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The main concern of this note is the Moore-Penrose inverse in the context ofBanach spaces and algebras. Especially attention will be given to a particularclass of elements with the aforementioned inverse, namely EP Banach spaceoperators and Banach algebra elements, which will be studied and characterizedextending well-known results obtained in the frame of Hilbert space operatorsand $C^*$-algebra elements.

On the Moore-Penrose inverse, EP Banach space operators and EP Banach algebra elements