| Title:
|
On the range of a closed operator in an $L_1$-space of vector-valued functions (English) |
| Author:
|
Sato, Ryotaro |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
2 |
| Year:
|
2005 |
| Pages:
|
349-367 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $X$ be a reflexive Banach space and $A$ be a closed operator in an $L_1$-space of $X$-valued functions. Then we characterize the range $R(A)$ of $A$ as follows. Let $0\neq \lambda_{n}\in \rho(A)$ for all $1\leq n < \infty$, where $\rho(A)$ denotes the resolvent set of $A$, and assume that $\lim_{n\rightarrow \infty} \lambda_{n}=0$ and $\sup_{n\geq 1} \|\lambda_{n}(\lambda_{n}-A)^{-1}\| < \infty$. Furthermore, assume that there exists $\lambda_{\infty}\in \rho(A)$ such that $\|\lambda_{\infty}(\lambda_{\infty}-A)^{-1}\|\leq 1$. Then $f\in R(A)$ is equivalent to $\sup_{n\geq 1} \|(\lambda_{n}-A)^{-1}f\|_{1}<\infty$. This generalizes Shaw's result for scalar-valued functions. (English) |
| Keyword:
|
reflexive Banach space |
| Keyword:
|
$L_1$-space of vector-valued functions |
| Keyword:
|
closed operator |
| Keyword:
|
resolvent set |
| Keyword:
|
range and domain |
| Keyword:
|
linear contraction |
| Keyword:
|
$C_0$-semigroup |
| Keyword:
|
strongly continuous cosine family of operators |
| MSC:
|
47A05 |
| MSC:
|
47A35 |
| MSC:
|
47B38 |
| MSC:
|
47D06 |
| MSC:
|
47D09 |
| idZBL:
|
Zbl 1123.47012 |
| idMR:
|
MR2176897 |
| . |
| Date available:
|
2009-05-05T16:51:29Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119529 |
| . |
| Reference:
|
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| . |
