The Banach Algebra B(x), Where x is a bk Space and Applications

Bruno de_Malafosse

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