Proyecciones Vol. 25, No 3, pp. 271-291, December 2006. Universidad Cat´olica del Norte Antofagasta - Chile
ON OPERATOR IDEALS DEFINED BY A REFLEXIVE ORLICZ SEQUENCE SPACE
J. LOPEZ´ UNIVERSIDAD POLITECNICA´ DE VALENCIA, ESPANA˜ M. RIVERA UNIVERSIDAD POLITECNICA´ DE VALENCIA, ESPANA˜ G. LOAIZA UNIVERSIDAD EAFIT, COLOMBIA
Received : March 2006. Accepted : September 2006
Abstract Classical theory of tensornorms and operator ideals studies mainly those defined by means of sequence spaces p. Considering Orlicz se- quence spaces as natural generalization of p spaces, in a previous paper [12] an Orlicz sequence space was used to define a tensornorm, and characterize minimal and maximal operator ideals associated, by using local techniques. Now, in this paper we give a new characteri- zation of the maximal operator ideal to continue our analysis of some coincidences among such operator ideals. Finally we prove some new metric properties of tensornorm mentioned above.
Key words : Maximal operator ideals. Ultraproducts of spaces, Orlicz spaces. AMS Mathematics Subject Classification : Primary 46M05, 46A32. 272 G.Loaiza,J.L´opez Molina and M. Rivera
1. Introduction
Using ideas from p spaces, Saphar introduced a tensornorm gp (see [18]) and given the great relationship between tensornorms and operator ideals, minimal and maximal operator ideals associated to the tensornorm in the sense of Defant and Floret were studied in the classical theory of tensornorm and operator ideals.
Since Orlicz spaces M are natural generalizations of p spaces, in [12] we c introduced a tensornorm gM by means of an Orlicz space M and using some aspects of local theory we characterized the minimal and maximal operator c ideals associated to gM . Now in this paper our aim is to give another characterization of maximal operator ideal associated to this tensornorm to study the coincidence between components of the two operator ideals c which in turn enables us to prove some metric properties of gM and its dual. Notation is standard. We will always consider Banach spaces over the real field, since we shall use results in the theory of Banach lattices. The canonical inclusion map from Banach space E into the bidual E00 will be denoted by JE.IngeneralifE is a subspace of F , the inclusion of E into F is denoted by IE,F .Thesetoffinite dimensional subspaces of a normed space E will be denoted by FIN(E). We recall the more relevant aspects on Banach lattices (we refer the reader to [1]). A Banach lattice E is order complete or Dedekind complete if every order bounded set in E has a least upper bound in E, and it is order continuous if every order convergent filter is norm convergent. Every dual Banach sequence lattice E0 is order complete and all reflexive spaces are even order continuous. A linear map T between Banach lattices E and F is said to be positive if T (x) 0inF for every x E,x 0.T is called order bounded if T (A) is order≥ bounded in F for every∈ order≥ bounded set A in E. Let ω be the vector space of all scalar sequences and ϕ its subspace of the sequences with finitely many non zero coordinates. A sequence space λ is a linear subspace of ω containing ϕ provided with a topology finer than the topology of coordinatewise convergence. A Banach sequence space will be a sequence space λ provided with a norm which makes it a Banach lattice and an ideal in ω, i.e. such that if x y with x ω and y λ, | | ≤ | | ∈ ∈ then x λ and x λ y λ. A sectional subspace Sk(λ),k N, is the ∈ k k ≤ k k ∈ topological subspace of λ of those sequences (αi) such that αi =0forevery i k. Clearly Sk(λ) is 1-complemented in λ. A Banach sequence space ≥ On operator ideals defined by a reflexive Orlicz sequence space 273
