Multipliers of sequence spaces (Q5906469)

Let \(A = (a_{nk})\) be a triangular nonnegative regular summability matrix. Denote by \(m_ A\) the linear space of sequences \(\{s_ n\}\) such that the \(A\)-transform is bounded. Let \((c_ 0)_ A\) be the closed subspace of sequences \(\{t_ n\}\) which are evaluated to 0 by \(A\). Denote by \(m_ A\) the Banach space \(\widetilde m_ A/(c_ 0)_ A\). In this paper the author studies the space \(M(A)\) of multipliers of the space \(m_ A\) into itself. For many matrices, the author shows that the condition \[ \text{if} \quad \mu \in M (A) \quad \text{and} \quad \text{GLB} | \mu | > 0, \quad \text{then} \quad 1/ \mu \in M (A), \tag{*} \] holds. If \((*)\) holds, the author gives some information regarding the maximal ideal space \(\Delta (A)\) of \(M(A)\). The paper ends with the following conjecture: If \(M(A)\) is semisimple and \((*)\) holds for the matrix \(A\), then \(\Delta (A)\) is totally disconnected.