Factorizations of weighted EP Banach space operators and Banach algebra elements. (English) Zbl 1334.46039

Let \(A\) be a complex unital Banach algebra. An element \(a\) of \(A\) is said to be positive if \(V(a)=\{ f(a) : f\in A^{\ast}, \| f \| \leq 1, f(1)=1 \} \subseteq \mathbb{R}^{+}\). Given an invertible element \(u\), we can consider the following norm on \(A\), \(\| x \|_{u} = \| u^{1/2} x u^{-1/2} \| \) for each \(x \in A\).
Let \(e\), \(f\) be invertible positive elements of \(A\). An element \(a\) is said to be weighted Moore-Penrose invertible with weights \(e\) and \(f\) if there exists a normalized generalized inverse \(b\) of \(a\) such that \(ab\) is a Hermitian element of \((A,\| \cdot \|_{e})\) and \(ba\) is a Hermitian element of \((A, \| \cdot \|_{f})\).
Finally, \(a \in A\) is said to be weighted EP with with weights \(e\) and \(f\) if \(a\) is weighted Moore-Penrose invertible with weights \(e\) and \(f\) and \(a\) commutes with its inverse.
In this paper, the authors characterize such elements using several factorizations.


46H05 General theory of topological algebras
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