Spectral analysis of difference and differential operators in weighted spaces
Sbornik. Mathematics, Tome 204 (2013) no. 11, pp. 1549-1564
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This paper is concerned with describing the spectrum of the difference operator
$$
\mathscr{K}\colon l_\alpha^p(\mathbb Z,X)\to l_\alpha^p(\mathbb Z,X),\quad
(\mathscr{K}x)(n)=Bx(n-1), \ \ n\in\mathbb{Z}, \ \ x\in l_\alpha^p(\mathbb Z,X),
$$
with a constant operator coefficient $B$, which is a bounded linear operator in a Banach space $X$.
It is assumed that $\mathscr{K}$
acts in the weighted space $l_\alpha^p(\mathbb Z,X)$,
$1\leq p\leq \infty$, of two-sided sequences of vectors from $X$.
The main results are obtained in terms of
the spectrum $\sigma(B)$ of the operator coefficient $B$
and properties of the weight function.
Applications to the study of the spectrum of a differential operator
with an unbounded operator coefficient (the generator of a strongly continuous semigroup of operators)
in weighted function spaces are given.
Bibliography: 23 titles.
Keywords:
difference operator, differential operator, spectrum of an operator, weighted spaces of sequences and functions.
@article{SM_2013_204_11_a0,
author = {M. S. Bichegkuev},
title = {Spectral analysis of difference and differential operators in weighted spaces},
journal = {Sbornik. Mathematics},
pages = {1549--1564},
publisher = {mathdoc},
volume = {204},
number = {11},
year = {2013},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2013_204_11_a0/}
}
M. S. Bichegkuev. Spectral analysis of difference and differential operators in weighted spaces. Sbornik. Mathematics, Tome 204 (2013) no. 11, pp. 1549-1564. http://geodesic.mathdoc.fr/item/SM_2013_204_11_a0/