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Keywords:
reflexive Banach space; $L_1$-space of vector-valued functions; closed operator; resolvent set; range and domain; linear contraction; $C_0$-semigroup; strongly continuous cosine family of operators
reflexive Banach space; $L_1$-space of vector-valued functions; closed operator; resolvent set; range and domain; linear contraction; $C_0$-semigroup; strongly continuous cosine family of operators
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.
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