REMARKS ON OPERATOR BMO SPACES (1) f(t) m I f 2 dt) 1/2 <, I T interval I I

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1 REMARKS ON OPERATOR MO SPACES OSCAR LASCO Abstract. Several spaces defined according to different formulations for operator-valued functions in MO are defined and studied.. ntroduction and notation Recall that a function f is said to belong to MO(T) if sup ( () f(t) m f 2 dt) /2 <, T interval where is an interval in T and m (f) stands for the averarage m f = f(t)dt. t is well known that there are many other equivalent characterizations of MO functions: First we can replace averaging over intervals by averaging respect to the Poisson kernel (see [Ga]), that is f MO(T) if and only if (2) sup ( f(t) P (f)(z) 2 P z (t)dt) /2 < z < T where P z (t) = z zt, t T and P (f) stands for the Poisson integral of f. Actually this is also equivalent to (3) sup P ( f 2 )(z) P (f)(z) 2 < z < Recall also that, due to John-Nirenberg s lemma, one can replace in () and (2) the L 2 -norm by the L p -norm for 0 <p<. Another possibility is to describe functions in MO by the Carleson condition: f MO(T) if and only if (f)(z) 2 ( z 2 ) is a Carleson measure on D, equivalently (4) sup ( w 2 ) f(w) 2 P z (w)da(w) < z < D where (f) stands for the gradient of f and da for the Lebesgue measure in the disc D (see [Ga, Z]). Of course, one of the first and main descriptions of MO is as the dual space of ReH (T), where ReH (T) stands for the space of functions f in L (T) such that the Hilbert transform Hf belong to L (T), endowed with the norm f H = f + Hf. There are other descriptions of functions in ReH (T) either in terms of maximal functions, as those functions f L (T) such that P f L (T), where P f(t) = Key words and phrases. operator MO, Carleson measures Mathematical Subjects Classifications. Primary 4230, 4235, Secondary 4735 The author gratefully acknowledges support by C..C.Y.T. MF

2 2 OSCAR LASCO sup 0<r< P r f(t) is the radial Poisson maximal function, or in terms of atomic decompositions, as those functions f in L (T) that can be decomposed as f = k N λ ka k,λ k C, where a k are atoms, and k N λ k <. Different proofs of the duality result (see [FS]) MO(T) =(ReH (T)), can be done using those formulations of MO and ReH. The reader is referred to [GR] for the general theory on Hardy spaces using real-variable techniques. There is a counterpart of Hardy spaces defined in terms of martingales (see [G]) and a particular and simpler case concerning dyadic martingales (see [Per]) which we will discuss here. Let D denote the collection of dyadic subintervals of the unit circle T, and let (h ), where h = (χ /2 + χ ), be the Haar basis of L 2 (T). f f L (T) and Dthen f denote the formal Haar coefficients f(t)h dt, and, as above, m f = f(t)dt denotes the average of f over. We say that f MO d (T), if (5) f MO d = sup( f(t) m f 2 dt) /2 <. Denote P (f) = J h Jf J and, using that (f m f)χ = P (f) one has f MO d (T) if and only if there exists a constant C>0such that, for all D, (6) P (f) 2 C /2. On the other hand, since P (f) L 2 =( J D,J f J 2 ) /2 for f L 2 (T). Hence f MO d (T) if and only if (7) sup J D,J f J 2 <. We shall use later on the following characterization of MO d in terms the boundedness of the paraproducts. t is well known that f MO d (T) if and only if (8) π g (f) 2 C f 2 and π g = g MO d norm (T), where π g (f) = g (m f)h (see [Per] for a survey on dyadic Harmonic Analysis and paraproducts). Throughout the paper we shall review some of the results on the vector-valued versions of the previously defined characterizations of MO and prove some new ones on operator dyadic MO. The paper is divided into three sections. The first one contains a survey of some results proved by the author about several vector valued versions of MO. The second section is devoted to operator-valued dyadic MO spaces. We concentrate on the connections with operator-valued Carleson measures and paraproducts. Several spaces associated to different formulations of the previous notions are introduced and studied. nclusions between them are analyzed. The third section contains new material. Some natural generalizations for functions taking values in L(H) where one replaces the action (T,h) Th in L(H) H H for a more general one L(H) A A where A is a subspace of linear operators in L(H) are introduced. This leads to definitons of new spaces whose properties and relations are studied.

