in this note‎, ‎we aim to show that several known generalizations of frames are equivalent to the continuous frame‎ ‎defined by ali et al‎. ‎in 1993‎. ‎indeed‎, ‎it is shown that these generalizations can be considered as an operator between two hilbert spaces‎.

Here, T(^p) denote^ the trace class of completely continuous operators in § and p(Hp) the resolvent set of Hp. The perturbation theory of absolutely continuous (abbr. a.c.) parts of Hp as well as the theory of wave and scattering operators has recently been studied independently by de Branges [2], Birman and Kreïn [ l ] , and Kato [3], In [ l ] and [3] the problem was considered from the viewpo...

Let $T$ be a bounded linear operator on a Hilbert space $mathscr{H}$. We say that $T$ has the hyponormal property if there exists a function $f$, continuous on an appropriate set so that $f(|T|)geq f(|T^ast|)$. We investigate the properties of such operators considering certain classes of functions on which our definition is constructed. For such a function $f$ we introduce the $f$-Aluthge tran...