On the commutant and orbits of conjugation

In this note, we analyse the relationship between the commutant of a bounded linear operator A and the algebra of similarity BA that was introduced in the late 70s as a characterization of nest algebras. Necessary and sufficient conditions are also obtained for an operator to commute with real scalar generalized operators in the sense of Colojoara-Foias in Banach spaces. In the second part, we analyse the relationship between the generalized inverse, the generalized commutant and the orbits of conjugation.