3 REMARKS ON OPERATOR-VALUED MO SPACES 3 2. Vector-valued MO Let X be a anach space and let f : T X be a ochner integrable function. We say that f belongs to MO(T,X) (respect. MO weak (T,X)) if sup ( (9) f(t) m f 2 dt) /2 <, T interval (respect. sup ( (0) f(t) m f,x 2 dt) /2 <, ) x =, interval where x X and m (f) stands for the averarage m f = f(t)dt. Same proof as in the scalar-valued case allows to get f MO(T,X) if and only if () sup ( f(t) P (f)(z) 2 P z (t)dt) /2 < z < T where P z (t) = z zt, t T and P (f) stands for the Poisson integral of f. Making use of the John-Nirenberg s lemma, which holds true in the vector-valued case, one can also replace the L 2 -norm by the L p -norm in (9), (0) and (). We say that f MO P (T,X) (see [Pa]) if sup P ( f 2 )(z) P (f)(z) 2 <. z < We say that f MO C (T,X) (see [4]) if ( w 2 ) f(w) 2 P z (w)da(w) < sup z < D where (f) stands for the gradient of f and da for the Lebesgue measure in the disc D. t is known that embeddings between the just defined spaces depend upon some geometrical properties of the underlying anach space. For instance, MO(T,X) MO C (T,X) implies X has cotype 2 and MO C (T,X) MO(T,X) implies X has type 2 (see [5], Theorem.2 ). On the other hand, if X is a 2-uniformly PL-convex space then MOA P (T,X) MOA C (T,X), where MOA P (T,X) and MOA C (T,X) stand for the analytic version of the spaces (see [Pa], Theorem 3.2 ). The reader is referred to [W] for the notions on Geometry of anach spaces and related questions to be used throughout the paper. The duality in the vector-valued setting is also very well understood. One can define certain vector-valued Hardy spaces (see [, 2] and [ou]) which will give the preduals of different versions of vector-valued MO spaces. Given a anach space X we write Hat(T,X) for the space of functions F L (T,X) such that F = k N λ ka k,λ k C, where k N λ k < and a k are X-valued atoms, that is to say a k L (T,X), supp(a k ) k for some interval k, a k k and k a k (t)dt = 0. We endow the space with the norm given by the infimum of k N λ k over all possible decompositions. We write H con(t,x) for the space of functions f L (T,X) such that Hf L (T,X), with the norm given by f con = f L (T,X) + Hf L (T,X).

4 4 OSCAR LASCO Proposition 2.. (see [3]) fx is a real anach space then MO weak (T,X) isometrically embedds into L(ReH (T),X). Remark 2.2. The reader is also referred to [3] for the definition of the space of vector-valued measures of bounded mean oscillation which characterizes L(ReH (T),X). Proposition 2.3. (see [RRT] or [], Th.3. and Prop. 3.3) f X is a real anach space then MO norm (T,X ) isometrically embedds into (H at(t,x)). Moreover MO norm (T,X )=(H at(t,x)) if and only if X has the RNP. Remark 2.4. The reader is referred to [, 4] for the definition of the space of vector-valued measures of bounded mean oscillation which leads to the duality result without conditions on the anach space X. Let = {, } D, equipped with the natural product measure which assigns measure 2 n to cylinder sets of length n. For each σ {, } D, define the dyadic martingale transform T σ : L 2 (T,X) L 2 (T,X), given by f = h f h σ f. n the case that X is a Hilbert space, T σ F L 2 (T,X) = F L 2 (T,X) for any (σ ) and then F L (,L 2 (T,X)) = F L 2 (T,X). Given F L (T,X) we write F the function defined in T, F (σ, t) =T σ F (t) = σ F h. Recall that X is said to be UMD space if there exists C>0such that sup T σ F 2 C F 2 σ for all F L 2 (T,X). n particular, for UMD spaces we have that T σ F L2 (T,X) F L2 (T,X) and F L 2 (,L 2 (T,X)) F L 2 (T,X). t is known that L p (µ) spaces for <p< are UMD. Also the Schatten classes S p are UMD spaces for <p< (see [GM]) while L(H) ors are never UMD spaces (unless H is finite dimesional). The reader is referred to [ur] for a general survey on the UMD property. We simply mention here that the UMD property is equivalent to the boundedness of the Hilbert transform on L 2 (T,X) and the following connection with vectorvalued MO and duality. Proposition 2.5. (see []) MO norm (T,X )=(H con (T,X)) if and only if X is a UMD space. n the case X = L(H) where H is a separable Hilbert space we shall use the notation MO norm (T, L(H)) for MO(T, L(H)), that is MO norm (T, L(H)) if sup ( (2) (t) m 2 dt) /2 <. T interval n this situation we can still consider two related notions. One by considering the weak -topology, using L(H) =(Hˆ H). Let : T L(H) be such that

