On some operator equivalences in Hilbert spaces

This paper is on some classes of operators which happen to be invariant under unitary and metric equivalences. In particular, we show that the classes of n-θ Operators, α-supraposinormal operators and S- operators are both isometrically and coisometrically and hence unitarily invariant while the class of supraposinormal is invariant under both unitary and metric equivalences.