## 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.

### MSC:

 46H05 General theory of topological algebras
Full Text:

### References:

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