The Banach algebra $B(X)$, where $X$ is a BK space and applications

Abstract

In this paper we give some properties of Banach algebras of boundedoperators $B(X)$, when $X$ is a BK space. We thenstudy the solvability of the equation $Ax=b$ for $bın\{s_{\alpha},s_{\alpha}^{{{}^{\circ}}},s_{\alpha }^{( c)},l_{p}(\alpha)\}$ with $\alphaın U^{+}$ and $1\leq p<ınfty$. Wethen deal with the equation $T_{a}x=b$, where $bın\chi(\Delta^{k})$ for $k\geq 1$ integer, $\chiın\{s_{\alpha },s_{\alpha}^{{{}^{\circ}}},s_{\alpha}^{(c)},l_{p}(\alpha)\}$, $1\leq p<ınfty$ and $T_{a}$ is a Toeplitz triangle matrix.Finally we apply the previous results to infinite tridiagonal matrices andexplicitly calculate the inverse of an infinite tridiagonal matrix. Theseresults generalize those given in [4,~9].