\mathscr {M}$ $M$ is invariant for every $T\in \mathscr {C}(A;\mathscr {M})$ $T \in C (A; M)$ . Every ultrainvariant subspace is necessarily hyperinvariant, though the converse does not always hold. In this article, first, we discuss some important properties of ultrainvariant subspaces of Hilbert space operators. Later, we describe ultrainvariant subspaces of unilateral right and left shift operators on the Hardy Hilbert space. Finally, we completely determine the ultrainvariant subspaces of an isometry on a Hilbert space.