Explicit representation of compact linear operators in Banach spaces via polar sets

. We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. It is shown that if the Gelfand numbers of T decay suﬃciently quickly, then the action of T is given by a series with calculable coeﬃcients. This provides a Banach space version of the well-known Hilbert space result of E. Schmidt.

1. Introduction.Apart from its intrinsic interest and wide applicability, the famous Schmidt decomposition of compact linear maps acting between Hilbert spaces may also be given credit for stimulating work on those compact linear maps from one Banach spaces to another, such as nuclear maps, that may also be represented in series form.Of course, it is not to be expected that an exact analogue of Schmidt's theorem will be generally true outside a Hilbert space setting.Our main result is that such a series representation with coefficients that are recursively calculable is possible for maps the Gelfand numbers of which decrease sufficiently rapidly.
More precisely, let X and Y be real Banach spaces, both of which are uniformly convex and uniformly smooth, and let T : X → Y be linear and compact, with trivial kernel and range dense in Y .We show that if the Gelfand numbers c n (T ) of T decay sufficiently quickly as n → ∞, then the action of T is given by a series: where α Y n ∈ X * and y n ∈ Y, for each n.The terms in this series originate in the construction of a decreasing sequence of linear subspaces X n of X with finite codimension and trivial intersection; the norm λ n of the restriction of T to X n is attained at x n and y n = T x n /λ n ; the coefficients α Y n (x) are recursively calculable by using projections and polar sets.
Banach space versions of the Schmidt result are also given in the recent papers [3]- [5], and in particular it has been shown that, under an additional assumption, there are representations of the form The additional assumption is that for each x ∈ X, the elements S X,T n x := n−1 j=1 ξ X j (x)x j should have uniformly bounded norms: This condition is automatically satisfied for every compact map T if X is a Hilbert space, no matter what Y is (within the class of spaces studied), and also in certain other special cases, but it remains unclear exactly how wide is the class of spaces and operators for which it holds.Here we cast some light on this question by showing that the Y -analogue of this assumption holds if the Gelfand number c n (T ) of T is bounded above by 2 −n+1 (2 n −1) −1 for all n ∈ N, and that consequently we have the series representation of T given in (1.1).Under these conditions on the Gelfand numbers the map T is nuclear, so that the existence of a series representation is guaranteed.
The main point of our work is to establish the particular decomposition given by (1.1), in which the coefficients are recursively calculable by procedures analogous to those used in the Hilbert space case.Note also that the requirements of uniform convexity and uniform smoothness imposed on X and Y are met by such commonly used spaces as L p (1 < p < ∞) and the associated Sobolev spaces.
A crucial part of the proof is the study of the projections In general these are nonlinear, but we show that because of the special properties of the Y n they have certain linearity properties that we are able to exploit to obtain the result.

