Equivalent Banach Operator Ideal Norms1

Abstract

Let X, Y be Banach spaces and consider the w′-topology (the dualweak operator topology) on the space (L(X, Y),‖.‖) of bounded linearoperators from X into X with the uniform operator norm.Lw′(X,Y) is the space of all T∈L(X, Y) for which there exists a sequence ofcompact linear operators (Tn)⊂K (X, Y) such thatT=w′−limnTn Two equivalent norms,

‖|T‖|:=inf{█(sup@n)┤‖Tn‖:Tn∈K(X,Y),Tnw′→T}and

‖T‖u:=inf{█(sup@n)┤ {max{‖Tn‖,‖T−2Tn‖}}:‖:Tn∈K(X,Y),Tnw→T}on Lw′(X, Y), are considered. We

show that (Lw′,|‖.‖|) and (Lw′,‖.‖u) are Banach operator ideals.