λ will be called regular whenever the sequence ei i∞=1 where ei := (δij)j (Kronecker’s delta) is a Schauder base in λ. { } We now discuss Orlicz spaces. A non degenerated Orlicz function M is a continuous, non decreasing and convex function defined in R+ such that M(t) = 0 if and only if t =0andlimt M(t)= .TheOrlicz →∞ ∞ sequence space M is the space of all sequences a =(ai)i∞=1 such that ∞ M( ai /c) < ,forsomec>0. The functional i=1 | | ∞ P ∞ ΠM (a)=inf c>0: M( ai /c) 1 { i=1 | | ≤ } X is a norm in M and ( M , ΠM (.)) is a Banach space. We say that an Orlicz function M has the ∆2 property at zero if M(2t)/M (t) is bounded in a neighbourhood of zero. In general M is not regular, but this is the case if and only if M satisfies the ∆2 property at zero. We say that the function M ∗ is the complementary of M if M ∗(u):= max ut M(t):0 ∞ a M =sup anbn : ΠM ((bn)) 1 k k {n=1 ≤ } X if a =(ai)i∞=1 which is equivalent to ΠM (.). Then if M has the ∆2-property at zero, ( M , ΠH (.)) =( H , . M ) as isometric spaces. Moreover M is 0 ∗ k k ∗ reflexive if and only if M and M ∗ have the ∆2-property at zero. For more information on Orlicz functions and Orlicz spaces we refer to [13]. All Orlicz spaces in this paper are considered regular and reflexive. Moreover we will always suppose that M(1) = 1. Let (Ω, Σ,µ)beameasurespace,wedenotebyL0(µ)thespaceof equivalence classes, modulo equality µ-almost everywhere, of µ-measurable real-valued functions, endowed with the topology of local convergence in measure. And the space of all equivalence classes of µ-measurable X-valued functions is denoted by L0(µ, X). By a K¨othe function space (µ)on K (Ω, Σ,µ), we shall mean an order dense ideal of L0(µ), which is equipped with a norm . (µ) that makes it a Banach lattice(if f L0(µ)and k kK ∈ g (µ) f g ,thenf (µ)with f (µ) g (µ)). Similar- ∈ K | | ≤ | | ∈ K k kK ≤ k kK ity, (µ, X)= f L0(µ, X): f(.) X (µ) , endowed with the norm K { ∈ k k ∈ K } f (µ,X) = f(.) X (µ). k kOnK theorykk of operatork kK ideals and tensor norms we refer to the books [16] and [4] of Pietsch and Defant and Floret respectively. 274 G.Loaiza,J.L´opez Molina and M. Rivera If E and F are Banach spaces and α is a tensor norm, then E α F represents the space E F endowed with the α-normed topology.⊗ The ⊗ completion of E α F is denoted by E ˆ αF ,andthenormofz in E ˆ αF by α(z; E F ).⊗ If there is no risk of confusion⊗ we write α(z) instead⊗ of α(z; E F⊗). ⊗ c 2. The tensor norm gM and M-nuclear operators associated to an Orlicz function M First we establish some notation. Given a Banach space E and an Or- licz function M with M(1)=1 such that M is reflexive, we say that a N sequence (xn) E is strongly M-summing if ( xn ) M and we n∞=1 ∈ k k ∈ write πM ((xi)) := ΠH(( xn )) and it is said to be weakly M-summing if ε ((x )) : = sup (k x k,x ) .Wedenoteby [E](resp. (E)) M i x0 1 n 0 M M M the space of all stronglyk k≤ k |h (resp.i| weakly)k M-summing in E with the norm πM (.)(resp.εM (.)). The more natural approach to define a tensornorm in analogy to Saphar’s tensornorm is as follows. Let E and F be Banach spaces and z E F , we define ∈ ⊗ gM (z; E F ):=infπM ((xn)) εM ((yn)) ⊗ ∗ m taking the infimum over all representations of z as xn yn.Wewill n=1 ⊗ write gM (z) instead of gM (z; E F ) if there is not possibility of confusion. ⊗ P It is possible that for some M the functional gM does not satisfy the triangle inequality, but it is always a reasonable quasi norm on E F ,see ˆ ⊗ [3] and [6]. We denote E gM F the corresponding quasi Banach space. ⊗ c To have a tensornorm gM in [12] we took the Minkowski functional, denoted gc (z; E F ), of the absolutely convex hull of the unit closed ball M ⊗ BgM := z E F/gM (z) 1 of the quasi norm gM in E F ,such that { ∈ ⊗ ≤ } ⊗ n c gM (z; E F ):=inf πM ((xij)) εM ∗ ((yij)) ⊗ i=1 X n m taking the infimum over all representations of z as i=1 j=1 xij yij. Again, we will write g (z) instead of g (z; E F ) if there is not possibility⊗ M M P P of confusion. ⊗ c It is easy to see that gM is a tensor norm on the class of all Banach spaces (using criterion 12.2 in [4] and bearing in mind that πM (ei)= c ei M =1foreveryi N), and that z E Fg(z; E F ) k k ∗ ∈ ∀ ∈ ⊗ M ⊗ ≤ gM (z; E F ). We denote E ˆ gc F the corresponding Banach space. ⊗ ⊗ M On operator ideals defined by a reflexive Orlicz sequence space 275 Proceeding as in [3] and [18], it is easy to see that if z E ˆ g F ,there ∈ ⊗ M are (xi) M [E]and(yi) H (F ) such that πM ((xi)) εM ((yi)) < i∞=1 ∈ i∞=1 ∈ ∗ ∗ and z = ∞ xi yi. Moreover the quasi norm of z in E ˆ g F (again ∞ i=1 ⊗ ⊗ M denoted by gM (z)) is given by gM (z)=infπM ((xi)) εM ∗ ((yi)) taking the P m infimum over all such representations of z as xn yn. Similarly, if n=1 ⊗ z E ˆ gc F then z can be represented