Products of Idempotents in Banach Algebras of Operators
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912120457658368 |
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| author | Jain, Surender K. Leroy, André Singh, Ajit Iqbal |
| author_facet | Jain, Surender K. Leroy, André Singh, Ajit Iqbal |
| contents | Let $X$ be a Banach space and $\mathcal A$ be the Banach algebra $B(X)$ of bounded (i.e. continuous) linear transformations (to be called operators) on
$X$ to itself. Let $\mathcal E$ be the set of idempotents in $\mathcal A$ and $\mathcal S$ be the semigroup generated by $\mathcal E$ under composition as multiplication. If $T\in \mathcal S$ with $0\ne T\ne I_{X}$ then $T$ has a local block representation of the form
$\begin{pmatrix}
T_1 & T_2
0 & 0
\end{pmatrix}$ on $X=Y\oplus Z$, a topological sum of non-zero closed subspaces $Y$ and $Z$ of $X$, and any $A\in \mathcal A$ has the form $\begin{pmatrix}
A_1 & A_2
A_3 & A_4
\end{pmatrix}$
with $T_1,A_1 \in \mathcal B(Y)$, $T_2,A_2\in B(Z,Y), A_3\in B(Y,Z)$, and $A_4 \in \mathcal B(Z)$. The purpose of this paper is to study conditions for $T$ to be in $\mathcal{S}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10118 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Products of Idempotents in Banach Algebras of Operators Jain, Surender K. Leroy, André Singh, Ajit Iqbal Functional Analysis Rings and Algebras 16S50, 16U40, 16W80, 46B28, 47B01 Let $X$ be a Banach space and $\mathcal A$ be the Banach algebra $B(X)$ of bounded (i.e. continuous) linear transformations (to be called operators) on $X$ to itself. Let $\mathcal E$ be the set of idempotents in $\mathcal A$ and $\mathcal S$ be the semigroup generated by $\mathcal E$ under composition as multiplication. If $T\in \mathcal S$ with $0\ne T\ne I_{X}$ then $T$ has a local block representation of the form $\begin{pmatrix} T_1 & T_2 0 & 0 \end{pmatrix}$ on $X=Y\oplus Z$, a topological sum of non-zero closed subspaces $Y$ and $Z$ of $X$, and any $A\in \mathcal A$ has the form $\begin{pmatrix} A_1 & A_2 A_3 & A_4 \end{pmatrix}$ with $T_1,A_1 \in \mathcal B(Y)$, $T_2,A_2\in B(Z,Y), A_3\in B(Y,Z)$, and $A_4 \in \mathcal B(Z)$. The purpose of this paper is to study conditions for $T$ to be in $\mathcal{S}$. |
| title | Products of Idempotents in Banach Algebras of Operators |
| topic | Functional Analysis Rings and Algebras 16S50, 16U40, 16W80, 46B28, 47B01 |
| url | https://arxiv.org/abs/2411.10118 |