5 REMARKS ON OPERATOR-VALUED MO SPACES 5 (t)e, h L 2 (T) for all e, h H. We say that WMO(T, L(H)) if and only if sup ( (3) (t)e m e,h 2 dt) /2 <. e = h =, T interval Of course MO weak (T, L(H)) WMO(T, L(H)). Note that belongs to MO norm (T, L(H)) or WMO(T, L(H)) if and only does. Another possiblity is the following (see [NTV]): Let : T L(H) be a function such that (t)e, (t)e L 2 (T, H) for all e H. We say that MO so (T, L(H)) if sup ( (4) ((t) m )e 2 dt) /2 < T, interval,e H, e = and sup ( (5) ( (t) m )e 2 dt) /2 <. T, interval,e H, e = t is not difficult to show the following chain of strict inclusions. MO norm (T, L(H)) MO so (T, L(H)) WMO(T, L(H)). Since the trace class operators can be described as S = l 2 ˆ l 2, where X ˆ Y stands for the completion of the projective tensor product of the spaces X and Y then (S ) = L(H). Hence it follows from Propositions 2. and 2.3 that (6) MO weak (T, L(H)) L(Hat(T),S ) and (7) MO norm (T, L(H)) (Hat(T,S )) ( Proposition 2.6. MO so (T, L(H)) H ˆ ( Hcon(H) Hcon(H) )). Proof. Using that (X ˆ Y ) = L(X, Y ) and Proposition 2.5, it suffices to see that MO so (T, L(H)) L(H,MO(H) MO(H)). Observe now that MO so (T, L(H)) implies that e ((t)e, (t)e) defines a bounded linear operator from H into MO(H) MO(H). 3. Dyadic versions of operator-valued MO. Let H be a separable finite or infinite-dimensional Hilbert space, and let : T L(H) such that ( )e, h L (T) for any e, h H. From the closed graph theorem H H L (T) given by (e, h) ( )e, h defines a bounded bilinear map. Hence, for D, we can define the Haar coefficients = (t)h (t)dt L(H), and the average of over, m = (t)dt L(H) as the operators given by e, f = (t)e, f h (t)dt and m e,f = (t)e, f dt. We can now give similar notions to those introduced in Section 2, but only for dyadic intervals. Thus, we write MO d norm(t, L(H)) if MO d norm = sup( (8) (t) m 2 dt) /2 <.