as z = ∞ ∞ xij yij where ∈ ⊗ M P i=1 j=1 ⊗ (xij) M [E]foreachj N,(yij) M (F )foreachj N and i∞=1 ∈ ∈ i∞=1 ∈ P∗ P ∈ π ((x )) ε ((y )) < . Moreover, the norm of z in E ˆ c F is j∞=1 M ij M ∗ ij gM gc (z)=inf π ((x )) ε ∞((y )) taking the infimum over all⊗ repre- PM j∞=1 M ij M ∗ ij sentations of z as ∞ ∞ xij yij. P i=1 j=1 ⊗ The topology defined by the quasi norm gM on E F is normable with P c P ⊗ norm equivalent to gM . In fact, being M areflexive Orlicz space and fol- lowing the arguments of proposition 16 of [3], we consider the bilinear onto ˆ map R : M [E] M ∗ (F ) E gM F such that R((xi), (yi)) = i∞=1 xi yi. R is continuous× with quasi→ norm⊗ less or equal one. Then there exists⊗ Pˆ a unique linear and continuous map M [E] π M∗ (F ) E gM F (see [20]). This map can be extended to a continuous⊗ linear→ and⊗ onto map M [E] ˆ π M (F ) E ˆ g F which is open by the open mapping theo- ⊗ ∗ → ⊗ M rem. Then E ˆ g F is isomorphic to a quotient of a Banach space and ⊗ M so it is a Banach space itself. In this way there is anormwM (.; E F ) ⊗ equivalent to the quasi norm gM (.; E F ) furthermore it is easy to see ⊗ c that wM (.; E F ), gM (.; E F )andgM (.; E F )areequivalentwith c ⊗ ⊗ ⊗ gM (.; E F ) wM (.; E F ). Given the last equivalence, gM seems appro- ⊗ ≤ ⊗ c priate for our purposes, but we need gM for our main results. To introduce M-nuclear operators, bearing in mind that every represen- tation of z E ˆ c F as x y defines a map T (E,F) 0 gM i∞=1 j∞=1 ij ij z such that ∈x ⊗E, T (x):= ⊗ x ,x y .Furthermore,∈ L T z P P i∞=1 j∞=1 ij0 ij z is well defined∀ ∈ and independent of the chosenh representationi for z.Let P P ΦEF : E0 ˆ gc F (E,F)bedefined by ΦEF (z):=Tz. ⊗ M → L Definition 1. An operator between Banach spaces T : E F is said to → be M- nuclear if T = ΦEF (z),forsomez E0 ˆ gc F . ∈ ⊗ M Given any pair of Banach spaces E and F , the space of the M-nuclear c operators T : E F endowed with the topology of the norm NM (T ):= c → inf gM (z) / ΦEF (z)=T or with the equivalent quasi-norm NM (T ):= { } c inf gM (z) / ΦEF (z)=T is denoted by H(E,F). Also ( M (E,F), NM ) { } N N c denotes a component of the minimal Banach operator ideal ( M , NM ) c N associated to the tensor norm gM . Analogously as in [12] we obtain the following result. 276 G.Loaiza,J.L´opez Molina and M. Rivera Theorem 2. Let E,F be any pair of Banach spaces and an operator T (E,F). Then the following are equivalent: ∈ L 1) T is M-nuclear. 2) T factors continuously in the following way: T E - F 6 A C ? - M ∞ B where B is a diagonal multiplication operator defined by a positive sequence (bi) M . ∈ Furthermore NM (T )=inf C B A ,infimum taken over all such factors. {k kk kk k} 3) T factors continuously in the following way: T E - F 6 A C ? - [ ] 1[ M ] ∞ ∞ B where B is a diagonal multiplication operator defined by a positive sequence (bi) 1[ M ]. ∈ c Furthermore NM (T )=inf C B A ,infimum taken over all such factors. {k kk kk k} c Associated to gM and gM , there are other important operator ideal. Definition 3. Let T (E,F),wesaythatT is M-absolutely summing ∈ L if exist a real number C>0, such that for all sequences (xi) in E,with εM ((xi)) < ,itsatisfies that ∞ (2.1) (T (xi)) M CεM ((xi)) k k ≤ On operator ideals defined by a reflexive Orlicz sequence space 277 For M (E,F) we denote the Banach ideal of the M-absolutely summing P operators T : E F endowed with the topology of the norm PM (T ):= inf C, taking the→ infimum over all C that satisfies (2.1) 0 Theorem 4. Let E and F be Banach spaces. E gc F = M (F, E0) ⊗ M P ∗ isometrically. ³ ´ 3. M-integral operators The characterization of maximal operator ideal obtained in [12] was given in terms of theory of finite representability of Banach spaces and/or Banach Lattices. In present paper, we give another characterization of such ideals by considering the structure of finite dimensional subspaces of Orlicz spaces involved. The behavior of the Orlicz sequences spaces under ultraproducts is also crucial. On ultraproducts of Banach spaces we refer to [8]. We only set the notation we will use. Let D be a non empty index set and a non-trivial U ultrafilter in D.Givenafamily Xd,d D of Banach spaces, (Xd) { ∈ } U denotes the corresponding ultraproduct Banach