A characterization of compact operators on $\ell^p$-spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913884690972672 |
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| author | Abtahi, Mortaza |
| author_facet | Abtahi, Mortaza |
| contents | Let $A$ be a Banach space, $p>1$, and $1/p+1/q=1$. If a sequence $a=(a_i)$ in $A$ has a finite $p$-sum, then the operator $Λ_a:\ell^q\to A$, defined by $Λ_a(β)=\sum_{i=1}^\infty β_i a_i, β=(β_i)\in \ell^q$, is compact. We present a characterization of compact operators $Λ:\ell^q\to A$, and prove that $Λ$ is compact if and only if $Λ=Λ_a$, for some sequence $a=(a_i)$ in $A$ with $\{(ϕ(a_i)): ϕ\in A^*, \|ϕ\|\leq 1\}$ being a totally bounded set in $\ell^p$. For a sequence $(T_i)$ of bounded operators on a Hilbert space $H$, the corresponding operator $T:\ell^q\to B(H)$, defined by $T(β) = \sum_{i=1}^\infty β_i T_i$, is compact if and only if the set $\{\langle T x,x\rangle:\|x\|=1\}$ is a totally bounded subset of $\ell^p$, where $\langle T x,x\rangle = (\langle T_1 x,x\rangle, \langle T_2 x,x\rangel, \dotsc)$, for $x\in H$. Similar results are established for $p=1$ and $p=\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06726 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A characterization of compact operators on $\ell^p$-spaces Abtahi, Mortaza Functional Analysis Primary 46B45, 47B37, Secondary 47A12, 47A30 Let $A$ be a Banach space, $p>1$, and $1/p+1/q=1$. If a sequence $a=(a_i)$ in $A$ has a finite $p$-sum, then the operator $Λ_a:\ell^q\to A$, defined by $Λ_a(β)=\sum_{i=1}^\infty β_i a_i, β=(β_i)\in \ell^q$, is compact. We present a characterization of compact operators $Λ:\ell^q\to A$, and prove that $Λ$ is compact if and only if $Λ=Λ_a$, for some sequence $a=(a_i)$ in $A$ with $\{(ϕ(a_i)): ϕ\in A^*, \|ϕ\|\leq 1\}$ being a totally bounded set in $\ell^p$. For a sequence $(T_i)$ of bounded operators on a Hilbert space $H$, the corresponding operator $T:\ell^q\to B(H)$, defined by $T(β) = \sum_{i=1}^\infty β_i T_i$, is compact if and only if the set $\{\langle T x,x\rangle:\|x\|=1\}$ is a totally bounded subset of $\ell^p$, where $\langle T x,x\rangle = (\langle T_1 x,x\rangle, \langle T_2 x,x\rangel, \dotsc)$, for $x\in H$. Similar results are established for $p=1$ and $p=\infty$. |
| title | A characterization of compact operators on $\ell^p$-spaces |
| topic | Functional Analysis Primary 46B45, 47B37, Secondary 47A12, 47A30 |
| url | https://arxiv.org/abs/2506.06726 |