Preliminaries.
Throughout the paper we shall suppose that X and Y are real, reflexive, infinite-dimensional Banach spaces with norms • X , • Y and duals X * , Y * that are strictly convex; the closed unit ball in X is denoted by B X and the family of all bounded linear maps from X to Y by B(X, Y ) (B(X) if X = Y ); T will be a compact linear map from X to Y.We denote the value of x * ∈ X * at x ∈ X by x, x * X , and given any closed linear subspaces M, N of X, X * respectively, their polar sets are A gauge function is a map µ : [0, ∞) → [0, ∞) that is continuous, strictly increasing and such that µ(0) = 0 and lim t→∞ µ(t) = ∞; the map J X : X → X * defined by is called a duality map on X with gauge function µ.For all x ∈ X, From now on we suppose that X and Y are equipped with duality maps corresponding to gauge functions µ X , µ Y respectively, normalised so that µ X (1) = µ Y (1) = 1.Let M be a closed linear subspace of X.Then if X is strictly convex, so are M and X \M ; if X * is strictly convex, so are (X \M ) * and M 0 .A semi-inner product is defined on X by (x, h) X = x X h, J X x X (x = 0), (0, h) X = 0.
Next we summarise some of the results of [3], [4] and [5], beginning with the elementary fact that, under the conditions on X, Y and T, there exists x 1 ∈ X, with x 1 X = 1, such that T = T x 1 Y , and that x 1 satisfies the equation or equivalently, Set X 1 = X, M 1 = sp{J X x 1 } (where sp denotes the linear span), Since X 2 and Y 2 are closed subspaces of reflexive spaces they are reflexive.Since and if it is not the zero operator we can repeat the above argument: there exists In this way we obtain elements x 1 , . . ., x n of X, all with unit norm, subspaces M 1 , . . ., M n of X * and N 1 , . . ., N n of Y * , where and decreasing families X 1 , . . ., X n and Y 1 , . . ., Y n of subspaces of X and Y respectively given by and so The process stops with λ n , x n and X n+1 if and only if the restriction of T to X n+1 is the zero operator while T n = 0.With respect to the semi-inner product on X the x n have the semi-orthogonality property and correspondingly the where δ r,s is the Kronecker delta.
The iterative process just described can be expressed in terms of the notion of orthogonality given by James [9], using the notation of [6] and [7].We say that an element x ∈ X is j-orthogonal (or orthogonal in the sense of James) to y ∈ X, and write x ⊥ j y, if In general, j-orthogonality is not symmetric, that is, x ⊥ j y need not imply y ⊥ j x; indeed, symmetry would imply that X is a Hilbert space.The connection between j-orthogonality and the semi-inner product defined earlier is given by the following result of [9]: if x, h ∈ X, then x ⊥ j h if and only if (x, h) X := x X h, J X x X = 0.

It now follows immediately that
x r ⊥ j x s and y r ⊥ j y s if r < s, and x r ⊥ j X r and y r ⊥ j Y r for all r.
A decomposition of X in terms of James orthogonality was given by Alber [1], who introduced the following terminology: given closed subsets M 1 , M 2 of X, the space X is said to be the James orthogonal direct sum of M 1 and M 2 , and we write Alber established the following Theorem 1.Let X be uniformly convex and uniformly smooth, and let M be a closed linear subspace of X; let J X be a duality map that is normalised in the sense that it has gauge function µ with µ(t) = t for all t ≥ 0. Then Returning to the properties of T we note that when the rank of T is infinite, For these results see [5].Next, we define , and introduce the family of maps determined by the condition that x − S X,T k x ∈ X k for all x ∈ X.When the meaning is clear we shall simply write S X k instead of S X,T k .It turns out (see [3]) that S X k is uniquely given by Let us note that ξ X j (•) is a linear functional; this guarantees the linearity of S X k .Maps S Y k : Y → Z Y k−1 are defined in an analogous fashion: where the ξ Y j (y) are given by expressions corresponding to (2.2).From the uniqueness it follows that (S X k ) 2 = S X k and hence (( Lemma 2. The spaces X and X * have the direct sum decompositions Corresponding statements hold for Y. Proof.We give the proofs for X only, and for simplicity omit the superscript X.The decomposition for X folllows from I = (I − S k ) + S k , where I is the identity map of X to itself, since I − S k maps X into X k by definition, and S k has range Z k−1 .It is unique in view of the construction of the X k .
From the definition of the S k and (2.2) we have and so, on setting From the decomposition of X we see that x k , and as (I − S k+1 )x ∈ X k+1 ⊂ X k , we have X k = X k+1 ⊕ sp{x k }, from which the final part of the lemma follows.Now let P X n be the projection of X onto X n (n ∈ N ∪ {∞}), by which we mean that P X n : X → X n takes x ∈ X to the point in X n nearest to x. Projections P Y n : Y → Y n are defined analogously.In [3] and [4] it is shown that for all x ∈ X, in the sense of weak convergence (strong if X is uniformly convex) and in the sense of strong convergence, even if X is not uniformly convex.More can be said if the additional assumption of uniform boundedness of the S X n is made: it then turns out (see [4,Corrigendum]) that if T has trivial kernel and (S X n ) n∈N is bounded, then (x n ) n∈N is a basis of X and accordingly there is a Schmidt-type representation of T. If the assumption that (S X n ) n∈N is bounded is omitted, this result no longer holds.Readers interested in decomposition of Banach spaces should look at [1, Section 3], [3] or [6] for more details.

