In this paper we study the Spectral Radius Algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {B}_{W}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">B</mi> <mi>W</mi> </msub> </math></EquationSource> </InlineEquation> associated with a bilateral weighted shift W and study its properties, and conditions under which <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {B}_{W}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">B</mi> <mi>W</mi> </msub> </math></EquationSource> </InlineEquation> can have non trivial invariant subspaces (n.i.s.). At first we consider the bilateral weighted shifts W on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\ell ^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> having positive weights <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\{\alpha _{n}\}_{n\in \mathbb {Z}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>α</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and show that W being quasinilpotent is a sufficient condition for having a n.i.s. for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {B}_{W}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">B</mi> <mi>W</mi> </msub> </math></EquationSource> </InlineEquation>. Next we consider bilateral operator weighted shift W on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {H}=\bigoplus _{i\in \mathbb {Z}}\mathcal {H}_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo>=</mo> <msub> <mo>⨁</mo> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </msub> <msub> <mi mathvariant="script">H</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> with operator weights <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\{A_n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$i,j\in \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>, we define <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {B}_W(i,j)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mi>W</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to be certain types of subspaces of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {L}(\mathcal {H}_j,\mathcal {H}_i)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">H</mi> <mi>j</mi> </msub> <mo>,</mo> <msub> <mi mathvariant="script">H</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Assuming that W is of finite multiplicity, if there exists <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$i\in \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {B}_W(i,i)\ne \mathcal {L}(\mathcal {H}_i,\mathcal {H}_i)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mi>W</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> <mo>≠</mo> <mi mathvariant="script">L</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">H</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi mathvariant="script">H</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> then <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {B}_W$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">B</mi> <mi>W</mi> </msub> </math></EquationSource> </InlineEquation> has n.i.s. in <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>; on the other hand if <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\mathcal {B}_W(i,i)=\mathcal {L}(\mathcal {H}_i,\mathcal {H}_i)\,\,\forall \,\,i\in \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mi>W</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="script">L</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">H</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi mathvariant="script">H</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mo>∀</mo> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mi>i</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> then <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\mathcal {B}_W$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">B</mi> <mi>W</mi> </msub> </math></EquationSource> </InlineEquation> has n.i.s. in <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> if and only if W is quasinilpotent. We also give an example of a bilateral weighted shift W for which the spectral radius algebra <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\mathcal {B}_{W}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">B</mi> <mi>W</mi> </msub> </math></EquationSource> </InlineEquation> and the associated Deddens algebra <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\mathcal {D}_{W}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">D</mi> <mi>W</mi> </msub> </math></EquationSource> </InlineEquation> are distinct. Additionally, we have discussed conditions under which <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\mathcal {B}_{W}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">B</mi> <mi>W</mi> </msub> </math></EquationSource> </InlineEquation> does not admit n.i.s. whereas <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\mathcal {D}_{W}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">D</mi> <mi>W</mi> </msub> </math></EquationSource> </InlineEquation> does.

Spectral Radius Algebras For Bilateral Weighted Shifts

期刊论文
发表日期：

2026年2月16日

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Munmun Hazarika,
Chinmoy Das

摘要

In this paper we study the Spectral Radius Algebra $\mathcal {B}_{W}$ $B_{W}$ associated with a bilateral weighted shift W and study its properties, and conditions under which $\mathcal {B}_{W}$ $B_{W}$ can have non trivial invariant subspaces (n.i.s.). At first we consider the bilateral weighted shifts W on $\ell ^{2}$ $ℓ^{2}$ having positive weights $\{\alpha _{n}\}_{n\in \mathbb {Z}}$ ${α_{n}}_{n \in Z}$ and show that W being quasinilpotent is a sufficient condition for having a n.i.s. for $\mathcal {B}_{W}$ $B_{W}$ . Next we consider bilateral operator weighted shift W on $\mathcal {H}=\bigoplus _{i\in \mathbb {Z}}\mathcal {H}_i$ $H = ⨁_{i \in Z} H_{i}$ with operator weights $\{A_n\}$ ${A_{n}}$ . For $i,j\in \mathbb {Z}$ $i, j \in Z$ , we define $\mathcal {B}_W(i,j)$ $B_{W} (i, j)$ to be certain types of subspaces of $\mathcal {L}(\mathcal {H}_j,\mathcal {H}_i)$ $L (H_{j}, H_{i})$ . Assuming that W is of finite multiplicity, if there exists $i\in \mathbb {Z}$ $i \in Z$ such that $\mathcal {B}_W(i,i)\ne \mathcal {L}(\mathcal {H}_i,\mathcal {H}_i)$ $B_{W} (i, i) \neq L (H_{i}, H_{i})$ then $\mathcal {B}_W$ $B_{W}$ has n.i.s. in $\mathcal {H}$ $H$ ; on the other hand if $\mathcal {B}_W(i,i)=\mathcal {L}(\mathcal {H}_i,\mathcal {H}_i)\,\,\forall \,\,i\in \mathbb {Z}$ $B_{W} (i, i) = L (H_{i}, H_{i}) \forall i \in Z$ then $\mathcal {B}_W$ $B_{W}$ has n.i.s. in $\mathcal {H}$ $H$ if and only if W is quasinilpotent. We also give an example of a bilateral weighted shift W for which the spectral radius algebra $\mathcal {B}_{W}$ $B_{W}$ and the associated Deddens algebra $\mathcal {D}_{W}$ $D_{W}$ are distinct. Additionally, we have discussed conditions under which $\mathcal {B}_{W}$ $B_{W}$ does not admit n.i.s. whereas $\mathcal {D}_{W}$ $D_{W}$ does.