These results are due to KWAPIEN [ll], PISIEB [18] and Borntom [3], BWKHOLDER [5], respectively.
We call an operator u:
bounded. Define &(u) := Ilu Q Hll. The claas of HT-operators forms an injective, surjective, completely symmetric and maximal BANACH ideal (in the sense of PJETSCH [17)}, see chapter 1. Furthermore, the HIP-operators are uniformely convexifying. In particular, as well-known the UHD-spaces are super-reflexive (ALDOUS [l], M.AuREY
By M. Rmsz [20] the real &BERT transform H: Lp(R) -+ Lp(R) is bounded for all 1 < p < 00. In the vector-valued case u Q H : E Q p LJR) + Lp(R, 3 ' ) is bounded for all 1 < p < 00 if and only if this holds for one 1 < p < do. Therefore for each HToperator u hp(u) := [[u @ H: E Q p Lp(R) -+ Lp(R, F)Il is finite. Examining the proof, which uses MARCINKIEWICZ interpolation theorem and a result of J. SCHWARTZ [22] For which operators U : E -+ P has u @ T:
ivhth. Neck. 141 (1989) 1 p < 00 :ha I ; h,, we confirm that there is a c 2 0 such that for all 2 sequently, see chapter 2, h&)p (2, 2 2) andh2(4
In chapter 3 we study the question: Is every HT-operator UMD-factorable? The estimates of h 2 ( 4 and an unpublished result of FIQIEL imply that the ideal of HToperators is not idempotent. Hence there is a HT-operator which is not UMD-factorable. But we axe also able to construct such a counter-example explicitly. For this we characterize sufficiently well those diagonal operators Dg: 1,+ b,,(x,,),,,~.),, which are HT-operators. The diagonal operator given by a := (In-2 (n + I)),, is a HToperator which is not UYD-factorable. If E is a (red or complex) ~E B T space, we show that /&,(id,) = h,,(idR) for all 1 < p < 00. The real norms hp(idR) were computed by PICHORIDES [l6]. Finally we prove that a UMD-space E is a HILBERT space if and only if h,(idE) = 1, see chapter 4. Prcliminaries. We use standard B ~A C H space notations. Let E be a BANACH space and 1 the LEBESQUE measure on R. M ( R , E ) is the set of all A-strongly measurable, E-valued functions on R. Lo(R, E ) denotes the space of all f E M ( R , E ) , bounded and vanishing outside of a set of finite measure. For 1 00 the BANAOH space L, ,(R,E)
R) with norm llollp in the usual manner. If we have the HAAR measure on the unit circle 17, we write similarly L,,(17) and L,(17, E). Let T: L,(R, E ) +-M(B, 3) be a linear mapping and I 5 p , q 2 00. T is of weak type ( p , q) with constant M , if for ell f E L,,(R, 1) cph,. Conc and a > 0 : A([llT(f)ll > a]) S (" IIfIl,,~. T is of strong type ( p , q) with constant M , U if for all f E Lo@, E ) : llT(f)]I,'S Ilfllp.'The standard references on operator ideals is [17]. The ideals of bounded, finite dimensional, weakly compact, (RADEMACHER-) type?, cotypeq operators are denoted by I, 3, W , S, and C, , respectively. For the class of uniformly convesifying operators W A we refer to [lo]. By a property P we mean a subclass of the class of all BANACH spaces BAN. A BANAOE space E has the property super P, if each BANACH space finitely representable in E has property P. A property P = super P is called super-property. * tions. Comm. on Pure and Appl. Math. 14 (1961) 785-799 Mathematiaches Seminar Universikit Kiel Olshau.senstra/le 40 -60 0-2300 Kiel I BRD