On the range of a closed operator in an L 1 -space of vector-valued functions

Ryotaro Sato

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 2, page 349-367
  • ISSN: 0010-2628

Abstract

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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 ≠ λ n ∈ ρ ( A ) for all 1 ≤ n < ∞ , where ρ ( A ) denotes the resolvent set of A , and assume that lim n → ∞ λ n = 0 and sup n ≥ 1 ∥ λ n ( λ n - A ) - 1 ∥ < ∞ . Furthermore, assume that there exists λ ∞ ∈ ρ ( A ) such that ∥ λ ∞ ( λ ∞ - A ) - 1 ∥ ≤ 1 . Then f ∈ R ( A ) is equivalent to sup n ≥ 1 ∥ ( λ n - A ) - 1 f ∥ 1 < ∞ . This generalizes Shaw’s result for scalar-valued functions.

How to cite

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Sato, Ryotaro. "On the range of a closed operator in an $L_1$-space of vector-valued functions." Commentationes Mathematicae Universitatis Carolinae 46.2 (2005): 349-367. <http://eudml.org/doc/249534>.

@article{Sato2005,
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 $0\ne \lambda _\{n\}\in \rho (A)$ for all $1\le n < \infty $, where $\rho (A)$ denotes the resolvent set of $A$, and assume that $\lim _\{n\rightarrow \infty \} \lambda _\{n\}=0$ and $\sup _\{n\ge 1\} \Vert \lambda _\{n\}(\lambda _\{n\}-A)^\{-1\}\Vert < \infty $. Furthermore, assume that there exists $\lambda _\{\infty \}\in \rho (A)$ such that $\Vert \lambda _\{\infty \}(\lambda _\{\infty \}-A)^\{-1\}\Vert \le 1$. Then $f\in R(A)$ is equivalent to $\sup _\{n\ge 1\} \Vert (\lambda _\{n\}-A)^\{-1\}f\Vert _\{1\}<\infty $. This generalizes Shaw’s result for scalar-valued functions.},
author = {Sato, Ryotaro},
journal = {Commentationes Mathematicae Universitatis Carolinae},
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; -space of vector-valued functions; closed operator; resolvent set},
language = {eng},
number = {2},
pages = {349-367},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the range of a closed operator in an $L_1$-space of vector-valued functions},
url = {http://eudml.org/doc/249534},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Sato, Ryotaro
TI - On the range of a closed operator in an $L_1$-space of vector-valued functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 2
SP - 349
EP - 367
AB - 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 $0\ne \lambda _{n}\in \rho (A)$ for all $1\le n < \infty $, where $\rho (A)$ denotes the resolvent set of $A$, and assume that $\lim _{n\rightarrow \infty } \lambda _{n}=0$ and $\sup _{n\ge 1} \Vert \lambda _{n}(\lambda _{n}-A)^{-1}\Vert < \infty $. Furthermore, assume that there exists $\lambda _{\infty }\in \rho (A)$ such that $\Vert \lambda _{\infty }(\lambda _{\infty }-A)^{-1}\Vert \le 1$. Then $f\in R(A)$ is equivalent to $\sup _{n\ge 1} \Vert (\lambda _{n}-A)^{-1}f\Vert _{1}<\infty $. This generalizes Shaw’s result for scalar-valued functions.
LA - eng
KW - 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; -space of vector-valued functions; closed operator; resolvent set
UR - http://eudml.org/doc/249534
ER -

References

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