space. If every Xd,d D, coincides with a fixed Banach space X the corresponding ultraproduct∈ is namedanultrapowerofX and is denoted by (X) . Recall that if every U Xd,d D is a Banach lattice, (Xd) has a canonical order which makes it a ∈ U Banach lattice. If we have another family of Banach spaces Yd,d D and { ∈ } a family of operators Td (Xd,Yd),d D such that supd D Td < , { ∈ L ∈ } ∈ k k ∞ then (Td) ((Xd) , (Yd) ) denotes the canonical ultraproduct operator. U ∈ L U U We now give a local definition which has been inspired in Gordon and Lewis definition of local unconditional structure. Definition 5. Given a sequence space λ, we say that a Banach space X has an Sk(λ)-local unconditional structure if there exists a real constant c>0 such that for every finite dimensional subspace F of X,thereisa section Sn(λ)ofλ and linear operators u : F Sn(λ)andv : Sn(λ) X → → such that u v c and vu= IF,X. k kk k ≤ The constant c which appears in above definition is called a Sk(λ)-local unconditional structure constant of X andinthiscasewesaythatX has c- Sk(λ)-local unconditional structure. If a Banach space X has c-Sk(λ)-local + unconditional structure for every c>CwesaythatithasC -Sk(λ)-local unconditional structure. 278 G.Loaiza,J.L´opez Molina and M. Rivera The following definition was introduced by Pelczy`nnski and Rosenthal [15] in 1975. Definition 6. ABanachspaceX has the uniform projection property if there is a b>0 such that for each natural number n there is a natural number m(n) such that for every n-dimensional subspace M X there exists a k-dimensional and b-complemented subspace Z of X containing⊂ M with k m(n). ≤ The constant b of the above definition is called a uniform projection property constant of X, and in this case we say that X has the b-uniform projection property. If X has the b-uniform projection property for every b>Bwe say that X has the B+-uniform projection property. We need to remark the following aspects involving Orlicz spaces. The class of Banach spaces with the uniform projection property is quite large and includes, for example the reflexive Orlicz spaces, see [14]. In particular they have the 1 + ε-uniform projection property for every ε>0. Furthermore If 1 p then, the Bochner space Lp(µ, E)and ≤ ≤∞ p(E)hastheb-uniform projection property if E does, see [8]. We highlight that the uniform projection property is stable under ultrapowers, see [8]. Moreover from [17]. Proposition 7. If M is reflexive, then every ultrapower of 1[ M ](of + + M )has1 -Sr( 1)[Sk( M )]-local unconditional structure (resp. 1 -Sk( M )- local unconditional structure). According to the general theory of tensor norms and operator ideals, the normed ideal of M-integral operators ( M , IM ) is the maximal operator c I ideal associated to the tensor norm gM in the sense of Defant and Floret [4], or in an equivalent way, the maximal normed operator ideal associated to the normed ideal of M-nuclear operators in the sense of Pietsch [16]. From [4], for every pair of Banach spaces E and F ,anoperatorT : E F → is M-integral if and only if JF T (E (gc ) F 0)0. ∈ ⊗ λ 0 For every pair of Banach spaces E,F we define the finitely generated tensor norm g such that if M FIN(E)and N FIN(F ), for every M0 ∈ ∈ z M N, gM0 (z; M N):=sup z,w /gM (w; M 0 N 0) 1 .Clearly ∈ ⊗c ⊗ {|h i| ⊗ ≤ } g0 =(g )0 since the unit ball in M 0 gc N 0 is the convex hull of the unit M M ⊗ M ball of M N .ButweremarkthatE c F (and no E F )is 0 gM 0 0 gM 0 0 gM 0 an isometric⊗ subspace of (E F ) because⊗ gc is finitely generated,⊗ see gM0 0 M [4], 15.3. ⊗ In this case we define IM (T ) to be the norm of JF T considered as an element of the topological dual of the Banach space E g F 0. Remark that ⊗ λ0 On operator ideals defined by a reflexive Orlicz sequence space 279 IM (T )=IM (JF T ) as a consequence of F 0 be canonically complemented in F 000. FirstwegiveanontrivialexampleofM-integral operators. Theorem 8. Let(Ω, Σ,µ) a measure space and let M be a reflexive Orlicz sequence space. Then every order bounded operator S : L (µ) ∞ → M and every order bounded operator S : L (µ) 1[ M ]areM-integral ∞ → with IM (S)= S . k k Proof. We will only give the proof if S : L (µ) M is an order bounded operator since the proof in the other case∞ is similar.