The main results.
Here we retain the notation of the last section and strengthen the assumptions (concerning X, Y and T ) by requiring that both X and Y should be uniformly convex and uniformly smooth, and that the compact map T should have trivial kernel and range dense in Y .We begin with a lemma connecting P X 2 and S X 2 .
Lemma 3.For all x ∈ X, x − P X 2 x = S X 2 x.Proof.Let x ∈ X.Since P X 2 x is the nearest element in X 2 to x, it follows that for all β ∈ R and all z ∈ X 2 , which means that x−P X 2 x ⊥ j X 2 .By Theorem 1, applied with M = X 2 , we see that X = X 2 sp{x 1 }, and in view of the uniqueness of decomposition we have x − P X 2 x = αx 1 for some α ∈ R. The uniqueness of S X 2 x now gives the result.
Repetition of this argument, with X, X 2 replaced by X 2 , X 3 and use of Theorem 1 to give X 2 = X 3 sp{x 2 }, shows that for all x ∈ X,

An obvious extension of this argument leads to
Note that even though the projections P X k and P Y k are nonlinear, in general, nevertheless the linearity of S X n and S Y n means that the products Lemma 5. Let n ∈ N and let K n be the convex hull of x 1 , . . ., x n , where the x i are defined in Section 2. Then The same holds with each x i replaced by y i .
In the same way it follows that for all l ∈ N, More generally we have, for each l ∈ N with l ≤ n, where the infimum on the left-hand side is taken over all α i ≥ 0 such that n i=1 α i = 1.When n = 2 we have to find inf 0≤t≤1 max(t, 1 − 2t).Since max(t, 1 − 2t) = 1 − 2t, 0 ≤ t ≤ 1/3, t, 1/3 ≤ t ≤ 1, we see that the infimum is attained when the two entries t and 1 − 2t are equal, that is, when t = 1/3, as otherwise one of the two entries would be greater than this value.For a general value of n, we need to find we can decrease α i by a sufficiently small ε > 0 while increasing each α j with j ∈ {i + 1, . . ., n} by ε/(n−i): this leads to a decrease of α i −α i−1 −• • •−α 1 by ε and an increase of every term α j −α j−1 −• • •−α 1 with j > i. Considerations like this show that as before the infimum is attained when the entries coincide: α 1 = α 2 − α 1 , so that α 2 = 2α 1 , α 3 = 2α 2 = 2 2 α 1 and so on up to α n−1 = 2 n−2 α 1 .Moreover, and establishes the claimed lower bound of the lemma.As the upper bound is obvious, the proof is complete.The y-version follows in exactly the same way.
We note that since the lower estimate in this lemma was obtained by quite crude means, there is a distinct possibility of improvement in particular spaces, such as L p (1 < p < ∞).
Next we show how an estimate involving the maps S Y n can be obtained when the λ k decay sufficiently rapidly.Lemma 6. Suppose that λ k ≤ 2 −k+1 for all k ∈ N. Then for all y ∈ T (B X ) and all n ∈ N, Hence y 1 := (y + y 0 )/2 ∈ T (B X ).As y 1 = 1 2 P Y 2 (y), we see that For if y ∈ Y 2 ∩ T (B X ), then y = T x for some x ∈ B X , and in view of the decomposition (2.3), x = u + v for some u ∈ X 2 and v ∈ Z X 1 , v = αx 1 , say.Thus y = T u + αT x 1 = T u + αλ 1 y 1 , so that αλ 1 y 1 = y − T u ∈ Y 2 .By the Y -form of (2.3) it follows that α = 0 and x ∈ X 2 , proving the claim.Thus if λ 2 ≤ 2 −1 , then sup{ y Y : y ∈ P Y 2 T (B X )} ≤ 1.Further use of Lemma 4 shows, by similar techniques, that More generally, the same procedure shows that for any n Together with Lemma 4 this finishes the proof.
