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Some counterexamples concerning strong \(M\)-bases of Banach spaces. (English) Zbl 0820.46002

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 D. R. Larson and 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.

MSC:

46A35 Summability and bases in topological vector spaces
47L10 Algebras of operators on Banach spaces and other topological linear spaces

Citations:

Zbl 0738.47045
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