Spectral analysis of difference and differential operators in weighted spaces

M. S. Bichegkuev

Sbornik. Mathematics, Tome 204 (2013) no. 11, pp. 1549-1564

Résumé

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.

Détail

Keywords: difference operator, differential operator, spectrum of an operator, weighted spaces of sequences and functions.

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