Abstract: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 $0\not =\lambda _{n}\in \rho (A)$ 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.
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
AMS Subject Classification: Primary 47A35; Secondary 47A05, 47D06, 47D09