→ The predual space of M is M , which is regular space because M has ∗ ∗ the ∆2 property at zero. Then, the linear span of the set ei,i N is T { ∈ } dense in M and by the representation theorem of maximal operator ideals (see 17.5 in∗ [4]) and the density lemma (theorem 13.4 in [4]) we only have to see that S (L (µ) ) . gM0 0 Given z ∈L (∞µ) ⊗ Tand ε>0, let X and Y be finite dimensional gM0 subspaces of∈L ∞(µ)and⊗ Trespectively such that z X Y and ∞ T ∈ ⊗ (3.1) gM0 (z; X Y ) gM0 (z; L (µ) )+ε. ⊗ ≤ ∞ ⊗ T m Let gs s=1 be a basis for Y and let k N be such that 1 s { }k ∈ ∀ ≤ ≤ m gs = csiei.Then f X, 1 s m i=1 ∀ ∈ ∀ ≤ ≤ P k S, f gs = f,S0(gs) = f, csi S0(ei) = h ⊗ i h i * Ãi=1 ! + X k k f csj ej, S0(ei) ei . * ⊗ j=1 i=1 ⊗ + X X k Then if U denotes the tensor U := i=1 S0(ei) ei L (µ)0 λ,by bilinearity we get z X Y z,S = U, z . ⊗ ∈ ∞ ⊗ ∀ ∈ ⊗ h i Ph i Given ν>0, for every 1 i k there is fi L (µ) such that fi 1 ≤ ≤ ∈ ∞ k k ≤ and S0(ei) S0(ei),fi + ν. Then f := sup1 i k fi lies in the closed k k ≤ |h i| ≤ ≤ unit ball of L (µ). On the other hand, H is a dual lattice and hence it is order complete.∞ By the Riesz-Kantorovich theorem (see theorem 1.13 in [1] for instance), the modulus S of the operator S exists in (L (µ), M ). | | L ∞ By the lattice properties of M we have k k πM ((S0(ei)) = πM S0(ei) ei πM S0(ei),fi ei Ãi=1 k k ! ≤ Ãi=1 |h i| ! X X 280 G.Loaiza,J.L´opez Molina and M. Rivera k k k +νπM ei πM S(fi), ei ei + νπM ei Ãi=1 ! ≤≤ Ãi=1 |h i| ! Ãi=1 ! X X X k k k πM S(fi) , ei + νπM ei πM S ( fi ), ei ei ≤ Ãi=1h| | i! Ãi=1 ! ≤ Ãi=1h| | | | i ! X X X k k k +νπM ei πM S ( f ), ei ei + νπM ei = Ãi=1 ! ≤ Ãi=1h| | | | i ! Ãi=1 ! X X X k k = πM ( S ( f )) + νπM ei S + νπM ei . | | | | Ãi=1 ! ≤ k| |k Ãi=1 ! X X k Moreover εM ((ei) ) 1. Hence, denoting by IX and IY the corre- ∗ i=1 ≤ sponding inclusion maps into L (µ)and H respectively, we have ∞ S, z = U, z = U, ((IX)0 (IY )0)(z) |h i| |h i| |h ⊗ i| ≤ c g (U; X Y ) g0 (((IX )0 (IY )0)(z); X0 Y 0) ≤ M ⊗ M ⊗ ⊗ ≤ gM (U; X Y ) g0 (((IX )0 (IY )0)(z); X0 Y 0) ≤ ⊗ M ⊗ ⊗ ≤ (gM (U; L ( M )) + ε)gM0 (z; L (µ) M ) ≤ ∞ ⊗ ∞ ⊗ ∗ ≤ gM0 (z; L (µ) M πM ((S0(ei)) εM ((ei)) + ε ≤ ∞ ⊗ ∗ ∗ ≤ ¡ k ¢ gM0 (z; L (µ) M ) S + νπM ei + ε ∞ ∗ ≤ ⊗ Ãk| |k Ãi=1 ! ! X and ν being arbitrary S, z gM0 (z; L (µ) M )( S +ε). Finally, |h i| ≤ ∞ ⊗ ∗ k| |k by since ε is arbitrary we get S, z gM0 (z; L (µ) M S .But |h i| ≤ ∞ ⊗ ∗ k| |k from [1] theorem 1.10, S (χΩ)=sup S(f) , f χΩ and as M is order continuous | | {| | | | ≤ } S = S (χΩ) =sup S(f) , f 1 = S . k| |k k| | k {k | |k k k ≤ } k k Then S is M integral with IM (S) S . But as ( M , IM )isaBanach − ≤ k k I operators ideal, S IM (S), hence IM (S)= S . k k ≤ k k On operator ideals defined by a reflexive Orlicz sequence space 281 Corollary 9. Let (Ω, Σ,µ)beameasurespaceandn, k N.Then ∈ every operator T : L (µ) Sk( M ) and every operator T : L (µ) ∞ → ∞ → Sn( 1)[Sk(M)] satisfy that IM (T )= T . k k Proof. The result follows easily from theorem 3, since every operator T : L (µ) Sk( M )(T : L (µ) Sn( 1)[Sk( M )] in the other case) is ∞ → ∞ → order bounded and Sk( M )(resp. Sn( 1)[Sk( M )]) is reflexive hence order continuous. For our next theorem we need a very deep technical result of Linden- strauss and Tzafriri [14] which gives us a kind of ”uniform approximation” of finite dimensional subspaces by finite dimensional sublattices in Banach lattices. Lemma 10. Let ε>0andn N be fixed. There is a natural number h(n, ε) such that for every Banach∈ lattice X and every subspace F X ⊂ of dimension dim(F )=n there are h(n, ε) disjoints elements zi, 1 i { ≤ ≤ h(n, ε) andanoperatorA from F into the linear span G of zi, 1 i h(n, ε)} such that { ≤ ≤ } x F A(x) x ε x . ∀ ∈ k − k ≤ k k Theorem 11. Let M be a regular Orlicz sequence space, G an abstract M-space, and X a Banach space with c-Sk( M )orc-Sk( 1)[Sn( M )]-local uniform structure . Then every operator T : G X is M-integral and −→ IM (T ) c T . ≤ k k Proof. We will prove the case where X has c-Sk( M )-local uncon- ditional structure since the other case is similar. By the representation theorem of maximal operator ideals (see 17.5 in [4]), we only need to show that JX T (G g X0)0. ∈ ⊗ M0 Given z G X0 and ε>0, let P G and Q X0 be finite dimensional ∈ ⊗ n ⊂ ⊂ subspaces and let z = i=1 fi xi0 be a fixed representation of z with fi P and x Q, i =1, 2,..,n such⊗ that ∈ i0 ∈ P g0 (z; G X0) g0 (z; P Q) g0 (z; G X0)+ε. M ⊗ ≤ M ⊗ ≤ M ⊗ Fromlemma10wehaveafinite dimensional sublattice P1 of G and an operator A : P P1 so that f P, A(f) f ε f . Then, if idG denotes the identity→ map on G∀ we∈ have k − k ≤ k k n n n JX T,z = T (fi),xi0 T (idG A)(fi),xi0 + TA(fi),xi0 |h i| ¯ h i¯ ≤ ¯ h − i¯ ¯ h i¯ ¯i=1 ¯ ¯i=1 ¯ ¯i=1 ¯ ¯X ¯ ¯X ¯ ¯X ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 282 G.Loaiza,J.L´opez Molina and M. Rivera n n ε T fi xi0 + TA(fi),xi0 . ≤ k k k kk k ¯ h i¯ i=1 ¯i=1 ¯ X ¯X ¯ ¯ ¯ Let X1 := T (P1). As X has Sk( M )-local¯ unconditional¯ structure, hence there are k N, u : X1 Sk( M )andv : Sk( M ) X such that ∈ → → IX ,X = vuand u v c.LetX2 := vu(X1)whichisafinite 1 k kkk ≤ dimensional subspace of X containing X1 and IX1,X2 = vu.PutK2 : X X = X /X be the canonical quotient map. Then 000 −→ 20 000 2◦ n n T (A(fi)),xi0 = IX1,X2 T (A(fi)),K2(xi0 ) = i=1h i i=1h i X X n vuT(A(fi)),K2(xi0 ) = i=1h i X n n = uT (A(fi)),v0 K2(xi0 ) = uT, A(fi) v0 K2(xi0 ) i=1h i h i=1 ⊗ i X X n with A(fi) v K2(x ) P1 (Sk( M )) and uT : P1 Sk( M ). i=1 ⊗ 0 i0 ∈ ⊗ 0 → Since P1 is a reflexive abstract M-space it is lattice isometric to some L (µ) P ∞ space, hence by corollary 9 this map is M-integral with IM (uT) u T . Then ≤ k kk k n n T (A(fi)),xi0 = uT, A(fi) v0 K2(xi0 ) ¯ h i¯ ¯* ⊗ +¯ ≤ ¯i=1 ¯ ¯ i=1 ¯ ¯X ¯ ¯ X ¯ ¯ n¯ ¯ ¯ ¯ ¯ ¯ ¯ IM (uT) gM0 ( A(fi) v0 K2(xi0 ); P1 Sk( M )) ≤ i=1 ⊗ ⊗ ≤ X u T g0 (A v0 K2)(z); P1 Sk( M ) ≤ k kk k M ⊗ ⊗ ≤ u T ¡A v0 K2 g0 (z; P Q) ¢ ≤ k kk kk kk kk k M ⊗ ≤ (1 + ε) c T g0 (z; P Q) (1 + ε) c T (g0 (z; G X0)+ε) ≤ k k M ⊗ ≤ k k M ⊗ and since ε is arbitrary we obtain JX T,z c T g (z; G X ) |h i| ≤ k k M0 ⊗ 0 Concerning to characterization theorem of M-integral operators we have: Theorem 12. Let M be a regular Orlicz sequence space and let E and F be Banach spaces. The following statements are equivalent: 1) T M (E,F). ∈ I 2) JF T factors continuously in the following way: On operator ideals defined by a reflexive Orlicz sequence space 283 JF T - E F 00 6 A C ? L (µ) - X ∞ B where X is an ultrapower of 1[ M ]andB is a lattice homomorphism. Fur- thermore IM (T )isequivalenttoinf D B A ,takingitoverallsuch factors. {k kk kk k} Proof. 1) = 2). Let D := (P, Q):P FIN(E),Q FIN(F 0) where FIN(Y )isthesetof⇒ finite dimensional{ ∈ subspace of a Banach∈ space} Y , endowed with the natural inclusion order (P1,Q1) (P2,Q2) P1 P2,Q1 Q2. ≤ ⇐⇒ ⊂ ⊂ For every (P0,Q0) D, R(P0,Q0):= (P, Q) D :(P0,Q0) (P, Q) and = R(P, Q), (P, Q∈) D . is filter{ basis in∈D,andaccordingtoZorn’s⊂ } R { ∈ } R lemma, let be an ultrafilter on D containing .Ifd D, Pd and Qd D R ∈ denote the finite dimensional subspaces of E and F 0 respectively so that d =(Pd,Qd). For every d D,ifz Pd Qd, JF T P Q (Pd g Qd)0 = ∈ ∈ ⊗ | d⊗ d ∈ ⊗ M0 M g Q = M (Pd,Q ). Then from theorem 2 of characterization of d0 ⊗ M d0 N d0 M-nuclear operators, JF T Pd Qd factors as | ⊗ JF T Pd Qd | ⊗ - Pd Qd0 6 Ad Cd ? - [ ] 1[ M ] ∞ ∞ Bd c where Bd is a positive diagonal operator and Ad Bd Cd NM (T P Q )+ k kk kk k ≤ | d⊗ d ε = IM (T Pd Qd )+ε.Then | ⊗ Ad Bd Cd IM (T Pd Qd )+ε IM (T )+ε k kk kk k ≤ | ⊗ ≤ Without loss of generality we can suppose that Ad = Cd =1. We k k k k define WE : E (Md) such that WE(x)=(xd) so that xd = x if x Md → D D ∈ 284 G.Loaiza,J.L´opez Molina and M. Rivera and xd =0ifx/Md.Inthesamewaywedefine WF : F 0 (Qd) such ∈ 0 → D that WF 0 (a)=(ad) so that ad = a if a Qd and ad =0ifa/Qd.Then we have the followingD commutative diagram:∈ ∈ JF T - E F 00 6 WE W F0 0 ? - - (Pd) (Qd0 ) ((Qd) )0 D (J T ) D I D F Pd Qd 6 | ⊗ D (Ad) (Cd) D D ? - ( [ ]) ( 1[ M ]) ∞ ∞ D (Bd) D D where I is the canonical inclusion map. As in [14] (( 1[ M ]) )00 is a 1- D complemented subspace of some ultrapower (( 1[ M ]) ) which from [19] D U is another ultrapower ( 1[ M ]) 1 with projection Q, the result follows with U A =(Ad) , B =((Bd) )00 which is a lattice homomorphism, D D C = PF (WF0 I (Cd) )00 Q,wherePF is the projection of F 0000 in F 00, 0000 0 D 0000 and X =( 1[λ]) 1 ,havinginmindthatas( [ ]) is an abstract M- U ∞ ∞ D space, there is a measure space such that L (µ)=(( [ ]) )00,where equality means that the spaces are lattice isometric.