It. Joural of Math. Aalysis, Vol. 6, 2012, o. 1, 19-27 Equivalet Baach Operator Ideal Norms 1 Musudi Sammy Chuka Uiversity College P.O. Box 109-60400, Keya sammusudi@yahoo.com Shem Aywa Maside Muliro Uiversity P.O. Box 190-50100, Keya shemaywa2000@yahoo.com Ja Fourie School of Computer, Statistical ad Mathematical Scieces North-West Uiversity (Potchefstroom Campus) Private bag X6001, Potchefstroom 2520, South Africa Ja.Fourie@wu.ac.za Abstract Let X, Y be Baach spaces ad cosider the -topology (the dual weak operator topology) o the space (L(X,Y ),. ) of bouded liear operators from X ito X with the uiform operator orm. L w (X, Y ) is the space of all T L(X, Y ) for which there exists a sequece of compact liear operators (T) K(X, Y ) such that T = lim T. 1 Fiacial support from the Natioal Coucil for Sciece ad Techology (NCST) is greatly ackowledged.
20 Musudi Sammy, Shem Aywa ad Ja Fourie Two equivalet orms, T := if{ sup T : T K(X, Y ),T w T } ad T u := if{ sup {max{ T, T 2T }} : : T K(X, Y ),T w T } o L w (X, Y ), are cosidered. We show that (L w,. ) ad (L w,. u ) are Baach operator ideals. Mathematics Subject Classificatio: 47B10; 46B10; 46A25 Keywords: -topology, compact liear operators, Baach operator ideals 1. Itroductio Throughout this paper X ad Y will be Baach spaces. The space of bouded liear operators from X to Y is deoted by L(X, Y ) ad the subspaces cosistig of all fiite rak bouded liear operators, all compact liear operators ad all weakly compact liear operators, are deoted by F (X, Y ), K(X, Y ) ad W (X, Y ), respectively. The closed uit ball of a Baach space X is deoted by B X ad the cotiuous dual space of X is deoted by X. We follow the authors of the paper [3], callig a subspace X of a Baach space Y a ideal i Y if the aihilator X of X is the kerel of a cotractive projectio P o the (cotiuous) dual space Y of Y, whose rage is isomorphic to X. Sice, by Hah Baach Theorem such a projectio has orm 1, it follows that id Y 2P 1. Moreover, if the projectio P exists o Y such that ker P = X ad id Y 2P 1 (i.e id Y 2P = 1 i this case), the X is called a u-ideal (or ucoditioal ideal) iy. This cocept was itroduced by Casazza ad Kalto (cf [2]) ad the equality id Y 2P = 1 is equivalet to requirig that if ξ X,φ V := P (Y ), the φ + ξ = φ ξ. The atural examples of u-ideals (with respect to their biduals ) are order
Equivalet Baach operator ideal orms 21 cotiuous Baach lattices-although there are may examples of u-ideals which are ot Baach lattices. I a subsequet paper, the authors of [3] further ivestigated u-ideals alog with so called h-ideals, i which case it is required that φ + ξ = φ + λξ for all ξ X,φ V ad all λ = 1. Much of the paper [3] is devoted to a geeral study of u-ideal ad h-ideals. However, i sectio 8 of that paper, the authors fid ecessary coditios o a Baach space X such that the space K(X) of compact operators is a u-ideal i the space L(X) of bouded liear operators, showig that this is the case if X is separable ad has (UKAP) (ucoditioal compact approximatio property, lim i.e. if there exists a sequece (K )ik(x) such that K x = x for all x X ad lim id X 2K = 1). Johso proved i [7] that if Y is a Baach space havig the bouded approximatio property the the aihilator K(X, Y ) i the (cotiuous) dual space L(X, Y ) is the kerel of a projectio o L(X, Y ). The rage space of the projectio is isomorphic to the dual space K(X, Y ). K. Joh showed i [5] that Johso s result is also true i case of ay separable Pisier space X = P ad its dual Y = P, both beig spaces which do ot have the approximatio property. This motivated his more geeral results i a later paper,(cf [6]). Followig Kalto [8] we deote by the dual weak operator topology o L(X, Y ) which is defied by the liear fuctioals T e (T f ), f Y,e X. Although the weak topology of L(X, Y ) is i geeral stroger tha, it is show by Kalto i [8] that -compact subsets of K(X, Y ) are weakly compact. I particular, If (T ) K(X, Y ) is a -coverget sequece which coverges to a T K(X, Y ), the T T i the weak topology of L(X, Y ).
