Fuglede-Putnam type commutativity theorems for $ EP $ operators

TL;DR

This paper investigates conditions under which Fuglede-Putnam type theorems hold for EP operators on Hilbert spaces, including versions involving Moore-Penrose inverses, and derives new results based on these theorems.

Contribution

It establishes new Fuglede-Putnam type theorems for EP operators, including variants with Moore-Penrose inverses, and presents novel results on EP operators using these theorems.

Findings

Fuglede-Putnam theorem does not hold universally for EP operators.

Under certain conditions, the theorem is valid for EP operators.

New results on EP operators are derived using these theorems.

Abstract

Fuglede-Putnam theorem is not true in general for $ EP $ operators on Hilbert spaces. We prove that under some conditions the theorem holds good. If the adjoint operation is replaced by Moore-Penrose inverse in the theorem, we get Fuglede-Putnam type theorem for $ EP $ operators -- however proofs are totally different. Finally, interesting results on $ EP $ operators have been proved using several versions of Fuglede-Putnam type theorems for $ EP $ operators on Hilbert spaces.