Abstract
Let $X$ be a Banach space which is continuously embedded in another Banach space $Y$ and is an invariant subspace for an $(a,k)$-regularized resolvent family $R(\cdot)$ of operators on $Y$. It is shown that the restriction of $R(\cdot)$ to $X$ is strongly continuous with respect to the norm of $X$ if and only if all its partial orbits are relatively weakly compact in $X$. This property is shared by many particular cases of $(a,k)$-regularized resolvent families, such as integrated solution families, integrated semigroups, and integrated cosine functions.
Citation
Sen-Yen Shaw. Hsiang Liu. "CONTINUITY OF RESTRICTIONS OF $(a, k)$-REGULARIZED RESOLVENT FAMILIES TO INVARIANT SUBSPACES." Taiwanese J. Math. 13 (2A) 535 - 544, 2009. https://doi.org/10.11650/twjm/1500405354
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