6 6 OSCAR LASCO WMO d (T, L(H)) if WMO d = sup ( (9) ((t) m )e, f 2 dt) /2 <.,e,f H, e = f = MO d so(t, L(H)) if γ() = sup ((t) m )e 2 dt <,, e = γ( ) = ( (t) m )h 2 dt <. We write (20) sup, h = MO d so = γ() /2 + γ( ) /2 As in the introduction we have P () = J h J J =( m )χ. Hence MO d norm(t, L(H)) if and only if there exists a constant C>0such that (2) P () L 2 (L(H)) C /2 for all D. WMO d (T, L(H)) if and only if there exists a constant C>0such that (22) P ()e, h L 2 C /2 e h for all Dand e, h H. MO d so(t, L(H)) if and only if there exists a constant C>0such that (23) max { P (e) L 2 (H)), P (h) L 2 (H))} C /2 e =, h = for all Dand e H. As before one can replace in (8), (9), (20), (2), (22) and (23) the L 2 -norm by any other L p -norm for 0 <p<. As in the scalar valued case we have P (f) L 2 =( f J 2 ) /2 sup J D,J for f L 2 (T,X)ifX is a Hilbert space, but not in the case X = L(H). This leads us to consider the following space in terms of Carleson measures. MO d Carl (T, L(H)) if (24) J 2 <. J D,J n the papers [GPTV, NTV, NPiTV] the study of the boundedness of the following version of the operator-valued paraproducts was iniciated and developped: The densely defined linear maps π : L 2 (T, H) L 2 (T, H), f = f h (m f)h, which is called the vector paraproduct with symbol, and Λ = π + π : L2 (T, H) L 2 (T, H), f (m f)h + (f ) χ.

7 REMARKS ON OPERATOR-VALUED MO SPACES 7 Recall that a sequence Φ L 2 (T, L(H)) for all Dis said to be an operatorvalued Haar multiplier (see [Per, Po3]) if there exists C>0 such that Φ (f )h L 2 (T,H) C( f 2 ) /2 for any finite family of elements (f ) H. n the papers [NTV, Po3] the spaces MO para (T, L(H)) and MO mult (T, L(H)) were introduced. A function is said to belong to MO para (T, L(H)) if π defines bounded linear operators on L 2 (T, H). A function is said to belong to MO mult (T, L(H)) if the sequence (P ()) defines a Haar multiplier. Due to the equality (25) Λ (f) = P ()(f )h. one has that MO mult (T, L(H)) if and only if Λ is bounded on L 2 (T, H). t was shown that (see [NTV, Po3]) MO d norm(t, L(H)) MO mult (T, L(H)) MO so (T, L(H)) WMO d (T, L(H)) and MO d Carl(T, L(H)) MO para (T, L(H)) MO d so(t, L(H)). The space MO d so(t, L(H)) can be understood as the space of functions satisfying a natural operator Carleson condition, namely max{sup J J, sup (26) J J } < J The result in [NTV] therefore represents a breakdown of the Carleson embedding theorem in the operator case. t was shown in [Po3] that the stronger condition (27) sup J J J J χ J J L (T,L(H) < implies that MO para (T, L(H)). The results on [Po3] heavily depends upon the use of the notion of sweep of a function defined by S = J D J J χ J J. n particular it was discovered that MO para (T, L(H)) if and only if S MO mult (T, L(H)) Another important difference for operator-valued functions is the no validity of John-Nirenberg theorems, meaning that S b MO C b 2 MO, in the operatorvalued case. Several replacements of the previous inequality were obtained in [Po3]. Let us mention to finish this section two new lines of research related to operatorvalued paraproducts which are now in progress. One is about connections between Hankel operators and Schatten classes that have been recently considered in [PSm] and another one about paraproducts and Haar multipliers on the bidisc. This last one is connected with operator-valued theory by taking H = L 2 (T). n this case L 2 (T,L 2 (T)) can be understood as a function in two variables, say b(t, s), ( ) J = b J where b J = b(t, s)h T 2 (t)h J (s)dtds and m J (m ) = m J b

