ON CLASS (BD) OPERATORS OF ORDER (n+k)

Abstract

In this paper,  we introduce  the class (BD) of order (n+k) operators  acting  on the classical Hilbert  space H.  An operator TEB(H) is said to belong to the class (BD) of order (n+k) if T^*2(T^D)² commutes with (T^*(n+k)T^D)², that is, [T^*2(T^D)², (T^*(n+k)T^D)² ]=0.  We investigate the  properties  of this  class and  analyze  its relation  to the (n+k)-power D-operator.  This  study  explores  various  aspects  such as unitary equivalence, restriction to  closed subspaces,  and  the  behavior  of these  operators  under  complex conjugation.  In addition, we examine the connection  between  (BD) operators  of order (n+k) and D-operators of the same order.