Taiwanese Journal of Mathematics


Sen-Yen Shaw and Hsiang Liu

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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.

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Taiwanese J. Math., Volume 13, Number 2A (2009), 535-544.

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 47D09: Operator sine and cosine functions and higher-order Cauchy problems [See also 34G10] 47D62: Integrated semigroups 45D05: Volterra integral equations [See also 34A12] 45N05: Abstract integral equations, integral equations in abstract spaces

$(a,k)$-regularized resolvent family cosine operator function partial orbit weak continuity weak compactness


Shaw, Sen-Yen; Liu, Hsiang. CONTINUITY OF RESTRICTIONS OF $(a, k)$-REGULARIZED RESOLVENT FAMILIES TO INVARIANT SUBSPACES. Taiwanese J. Math. 13 (2009), no. 2A, 535--544. doi:10.11650/twjm/1500405354. https://projecteuclid.org/euclid.twjm/1500405354

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