∞ ∞ ∞ D 2) = 1) As ( M , IM ) is a operator ideal, it follows easily from theorem 3 and proposition⇒ I 3 The following new formulation of the preceding characterization theo- rem is needed in our context: Theorem 13. Let M be an Orlicz space. For every pair of Banach spaces E and F , the following statements are equivalent: 1) T M (E,F). ∈ I 2) There exists a σ-finite measure space ( , ,ν)andaK¨othe function O S space (ν) which is complemented in a space with Sk( 1)[Sn( M )]-local K unconditional structure, such that JF T factors continuously in the following way: On operator ideals defined by a reflexive Orlicz sequence space 285 JF T - E F 00 6 A C ? L (ν) - (ν) ∞ B K where B is a multiplication operator for a positive function of (ν). Fur- K thermore IM (T )=inf C B A , taking the infimum over all such factors. {k kk kk k} Proof. Starting from the theorem 12, as 1[ M ]hasfinite cotype, it is order continuous ([7], 4.6), and for [13], theorem 1.a.9 1[ M ]canbede- composed into an unconditional direct sum of a family of mutually disjoint ideals Xh,h H having a positive weak unit, and then from 1.b.14 in { ∈ } [13], as every Xh is order isometric to a K¨othe space of functions defined on a probability space ( h, h,νh), then ( 1[ M ]) is order isometric to a K¨othe function space O(ν1)overameasurespace(S U 1, 1,ν1), hence we K 1 O S can substitute ( 1[ M ]) for (ν ) in 12. If we denote z := B(χΩ)with U K z = i∞=1 yhi with yhi Xhi for every i N,thenB(L (µ)) is contained in the unconditional direct∈ sum of X ∈,i N which∞ is is order isomet- P hi ric to a space of K¨othe function space{ (ν∈)overa} σ-finite measure space ( , ,ν) which is 1-complemented in K(ν1). O S K Now, since (ν) is order complete, there exists g := sup B(f)in f L (µ) K k k ∞ (ν). Then the operators B1 : L (µ) L (ν)andB2 : L (ν) (ν), K ∞ → ∞ ∞ → K such that B1(f)(ω):=B(f)(ω)/g(ω), for all f L (µ), ω with ∈ ∞ ∈ O g(ω) =0andB1(f)(ω)=0otherwise,andB2(h)(ω):=g(ω)h(ω)forall 6 h L (ν), ω , satisfy that B = B2B1 and B2 is a multiplication operator∈ ∞ for a positive∈ O element g (ν). ∈ K 4. On equality between M-nuclear and M-integral operators Finally, using the preceding characterization theorems we give some prop- erties of M-nuclear and M-integral operators. Let us establish now a nec- essary condition for equality between components of M-nuclear and M- integral operator ideals. First, we introduce a new operator ideal, which is contained in the ideal of the M-integral operators. 286 G.Loaiza,J.L´opez Molina and M. Rivera Definition 14. Given E and F Banach spaces, let M be a Orlicz sequence. We say that T (E,F)isstrictly M-integral if exist a σ-finite measure space ( , ∈,νL)andaK¨othe function space (ν)whichis O S K complemented in a space with Sk( 1)[Sn( M )]-local unconditional structure, such that T factors continuously in the following way: T E - F 6 A C ? L (ν) - (ν) ∞ B K where B is a multiplication operator for a positive function of (ν). en- K dowed with the topology of the norm SIM (T )=IM (T ). Obviously, if F is a dual space, or it is complemented in its bidual space, then M (E,F)= M (E,F). SI I Theorem 15. Let H be a Orlicz sequence space, and let E and F be Banach spaces, such that E0 satisfies the Radon-Nikod´ym property then, M (E,F)= M (E,F). N SI Proof. Let T λc (E,F)WereE0 has the Radon-Nikod´ym property and. ∈ SI a) First, we suppose that B is an multiplication operator for a function g (ν)withfinite measure support D.WedenoteνD the restriction of ν ∈toKD. As (χDA):E L (νD), then (χDA)0 :(L (νD))0 E0 and the → ∞ ∞ → restriction of (χDA)0 : L1(νD) E0,thus,foreveryx E and L1(νD) → ∈ f L1(νD) ∈ x, (χDA)0 (f) = χDA (x) ,f = χDA (x) fd(νD). h i D Z ® As E0 has the Radon-Nikod´ym property, by III(5) of [2], we have that (χDA) has a Riesz representation, therefore exist a function φ 0 ∈ L (νD,E0) such that for every f L1(νD) ∞ ∈ (χDA)0 (f)= fφd(νD). ZD On operator ideals defined by a reflexive Orlicz sequence space 287 Then, for every x E,wehavethatχDA(x)(t)= φ(t),x , νD-almost ev- ∈ h i erywhere in D, and then B(χDA)(x)= mk x0 ,x S1(x)= h kj iχ k x Akj j=1 kj0 X k k 2 2 mk and let Sk : L (ν) (ν) be such that Sk(f)= j=1 xkj0 fχAkj . 