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The Banach algebra \(B(X\)), where \(X\) is a BK space and applications. (English) Zbl 1112.46004

Let \(A=(a_{nk})_{n,k=1}^{\infty}\) be an infinite matrix of complex numbers and \(x=(x_k)_{k=1}^{\infty}\), \(b=(b_{n})_{n=1}^{\infty}\) be sequences of complex numbers. The solvability of infinite systems of linear equations \(A_nx=\sum_{k=1}a_{nk}x_k=b_n\) \((n=1,2,\dots)\) is studied in sequence spaces closely related to the spaces of bounded, convergent and null sequences, and of sequences that are absolutely \(p\)-summable for \(1\leq p<\infty\). Furthermore, the special cases are considered when the matrix of the infinite system is a Toeplitz triangle. As an application, the inverse of an infinite tridiagonal matrix in the spaces of bounded, convergent and null sequences is explicitly given. The results are obtained by combining the theories of BK spaces, matrix transformations, and Banach algebras. The results are new and interesting.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
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