8 8 OSCAR LASCO where m J b = J b(t, s)(s)dtds. The reader is referred to [Po, Po2] for J results about paraproducts and Haar multipliers on the bidisc. 4. Dyadic A-valued MO spaces Throughout this section A denotes an operator ideal, that is A a anach space such that there exist two continuous embeddings maps : A L(H) and 2 : A L(H) in such a way that the composition maps L(H) (A) (A) and 2 (A) L(H) 2 (A) are bounded, i.e. u A,v L(H) = v (u) (A), 2 (u)v 2 (A) and max{ v (u) A, 2 (u)v A } C v u A, where we write u i A also the norm of the corresponding u Awhere u i = i (u) i (A) for i =, 2. We shall use simply vu and uv for v (u) and 2 (u)v in the sequel. We use the notation e h for the operator e h(x) = h, x e for x, y, e H. Clearly one has (28) T (e h) =Te h, (e h)t = e T h (29) (e h)(e h )= h, e (e h ) (30) (e h) = h e Proposition 4.. Let e 0 Hwith e 0 =. Then H is an operator ideal by selecting : H L(H) given by h h e 0 and 2 : H L(H) given by h e 0 h. Proof. Observe that i (h) = h for i =, 2 and T (h e 0 )=(Th) e 0 and (e 0 h)t = e 0 T (h). This corresponds to the actions L(H) H Hgiven by (T,h) Th and H L(H) Hgiven by (h, T ) T h. The properties are now straighforward. Remark 4.2. A = L(H) and the Schatten classes A = S p, p < are operator ideals for = 2 the inclusion maps. Recall that S = H ˆ H, (S p ) = S p,/p +/p = and that (S ) = L(H), with the duality given by (u, e h) = u(h),e, where u L(H), and e, h H. As in the previous sections we write MO d norm(t, A) if MO d norm (T,A) = sup( (3) (t) m 2 Adt) /2 <. and MO d Carl (T, A) if (32) J 2 A <. sup J D,J t was shown in [Po3] that L (T, L(H)) was not contained into MO d Carl (T, L(H)). Let us see that if we replace L(H) bya the situation becomes different. As usual r k denote the Rademacher functions. Recall that a anach space X is said to have type p, for some <p 2, if there exists a constant C>0such that N N x k r k L2 (T,X) C( x k p ) /p

9 REMARKS ON OPERATOR-VALUED MO SPACES 9 for all x,..., x n X. Similarly, a anach space X is said to have cotype q, for some 2 q<, if there exists a constant C>0 such that for all x,..., x n X. N N ( x k p ) /p C x k r k L 2 (T,X) Proposition 4.3. (i) f MO d Carl (T, A) MOd norm(t, A) then A has type 2. n particular, MO d Carl (T,S p) MO d norm(t,s p ) for p>2. (ii) f A has cotype 2 then MO d norm(t, A) MO d Carl (T, A). n particular, MO d norm(t,s p ) MO d Carl (T,S p) for p 2. Proof. t was shown (see [4],Theorem.) that for any anach space X (33) N N x k r k MO(T,X) x k r k L 2 (T,X) for any x k be a sequence of elements in X. Take = k /2 for =2 k. Then h = kr k. Note that sup J D,J J 2 A = sup 2 k k 2 A( J, J =2 k J )= k 2 A Applying (33) one gets N k r k 2 L 2 (T,A) C h 2 MO d Carl = C k 2 A Now the assumption in (i) gives type 2. Similarly the part (ii). We can use martingale transforms (see Section 2 for the notation) to analyze the validity of John-Nirenberg s lemma in our situation, that is to say to study whether MO d norm implies S MO d norm. Let us rewrite the sweep S = J D J J χ J J by the formula (34) S = T σ T σ dσ, Theorem 4.4. Let L (T, A). Then (i) S MO d norm (A) T σ 2 MO d (A)dσ. norm (ii) S MO d norm (S p/2 ) C 2 MO d norm (Sp) for 2 p<. Proof. t is not difficult to show that P (S )=P S P (see [Po3]). Hence, using (34), one gets P (S )= T σp T σ P dσ.

10 0 OSCAR LASCO Therefore P (S ) L (T,A) (T σ P )(T σ P )dσ L (T,A) (P T σ )(P T σ ) L (T,A)dσ P T σ L 2 (T,L(H)) P T σ L 2 (T,A)dσ P T σ 2 L 2 (T,A) dσ ( T σ 2 MO dσ). d norm Use now John-Nirenberg s lemma to obtain S MO d norm (A) C sup P (S ) L (T,A) C( T σ 2 MO dσ). d norm (ii) Use the argument above, together with the estimate uv Sp/2 u Sp v Sp, to get that P (S ) L (T,S p/2 ) P T σ 2 L 2 (T,S dσ p) ( T σ 2 MO d (Sp)dσ) norm sup T σ 2 MO d norm σ (Sp) C 2 MO d (Sp), norm where the last inequality follows from the fact that S p is a UMD space. Now finish the proof applying John-Nirenberg s lemma again. Definition 4.5. Let A be an operator ideal and let : T L(H) such that (t)u, v(t) L 2 (T, A) for any u, v A. We say that MO d so,a (T, L(H)),if γ r,a () = sup ( (35) ((t) m )u 2,u A, u A = Adt) /2 < and γ l,a () = sup ( (36) v((t) m ) 2,v A, v A = Adt) /2 <. The norm MO d norm (T,A) = γ r,a ()+γ l,a (). Definition 4.6. Let A be an operator ideal and let : T L(H) such that v(t)u L 2 (T, A) for any u, v A. We say that WMO d A(T, L(H)), if γ A () = sup ( (37) v((t) m )u 2,u,v A, v = u A = Adt) /2 <. Of course for A = H (see Proposition 4.), one has MO so,h (T, L(H)) = MO so (T, L(H)), WMO d H(T, L(H) = WMO d (T, L(H).