22 Musudi Sammy, Shem Aywa ad Ja Fourie This result was used by K. Joh (i [6]) to show that if for each T L(X, Y ) there exists a sequece T K(X, Y ) such that T T i the dual weak operator topology, the the aihilator K(X, Y ) i L(X, Y ) is the kerel of a projectio o L(X, Y ). I the paper [1] a alterative (operator ideal) approach is followed to prove similar (ad more geeral) versios of Joh s results. I this paper we build o the results i [1] to show that (L w,. ) ad (L w,. u ) are Baach operator ideals. 2. Operator ideal properties. Defiitio 2.1: Let T L(X, Y ). T is said to have the -compact approximatio property ( -cap) if there is a sequece (T ) K(X, Y ) such that T T }. Let Lw (X, Y ) be the family of all T L(X, Y ) which have the -compact approximatio property. A easy applicatio of the Uiform Boudedess Theorem shows that Lemma 2.2: If T T i the -topology of L(X, Y ) the (T ) is orm bouded. Let X, Y be fixed Baach spaces. For T L w (X, Y ) we put ( ) T := if{ sup T : T K(X, Y ),T w T }. Clearly, if T K(X, Y ), the T = T. Refer to [9] ad [4] for iformatio i coectio with operator ideals. I particular we recall the followig criteria for a subclass of the operator ideal (L,. ) to be a complete operator ideal o the family of all Baach spaces. Theorem 2.3: (cf. [9], 6.2.3, pp.91) Let U be a subclass of L with a R + -valued fuctio α such that the followig coditios are satisfied: (i) If X, Y are Baach spaces, the a y U(X, Y ) for all a X,y Y ad α(a y) = a y. (ii) RST U(X, Y ) ad α(rst ) R α(s) T wheever T L(X, X 0 ),S U(X 0,Y 0 ) ad R L(Y 0,Y).
Equivalet Baach operator ideal orms 23 (iii) If S 1,S 2,... U(X, Y ) ad α(s i) <, the S = S i =. lim S i U(X, Y ). Adα( S i) α(s i). The (U, α) is a complete ormed operator ideal. This importat result is istrumetal i provig that (L w,. ) is a Baach operator ideal. This fact is proved i [2]. Both for the sake of completeess ad later referece, we discuss the proof here. Theorem 2.4: ([1], Theorem 2.4) Let L w deote the assigmet which associates with each pair of Baach spaces X, Y the vector space L w (X, Y ). Ad let. be the assigmet that associates with every pair of Baach spaces X, Y ad with every operator S belogig to L w (X, Y ) the real umber S i ( ). The (L w,. ) is a Baach operator ideal. Proof: Notice that.. o L w (X, Y ), where. is the uiform operator orm o L(X, Y ). I fact for ay ɛ>0, let x 1, y 1such that T ɛ y (Tx) = lim y (T x) sup T where (T ) K(X, Y ) such that T T. Clearly T T +ɛ. To prove that (Lw,. ) is a complete ormed ideal we make use of Theorem 2.3: (i) I K = 1 where I K L w (K) is the idetity map o the 1-dimesioal Baach space K. (ii) Let T L(X, X 0 ), S L w (X, Y 0 ) ad R L(Y 0,Y). The if S S, S K(X, Y ) arbitrary, the RS T w RST. Hece RST sup RS T R ( sup S ) T. Sice (S ) was arbitrary chose, it is clear that RST R S T. (iii) Now suppose that (T ) L w (X, Y ) with T <. We have to show that T i =. lim T i exists ad is i L w (X, Y ) with