1 ∞ → K 2 k2 1k Then Sk 1and Sk Sk (ν,E ) and Sk = Sk Sk.Butas (ν) k k ≤ k k ≤ k kK 0 P K is a complemented subspace of space with Sk( 1)(Sn( M ))-local uncondi- 2 2 tional structure, from 11, there is K>0 such that IM (Sk) K Sk c 2 2 ≤ c k k ≤ K Sk (ν,E ),henceNM (Sk) K Sk Sk (ν,E ),henceNM (Sk) k kK 0 ≤ k k ≤ k kK 0 ≤ K Sk (ν,E ). k kK 0 Then, as (Sk)k∞=1 converges in the (νD,E0)space,itisaCauchyse- quenceK in M (E, (νD)), and since this is complete, (Sk) converges to gφ,thatis N K k∞=1 to say, gφ M (E, (νD)). Therefore, gφ = BχDA is M-nuclear and so T is also M∈-nuclear.N K b) Now, if g is any element of (ν), g it can be approximated in norm by K means of a sequence (tn)n∞=1 of simple functions with finite measure support, and therefore by a), the sequence Tn = CBtn A is a Cauchy sequence in M (E,F)convergingtoT in (E,F), and then T M (E,F). N L ∈ N As consequence of the former result and of the factorization theorems c c 13 and 2, we obtain the following metric properties of gM and (gM )0. c Theorem 16. (gM )0 is a totally accessible tensor norm. c Proof. Since (gM )0 is finitely generated, it is sufficient to prove that the map F (gc ) E M (E0,F00), is a isometric. ⊗ M 0 → P ∗ n li Let z = yij xij F c E,andletHz (E ,F ) i=1 j=1 (gM )0 M ∗ 0 00 be the canonical map associated⊗ ∈ to z,thatistosay,⊗ ∈ P P P 288 G.Loaiza,J.L´opez Molina and M. Rivera n li Hz (x )= xij,x yij for all x E ,conHz (E ,F) 0 i=1 j=1 h 0i 0 ∈ 0 ∈ L 0 ⊂ (E0,F00). L P P c Applying the theorem 15.5 of [4] for α =(gλ)0, the theorem 4, and the c c c equality (gM )00 = gM since gM finitely generated, we have that inclusion F E F c E 0 E ,F c 0 gM 0 M ∗ 0 00 ⊗←(g−−M−)0 → ⊗ →P ³ ´ ¡ ¢ is an isometry, and therefore by proposition 12.4 in [4] we obtain c c PM (Hz)=(←g−−−)0 (z; F E) (g )0 (z; F E) . ∗ M ⊗ ≤ M ⊗ Now, given N,afinite dimensional subspace of F such that z N (gc ) ∈ ⊗ M 0 E, there exists V (N c E) = (N,E ) such that I (V ) 1and (g )0 0 0 M c ∈ ⊗ M I ≤ (g ) (z; N E)= z,V . Clearly enough V M (N,E )= M (N,E ) M 0 ⊗ h i ∈ SI 0 I 0 because E0 is a dual space, and N 0,beingfinite dimensional, has the Radon- Nikod´ym property. Therefore by theorem 15, V M (N,E0)andby theorem 2, given >0, there is a factorization V in∈ theN way T - N E0 6 A C ? - [ ] 1[ M ] ∞ ∞ B c c such that C B A NM (V )+ = IM (V )+ 1+ . As [k kk] haskk thek ≤ extension metric property,≤ (to see proposition 1, C.3.2. in∞ [16]),∞ A can be extended to a continuous map A (F, [ ]) ∈ L ∞ ∞ such that A = A . By theorem 2 again, W := CBA is in M (F, E ), k k N 0 so there is° a° representation w =: ∞ ∞ y x F gc E of W ° ° i=1 j=1 ij0 ij0 0 M 0 verifying ° ° ⊗ ∈ ⊗ P P b ∞ c πM y0 εM x0 N (W )+ C B A + 1+2 . ij ∗ ij ≤ M ≤ k kk k ≤ i=1 ³³ ´´ ³³ ´´ ° ° X ° ° Then, (gc ) (z; F E) (gc ) (z; N E)= z,V = z,W° ° it follows that M 0 ⊗ ≤ M 0 ⊗ h i h i c c (g )0(z; F E) g (w)PM (Hz) (1 + 2 )PM (Hz) M ⊗ ≤ M ∗ ≤ ∗ c whence (g ) (z; F E) PM (Hz), and the equality is obvious. M 0 ⊗ ≤ ∗ On operator ideals defined by a reflexive Orlicz sequence space 289 Finally, as consequence of the former theorem and of proposition 15.6 of [4], we have: c Corollary 17. gM is an accessible tensor norm. References [1] Aliprantis, C. D., Burkinshaw, O.: Positive operators. Pure and Ap- plied Mathematics 119. Academic Press, Newe York, (1985). 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Preprint L L L [18] Saphar, P.: Produits tensoriels topologiques et classes d’applications line´aires. Studia Math. 38, pp. 71-100, (1972). [19] Sims, B.: ”Ultra”-techniques in Banach space theory, Queen’s Papers in Pure and Applied Mathematics, 60. Ontario, (1982). [20] Tom´a˘sek, S: Projectively generated topologies on tensor products, Co- mentations Math. Univ. Carolinae, 11, 4 (1970). J. A. L´opez Molina Universidad Polit´ecnica de Valencia E.T.S. Ingenieros Agr´onomos Camino de Vera 46072 Valencia Spain e-mail : [email protected] M. J. Rivera Universidad Polit´ecnica de Valencia E. T. S. Ingenieros Agr´onomos Camino de Vera 46072 Valencia Spain e-mail : [email protected], On operator ideals defined by a reflexive Orlicz sequence space 291 G. Loaiza Universidad EAFIT Departamento de Ciencias B´asicas Carrera 49 n- 7 sur-50 Medell´in. Colombia e-mail: gloaiza@eafit.edu.co