Linear difference and differential operators with unbounded operator coefficients in weight spaces (Q845191)

Let \(X\) be a complex Banach algebra, \(B(X)\) the set of all bounded linear operators on \(X\), \(\alpha : \mathbb{Z}_+ \to (0, \infty)\) a weight function, \(\ell_{\alpha}^p\) the Banach space of sequences \(x : \mathbb{Z}_+ \to X\) from \(X\), summable with weight \(\alpha\), \(\|x\| = \|x\|_{p,\alpha} = (\sum_{n\geq 0}(\|x(n)\|/ \alpha (n))^p)^{1/p}\), \(p \in [1, \infty)\), and \(\|x\|_{\infty,\alpha} = \sup_{n\geq 0}(\|x(n)\|/\alpha (n)) < \infty\). It is also assumed that the weight \(\alpha\) satisfies the condition \(\sup_{n\geq 1}(\alpha(n - 1)/\alpha (n)) < \infty\). The author determines the spectra of the operators \(\mathcal{K}\) and \(\mathcal{D}\), where \((\mathcal{K}x)(0) = 0 \) and \((\mathcal{K}x)(n) = Bx(n-1)\) for \(n \geq 1\), for some \(B \in B(X)\), and \(\mathcal{D} := I - \mathcal{H}\). These results are then used to determine the spectrum of a certain differential operator.