11 REMARKS ON OPERATOR-VALUED MO SPACES t is also elementary to show MO d norm(t, A) MO so,a (T, L(H)) WMO d A(T, L(H)). Although for A = H it was shown (see [Po3]) that the inclusions are strict, however for A = L(H) one obviously has MO d norm(t, L(H)) = MO so,l(h) (T, L(H)) = WMO d L(H)(T, L(H)). Let us study the situation for A = S. Proposition 4.7. (i) MO so,s (T, L(H)) = MO so (T, L(H)). (ii) WMO d S (T, L(H)) = WMO d (T, L(H)). Proof. (i) Let MO so,s (T, L(H)). Take u = e h with e = h = and observe that ((t) m )u =((t) m )e h and u((t) m )=e ( (t) m )h. Hence ((t) m )u S = ((t) m )e and u((t) m ) S = ( (t) m )h. This implies that MO so,s (T, L(H)) MO so (T, L(H)). Conversely, let MO so (T, L(H) and let u = λ ke k h k where λ k < and e k = h k =. Now ((t) m )u S = Then, for Dand u S, one gets ((t) m )u S dt λ k ((t) m )e k h k S λ k ((t) m )e k λ k ((t) m )e k dt λ k MOso Hence using John-Nirenberg s lemma one gets γ r,a () u S MOso. Similarly one obtains γ l,a () u S MOso. (ii) Note that for e = h = e = h =,(e h)((t) m )(e h )= h, ((t) m )e e h. Hence (e h)((t) m )(e h ) S = h, ((t) m )e. Now similar arguments to the ones used in (i) allow to get the result. Let F 00 (A) denote the subspace of A-valued functions on T with finite formal Haar expansion (we keep the notation F 00 in the case A = L(H)) and write L 2 0(T, A) the closure of F 00 (A) inl 2 (T, A). Definition 4.8. Let (Φ ) L 2 (T, L(H)) be a sequence of operators. t is said to be an A-Haar multiplier if there exists C>0 such that Φ F h L 2 (T,A) C F h L 2 (T,A) for any F F 00 (A). We write (Φ ) mult,a L 2 0(T, A). for the norm of the extension of the operator to

12 2 OSCAR LASCO Definition 4.9. Let L 2 (T, L(H)). We say that MO multa (T, A) if (P ) defines a A-Haar multiplier and we write MOmult,A = (P ) mult,a. t was shown in [Po3] that MO d norm(t, L(H)) MO mult (T, L(H)). Now one has the following Proposition 4.0. MO mult,l(h) MO d norm(t, L(H)). Proof. Let MO mult,l(h). Take F = h J for fixed J D. Observe that P ()F h L 2 0 (L(H)) = P J ()h J L 2 0 (L(H)) = J P J() /2 L 2 (L(H)) MOmultL(H). Definition 4.. Let L 2 (T, L(H)). We define the A-paraproduct with symbol by π A : F 00 (A) F 00 (A) given by F = F h m Fh, where m F stands for the composition of operators. Definition 4.2. Let L 2 (T, L(H)). We say that MO para, A (T, L(H)) if π A extends to a bounded operator from L2 0(T, A) to L 2 0(T, A). We write MOpara, A = π A L 2 0 (T,A) L 2 0 (T,A), Theorem 4.3. MO para, L(H) MO d norm(t, L(H)). Moreover 2 MO d norm (L(H)) MOpara,L(H) 2 Proof. Applying the assumption on functions F =(e h)φ for fixed e = h = and φ L 2 (T) =, one easily obtains that WMO d π L(H). n particular J π L(H) J /2 for all J D. Consider F (t) =χ J (t) for some J Dwhere stands for the identity operator. Now π L(H) (F )= h + J h = P J ()+ 2 k h. J J Clearly =2 k J,k 2 k h L 2 (L(H)) =2 k J,k =2 k J,k 2 k J /2 π L(H) 2 k/2 π L(H) J /2. k 2

