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Finite Dimensional Subspaces of Lp

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Finite Dimensional Subspaces of Lp Finite dimensional subspaces of L p William B Johnson Department of Mathematics Texas AM University Col lege Station Texas USA Gideon Schechtman Department of Mathematics Weizmann Institute of Science Rehovot Israel We discuss the nite dimensional structure theory of L in particular p n the theory of restricted invertibility and classication of subspaces of p Contents Intro duction The go o d the bad the natural and the complemented The role of change of density n Subspaces of p n Fine emb eddings of subspaces of L into p p k n Natural emb eddings of into r p k n subspaces of mdimensional subspaces and quotients of r p Finite dimensional subspaces of L with sp ecial structure p Supp orted in part by NSF DMS and DMS Texas Advanced Research Program and the USIsrael Binational Science Foundation Supp orted in part by the USIsrael Binational Science Foundation Preprint submitted to Elsevier Preprint January Subspaces with symmetric basis Subspaces with bad gl constant Restricted invertibility and nite dimensional subspaces of L of p maximal distance to Euclidean spaces n Restricted invertibilityof op erators on p Subspaces of L with maximal distance to Hilb ert spaces p Complemented subspaces n Fine emb eddings of complemented subspaces of L in p p Finite decomp osition and uniqueness of complements References Intro duction The good the bad the natural and the complemented innite dimensional subspaces of There are many interesting problems ab out L L which have nite dimensional analogues For example it has p p long been a central problem in Banach space theory to classify the comple mented subspaces of L up to isomorphism the nite dimensional analogue p is to nd for anygiven C a description of the nite dimensional spaces which are C isomorphic to C complemented subspaces of L A lot is known ab out p b oth the innite dimensional see and nite dimensional see section versions of this complemented subspaces of L problem but in neither case p do es a classication seem to b e close at hand It sometimes happ ens that the nite dimensional version of an innite di mensional problem leads to a theory which is much more interesting than the innite dimensional theory Take for example the problem of describing the subspaces of L which embed isomorphically into a smaller L space p p namely for which there is a fairly simple answer see Now it is clear p that a nite dimensional subspace X of L emb eds with isomorphism constant p N if N N X is suciently large The attempt to estimate into p well N in terms of and X or the dimension of X has led to adeep theory see sections and A sideline of this investigation also led to deep er understanding of how certain natural subspaces of L such as the span of a p sequence of indep endent Gaussian random variables are situated in L see p section Besides b eing an interesting sub ject in its own right the study of nite di mensional subspaces of L is often needed in order to understand prop erties p of innite dimensional subspaces For example the easiest and best way to obtain subspaces of L p which fail GLlust cf section p n is to show that random large dimensional subspaces of are bad in a certain p sense see section The topic discussed herein which has the most applications is that of restricted invertibility see section Basically the theorem says that an n b y n matrix n which has ones on the diagonal and is of norm M say as op erator on must p be invertible on a co ordinate subspace of dimension at least M n One of the many consequences of this result is that certain nite dimensional subspaces n X of L contain wellcomplemented subspaces with n prop ortional to the p p dimension of X The role of change of density Generally the structure of the L spaces is describ ed for a xed value of p p However pro ofs of many of the results ab out L for a xed p use the entire p scale of L spaces p Consider for example the pro of section r that a subspace X of L p is either isomorphic to a Hilb ert space p or contains a subspace which is complemented in L and is isomorphic to p p There one needs only to compare the L norm and the L norm on X p In pro ofs of other theorems ab out L it is necessary to change the measure p b efore making a comparison between the L norm and another norm Since p the technique of changing the measure making a change of density is used in the pro ofs of most of the results we discuss in this article wechose to devote this section to describing the change of densities that arise later It turns out that the framework in which this technique is most naturally used is that of an L space when is a probabilityFor us there is no loss of generalityin p N restricting to that case since the space is isometric to L when is any p p probabilityon fNg for which fng for each n N For such N N a measure we denote L by L or just L if assigns mass N to p p p each integer n n N A density on a probability space is a strictly p ositive measurable R function g on for which g d Such a density g induces for xed p an isometry M M from L onto L g d dened by gp p p p Mf g f Sometimes a gain is achieved b y making such a change of density that is by replacing L by its isometric copy L g d The gain p p usually occurs b ecause for some subspace E of L and some value of r p dierent from p the space M E is b etter situated with resp ect to L g d gp r than E is to L For example it follows from the Pietsch factorization r theorem section that if an op erator T from some space X into L has psumming adjoint then by replacing the original L space by another L space TX is actually contained in L Formally isometrically equivalent p Prop osition If T X L a probability measure has psumming adjoint then there is a change of density g and an operator T X L g d p so that M T is the composition of T fol lowed by the canonical injection from gp L g d into L g d Moreover kT k T as long as TX has ful l sup p p port ie thereisnosubset with so that Tx Tx ae for every x in X To prove this factorization theorem assume for simplicitythat is a regular Borel measure on a compact space and that TX has full supp ort Get a Pietsch measure for the restriction of T to C K which means that is a R p p p regular probability measure on K such that kT f k T jf j d for all p f in C K This same inequality is true if is replace by its RadonNikodym derivative with resp ect to so one can assume that is of the form g d R with g and g d It is easily checked that this g is the desired density provided that g is strictly p ositive ae which it must be since TX has full supp ort When TX do es not have full supp ort reason the same way but at the end add a small constant function to g and renormalize The next change of density result due to D Lewis gives useful information ab out nite dimensional subspaces of L p Theorem Let be a probability measure and let E be a k dimensional subspace of L p with ful l support Then there is a density g p so that M E has a basis ff f g which is orthonormal in L g d and gp k k P such that jf j k i n For a pro of when p see The rst step is to apply Lewis lemma N section to get an op erator T from onto E for which T p T N The rest of the pro of involves checking that the choices and I p p P p g jTe j and f kg Te satisfy the conditions of Theorem i i i Another pro of for the entire range p is contained in Recall that if T X Y is an op erator T is the inmum of kT kkU k over all factorizations T SU with U X and S Y Theorem If E is a k dimensional subspaceofL then thereisaprojec p pj j tion P from L onto E with P k p k k jpj Since the BanachMazur distance from to is k section p Theorem implies that the distance of a k dimensional subspace of L to p k k is maximized when the subspace is p To prove Theorem observe rst that the case p follows trivially from p the fact proved in section that I k for every k dimensional E space E When p in view of the comments made in section there is no loss in generality in assuming the is a probability measure and that E has full supp ort Theorem says that we can further assume that E has a k P jf j k basis ff f g which is orthonormal in L and such that i k n Let P be the orthogonal pro jection onto E If p a simple computation p shows that kP L L k k and so also P L p p p L k The case p is even easier p For a generalization of Theorem to spaces of typ e p see The change of density in Theorem is used in section to showthatak di n mensional subspace of L well emb eds into with n not to o large There what p p is needed is the relation between the L and the L g d norms on M E p gp The relevant estimate follows from a trivial observation which we record for later reference Lemma Let be a probability measure and let E be a k dimensional sub space of L which has a basis ff f g which is orthonormal in L p k k P p ach f in E kf k k kf k if jf j k Then for e and such that p i n p and kf k k kf k if p p There is a prettycharacterization due to Maurey I I IH of subsets of L which are b ounded in L after a change of density p q Theorem Let be a probability measure p q and S a subset of L of ful l support Then there is a density g so that M S B if p gp L g d q and only if for al l nite subsets S of S and fa x S g x X X q q q q k ja xj k ja j x x L p xS xS Assuming the Pietsch factorization
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