24 Musudi Sammy, Shem Aywa ad Ja Fourie T i w T i : Let T,i K(X, Y ) such that T,i T i, sup T,i T i + ɛ. For arbitrary x 1, y 1 2 i we have x (T,i y ) T i + ɛ, i ad. 2 i Hece x (T,i y ) coverges uiformly i N, thus showig that ( ) x (T i y )= lim x (T i y ). It follows from the completeess of (L(X, Y ),. ) ad (K(X, Y ),. ) ad the iequalities T i T i for all i ad T,i T i + ɛ for all i, that 2 i T i L(X, Y ) ad T,i K(X, Y ) for all. Sice ( ) holds for arbitrary x B X ad y B Y, it follows that T,i Hece T i is i L w (X, Y ) ad T i sup T,i sup T,i ɛ + T i. T i. This shows that T i T i. By theorem 2.3, (L w,. ) isa Baach ideal of operators. Defiitio 2.5: Let T L w (X, Y ) ad suppose (T ) K(X, Y ) coverges i the dual weak operator topology of T. We deote by K u ((T )) the umber give by K u ((T )) := sup N {max{ T, T 2T }}, which is a fiite umber because of the Uiform Boudeddess Theorem. The u-orm o L w (X, Y ) is the give by T u := if{k u ((T )) : T = lim T, T K(X, Y )}. It is clear from the defiitio that T T u for all T L w (X, Y ). Also, if T K(X, Y ) the we may put T = T for all, i which case K u ((T )) = T,
Equivalet Baach operator ideal orms 25 showig that T u T ; therefore we have T = T u = T for all T K(X, Y ). Theorem 2.6: (L w,. u ) is a Baach operator ideal. Proof: (i) It is clear that... u o L w (X, Y ) for all Baach spaces X, Y ad that the idetity map o ay 1-dimesioal Baach space has u-orm 1. (ii) For T L(X, X 0 ),R L(Y 0,Y), S L w (X 0,Y 0 ) ad (S ) K(X 0,Y 0 ) such that S S, we have RST u K u ((RS T )) R T sup {max{ S, S 2S }} R T K u ((S )). The sequece S K(X 0,Y 0 ) beig arbitrarily chose to satisfy S follows that RST u R S u T. S,it (iii) Now suppose that (T ) L w (X, Y ) with T u <. Sice this implies that T u < ad (L w,. ) is a Baach operator ideal, it follows that T L w (X, Y ). We still have to prove that T u T u. To do so, we choose for arbitrary ɛ>0ad each fixed i N, a sequece (T,i ) K(X, Y ) such that T,i T i if ad K u ((T,i )) T i u + ɛ 2. i As i the proof of Theorem 2.4 it follows that T i = lim T,i L w (X, Y ).
26 Musudi Sammy, Shem Aywa ad Ja Fourie Therefore, we have = sup sup T i u K u (( T,i ) ) { { max T,i, }} T i 2 T,i { { max T,i, }} T i 2 T,i K u ((T,i )) T i u + ɛ, which proves that T i u T i u. We coclude that (L w,. u ) is a Baach operator ideal. Corollary 2.7: The orms. ad. u are equivalet o L w (X, Y ) for all Baach spaces X, Y. Refereces [1] A. Shem ad J.H Fourie, Spaces of compact operators ad their dual spaces, Redicoti del Circolo Materatico di Palermo. Serie II Tomo LIII(2004), 205-224. [2] J. Casazza ad J. Kalto, Approximatio properties o separable Baach spaces, i : Geometry of Baach spaces, Lodo Math. Soc., 158(1990), 49-63. [3] G. Godefroy, N.J. Kalto ad P.D. Saphar, Ucoditioal ideals i Baach spaces, Studia Mathematica, 104(1) (1993), 13-59. [4] H. Jarchow Locally covex spaces, B. G. Teuber Stuttgart, 1981. [5] K. Joh O the space K(P, P ) of compact operators o Pisier space P, Note di Mathematica, XII:1992, 69-75.
Equivalet Baach operator ideal orms 27 [6] K. Joh O a result of J. Johso. Czech. Math. J. 45(120) 1995, 235-240. [7] J. Johso Remarks o Baach spaces of compact operators, J. Fuctioal Aalysis, textbf32(1979), 302-311. [8] N.J. Kalto Spaces of compact operators, Math. A., 208(1974), 267-278. [9] A. Pietsch Operator ideals, North-Hollad, 1980. Received: July, 2011