13 REMARKS ON OPERATOR-VALUED MO SPACES 3 Therefore P J () L2 (L(H)) π L(H) + Thus the proof is complete. 2 WMO d J /2 2 2 π L(H) J /2. Definition 4.4. Let L 2 (T, L(H)). We can define A : F 00 (A) L 2 (T, A) given by F = F h χ F. Let us denote Λ A = πa + A. We also define Γ A : F 00 (A) F 00 (A) given by F = F h F h /2. Remark 4.5. Clearly if Γ A is bounded on L2 (T, A) then sup u A = u C /2. For A = H one has Γ A is bounded if and only if C /2. Proposition 4.6. MO multa (T, L(H)) if and only if Λ A bounded operator on L 2 0(T, A). Moreover MOmult,A = Λ A L 2 0 (T,A) L (T,A). 2 Proof. t follows from the formula (38) Λ A F = F (m )F h = (P )F h. extends to a We observe that the boundedness of A MO d norm(t, A). on L2 0(T, A) can be pushed to Proposition 4.7. (39) A MO d norm (T,A) MO d norm (T,A) 2 A L 2 0 (T,A) L 2 0 (T,A). Proof. Assume A is bounded on L2 0(T, A). Let F MO d norm(t, A) of norm, that is P F L 2 (T, L(H)) with norm bounded by /2. t is not difficult to see that P A (F ) L 2 (T,A) = P A (P F ) L2 (T,A). Since P G =(G m G)χ then we have P A (P F ) L 2 (T,A) 2 A (P F ) L 2 (T,A) 2 A L 2 0 (T,A) L 2 0 (T,A) /2. Hence one gets the desired estimate. Definition 4.8. We write :F 00 F 00 L 2 (T, L(H)) for the map (,F) = L(H) (F )= χ F.

14 4 OSCAR LASCO And we denote Γ:F 00 F 00 F 00 given by Γ(,F) = F h. /2 Of course Γ(,F) =Γ L(H) (F ). n particular, the dyadic sweep of F 00 is given by (40) S = χ = L(H) () = (,). Let us finish by giving the formulation of the main connection between MO para and MO mult (see [Po3]) in the new situation. Next result is the extension of the similar one shown in [Po3], and the proof presented here is different from the one given there for A = H. Theorem 4.9. Let,F F 00. Then A F π A =Λ A (F,) ΓA Γ(F,). Proof. A F π A (G) = A F ( m (G)h ) = F m (G) χ = χ F G J h J = ( J D J = ( J D J J F χ )G Jh J F χ )G Jh J J D = Λ A (F,) (G) ΓA Γ(F,) (G). F J J G J h J J Corollary Let F 00. Then L(H) πl(h) =Λ L(H) S Γ L(H) where =Γ(,) = /2 h. The author thanks T. Hytonen for pointing out that Proposition 4.7 holds for any ideal of operators. References [GM] E. erkson, T.A. Gillespi, P.S. Muhly Abstract spectral decomposition guaranteed by the Hilbert transform, Proc. London Math. Soc. 53, no. 3 (986), [] O. lasco, Hardy spaces ofvector-valued functions: Duality, Trans. Am. Math. Soc. 308 (988), no.2, [2] O. lasco, oundary values offunctions in vector-valued Hardy spaces and geometry of anach spaces, J. Funct. Anal. 78 (988), [3] O. lasco, Operators from H into a anach space and vector-valued measures, London Math. Soc. Lecture Note series vol 40 (989),

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