From this it follows that the S Y n , regarded as maps from the closure in Y of T (X) to itself, have uniformly bounded norms.However, all this is based on the assumption that the λ n decay quickly enough, and it would clearly be desirable if the result were to hinge instead on the decay of more familiar objects associated with T. The Gelfand numbers come immediately to mind (see, for example, [14]): recall that the nth Gelfand number of T is c n (T ) := inf{ T J X M : codim M < n} (n ∈ N), where J X M is the natural embedding from the closed linear subspace M of X into X.It is plain that Companion to these numbers are the Gelfand widths c n (T ) defined by where the infimum is taken over all closed linear subspaces L n of Y with codimension at most n − 1; equivalently, Clearly c n (T ) ≤ λ n and c n (T ) ≤ c n (T ) for every n.However, to be able to use Lemma 6 we need inequalities in the reverse direction, and to establish these we proceed as follows.
Lemma 7.For every n ∈ N, Proof.Let n ∈ N. Then T (B X ) contains the convex hull H n of ±λ n y 1 , . . . . . ., ±λ n y n .Let ε > 0. We claim that , so that if the claim were false, the definition of c n (T ) would be contradicted and then ĉn (T ) ≤ λ n .By Lemma 5 applied to the y i , We remind the reader of our standing assumption in this section that both X and Y are uniformly convex and uniformly smooth, and that the compact map T should have trivial kernel and range dense in Y .For the Gelfand numbers we need the following result established in [8]: We emphasise that this result is not generally true for arbitrary compact maps between arbitrary Banach spaces: the proof uses in a crucial way the uniform convexity and smoothness of the spaces together with the assumptions that T (X) is dense in Y and the kernel of T is trivial.
From this and Lemma 6 we immediately have As mentioned in the Introduction, the rapid decay imposed on the Gelfand numbers of T in Lemma 9 implies that T is nuclear.Indeed, since the approximation numbers a n (T ) of T satisfy a n (T ) ≤ 2 √ n c n (T ) (see Pietsch [14, Section 6.2.3.14]), it follows that (a n (T )) n∈N ∈ l 1 ; the nuclearity of T now results from [15].To begin with, we claim that if x ∈ X and T x ∈ Y n+1 , then x ∈ X n+1 .For by (2.3), x = x Z + x X , where x Z ∈ Z X n and x X ∈ X n+1 .Since T (Z X n ) ⊂ Z Y n and T (X n+1 ) ⊂ Y n+1 , we see that if x / ∈ X n+1 , then x Z = 0 and y := T x = T (x Z ) + T (x X ), where T (x Z ) = 0 as ker(T ) = {0}.Since T (x Z ) ∈ Z Y n and T (x X ) ∈ Y n+1 , application of the decomposition of Y corresponding to (2.3) shows that T x / ∈ Y n+1 , and we have a contradiction.Now suppose that there exists z ∈ ( ∞ n=1 Y n )∩T (X), z = 0. Then z = T x for some x = 0. Since the X n are decreasing and have trivial intersection, there exists N ∈ N such that for all n > N, x / ∈ X N .But T x = z ∈ Y n for all n ∈ N, so that by our claim, x ∈ X n for all n ∈ N.This contradiction establishes (3.2).
To finish, simply note that from which the result follows directly.

2 .
It follows by induction that S * k and I * − S * k have ranges M k−1 and Z 0 k−1 respectively, and hence X