Isomorphic Embeddings and Harmonic Behaviour of Smooth Operators

Let Y be a Banach space, 1 < p < ∞. We give a simple criterion for embedding lp ⊂ Y , namely it suffices that the positive cone l+p ⊂ Y . This result is applied to the study of highly smooth operators from lp into Y (p is not an even integer). The main result is that every such operator has a harmonic behaviour unless l p K ⊂ Y for some K ∈ N. In this note we establish a natural criterion for embedding of lp or c0 into a given Banach space and apply it to smooth operators with harmonic behaviour from lp spaces. Recall that the well-known summing basis {ei} of c0 has the property that ‖ ∑ aiei‖ = ∑ ai provided that ai ≥ 0, which means (in our notation) that l + 1 ⊂ c0. In fact, and more surprisingly, l + 1 ⊂ Y for any Banach space Y . Moreover, if Y is separable, then there exists a minimal and fundamental system in Y whose positive cone is isomorphic to l1 ([S1], [S2], [DJ]). In our paper we prove a result going in the opposite direction, that Z ⊂ Y already implies Z ⊂ Y for Z = lp, 1 < p < ∞, or c0. This simple and somewhat unexpected criterion allows us to completely characterize Banach spaces Y , for which there exist separating polynomial (or smooth enough) operators from lp into Y , as those for which l p k ⊂ Y for some integer k. 1. Embedding of the Positive Cone Let Y be a Banach space, Z be a Banach space with a Schauder basis {ei}. Let us denote the positive cone of Z by Z = { z ∈ Z; z = ∑ aiei, ai ≥ 0 }. We say that Z + ⊂ Y if there is a basic sequence {yi} in Y such that ‖ ∑ aiei‖Z ≤ ‖ ∑ aiyi‖Y ≤ C ‖ ∑ aiei‖Z for any ∑ aiei ∈ Z . We say that C is an isomorphism constant. For a ∈ R, let a = max{a, 0} and a = max{−a, 0}. Theorem 1. Let Y be a Banach space. If c0 ⊂ Y then c0 ⊂ Y . Moreover, {yi} is equivalent to the canonical basis of c0. Proof. Let ∑ aiyi ∈ Y . Then by assumption ∥∥ ∑ aiyi ∥∥ = ∥∥ ∑ a+i yi − ∑ a−i yi ∥∥ ≤ ∥∥ ∑ a+i yi ∥∥ + ∥∥ ∑ a−i yi ∥∥ ≤ C max{a+i } + C max{a − i } ≤ 2C max { |ai| } . But, as {yi} is a basic sequence, ∥∥ ∑ aiyi ∥∥ ≥ 1 2K max { ‖aiyi‖ } ≥ 1 2K max { |ai| } , where K is a basis constant of {yi}. ⊓⊔ Theorem 2. Let Y be a Banach space, 1 < p < ∞. If lp ⊂ Y then lp ⊂ Y . First notice the following lemma: Lemma 3. Let Z be a Banach space with an unconditional Schauder basis {ei}, Y be a Banach space and Z ⊂ Y such that {yi} is an unconditional basic sequence. Then Z ⊂ Y (in fact {yi} is equivalent to {ei}). Date: November 2003. Supported by grants GAUK 277/2001, GAČR 201-01-1198, A1019205. 1 ISOMORPHIC EMBEDDINGS AND HARMONIC BEHAVIOUR OF SMOOTH OPERATORS 2 Proof. There is a K1 ≥ 1 such that K −1 1 ∥∥∑ |ai| yi ∥∥ Y ≤ ∥∥∑ aiyi ∥∥ Y ≤ K1 ∥∥∑ |ai| yi ∥∥ Y for any ∑ aiyi ∈ Y and a K2 ≥ 1 such that K −1 2 ∥∥∑ |ai| ei ∥∥ Z ≤ ∥∥∑ aiei ∥∥ Z ≤ K2 ∥∥∑ |ai| ei ∥∥ Z for any ∑ aiei ∈ Z. Thus K 1 K −1 2 ∥∥∑ aiei ∥∥ Z ≤ ∥∥∑ aiyi ∥∥ Y ≤ K1CK2 ∥∥∑ aiei ∥∥ Z for any ∑ aiei ∈ Z. ⊓⊔ Proof of Theorem 2. We claim that there is an unconditional normalized block basic sequence of {yi} such that all its vectors have nonnegative coordinates with respect to {yi}. Then it is easily seen by Lemma 3 that this block basic sequence is equivalent to the canonical basis of lp. For x = ∑ aiyi ∈ Y we denote x + = ∑ a+i yi, x − = ∑ a−i yi and x̂ = ∑ aiei ∈ lp. Suppose that {yi} is not unconditional and l + p ⊂ Y with isomorphism constant C. Then for any ε > 0 there is y ∈ span{yi} with finite support such that ‖y ‖ = 1 and ‖y‖ < ε. If this was not true for some ε > 0, then for any x ∈ span{yi} ‖x‖ ≥ εmax ∥∥x+ ∥∥ , ∥x− ∥∥ } ≥ ε 2 ∥∥x+ ∥∥ + ∥x− ∥∥ ) ≥ ε 2 ∥x+ + x ∥∥ . On the other hand ‖x‖ = ∥x+ − x ∥∥ ≤ ∥x+ ∥∥ + ∥x− ∥∥ ≤ C ∥∥x̂+ ∥∥ p + ∥x̂− ∥∥ p ) ≤ C2 1 p ∥x̂+ + x̂− ∥∥ p ≤ C2 1 p ∥x+ + x ∥∥ , which means that {yi} would be unconditional. Thus we can construct a block basic sequence {vi} of {yi} such that ‖vi‖ < 1 2 1 2i and ∥v̂+ i ∥∥ p = 1. Let {aj} n j=1 be a finite sequence of nonnegative real numbers. Then ∥∥∥∥ n ∑