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 $\alpha ın{U}^{+}$ and $1\le p<ınfty$. Wethen deal with the equation $T_{a}x=b$, where $bın\chi ({\mathrm{\Delta }}^k)$ for $k\ge 1$ integer, $\chi ın\{s_{\alpha },s_{\alpha }^{{}^{\circ }},s_{\alpha }^{(c)},l_p(\alpha )\}$, $1\le p<ınfty$ 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].