Some counterexamples concerning strong \(M\)-bases of Banach spaces (Q1340310)

Let \(X\) be a Banach space and let \((f_ n)_ 1^ \infty\) be an \(M\)- basis of \(X\). Strong \(M\)-bases or \(M\)-bases which are \(k\)-series summable \((k\in \mathbb{Z}^ +)\) are also considered. It is known that \((f_ n)^ \infty_ 1\) is a strong \(M\)-basis if and only if it is 1-series summable. The authors prove that on any Banach space with a Schauder basis there exists a strong \(M\)-basis which is not 2-series summable. A special case of this result is given by \textit{D. R. Larson} and \textit{W. R. Wogen} [J. Funct. Anal. 92, No. 2, 448-467 (1990; Zbl 0738.47045)]. The strong \(M\)- basis constructed in the proof is used to settle in the negative an open question in the theory of nonselfadjoint operator algebras. Finally, in Banach spaces \(c_ 0\) and \(c\) a strong \(M\)-basis \((f_ n)^ \infty_ 1\) is constructed whose biorthogonal sequence \((f^*_ n)^ \infty_ 1\) (with \(\bigvee^ \infty_{n=1} f^*_ n= X^*\)) fails to be a strong \(M\)-basis.