Compact and Limited Operators
aa r X i v : . [ m a t h . F A ] O c t COMPACT AND LIMITED OPERATORS.
M. BACHIR ∗ , G. FLORES † AND S. TAPIA-GARC´IA † Abstract.
Let T : Y → X be a bounded linear operator between two normed spaces. We characterizecompactness of T in terms of differentiability of the Lipschitz functions defined on X with values in anothernormed space Z . Furthermore, using a similar technique we can also characterize finite rank operators interms of differentiability of a wider class of functions but still with Lipschitz flavour. As an application weobtain a Banach-Stone-like theorem. On the other hand, we give an extension of a result of Bourgain andDiestel related to limited operators and cosingularity. Keywords:
Compact Operator, Limited operator, Gˆateaux differentiability, Fr´echet differentiability, co-Lipschitz function, Delta-convex function. 1.
Introduction
It is a well-known fact that differentiability in the sense of different bornologies implies distinct propertiesof the functions depending on the chosen bornology. In this sense, the most common bornologies are thoseof finite, compact and bounded sets. Each one of them is related to some type of differentiability, moreprecisely Gˆateaux, Hadamard and Fr´echet, respectively (see [19]). It is shown in [4] that we can characterizelimited operators in terms of differentiability of convex functions via the composition with the operator (seeTheorem 1.3 below). To understand the mentioned result, recall the following definitions introduced byBourgain and Diestel in [8].
Definition 1.1.
Let X be a Banach space. A subset A of X is called limited if p n ∗ ⇀ ⇒ lim n →∞ sup x ∈ A |h p n , x i| = 0 , that is, if every weak- ∗ null sequence converges uniformly on A . Definition 1.2.
Let X , Y be Banach spaces and T : Y → X a bounded linear operator. T is called limitedif T ( B ) is limited for every bounded B ⊂ Y . Equivalently, if k T ∗ p n k n →∞ −→ whenever p n ∗ ⇀ . We know that every relatively compact subset of X is limited, but the converse is false in general. Foruseful properties of limited sets and limited operators we refer to [8]. Considering this, the result in [4] goesas follows. Theorem 1.3.
Let X , Y be Banach spaces, U be a convex open subset of X and T : Y → X be a boundedlinear operator. Then, T is limited if and only if for every convex continuous function f : U → R , f ◦ T isFr´echet differentiable at y ∈ Y whenever f is Gˆateaux differentiable at T y ∈ U . In this sense, a limited operator transforms (for convex functions) Gˆateaux differentiability (the weakertype) into Fr´echet differentiability (the stronger type) via composition. But considering this, we can gofurther. In this article we prove (see Theorem 3.1 and Theorem 3.10) an analogue to Theorem 1.3, by replacingconvex continuous functions (resp. limited operator) by Lipschitz functions (resp. compact operators). Wealso extend this idea to characterize finite-rank operators in the same spirit (see Theorem 5.3). Another wayto express these results is the fact that in infinite dimensions, what prevents a continuous convex function f : X → R which is Gˆateaux differentiable at some point, from being Fr´echet differentiable at the samepoint, is the fact that the identity operator on X is not limited, whereas what prevents a general Lipschitz Date : 8th July 2019.1991
Mathematics Subject Classification. ∗ , G. FLORES † AND S. TAPIA-GARC´IA † function which is Gˆateaux differentiable to be Fr´echet differentiable is the fact that the identity on X is nota compact operator. The non-compactness of the identity operator in infinite dimension is the well knownRiesz theorem. On the other hand, the fact that the identity operator is not limited in infinite dimensionhas been discovered independently by Josefson in [15] and Nissenzweig in [18].This paper is organized as follows. In Section 2, we give some useful notation and definitions. In Section 3,we prove our first main result Theorem 3.1 (and its smooth version Theorem 3.10) which gives a connectionbetween compact operators and differentiability of Lipschitz functions. As an application, we obtain aBanach-Stone-type theorem in Theorem 3.12. In Section 4, we give our second main result Theorem 4.5,which is an extension of the Bourgain-Diestel result in [8]. In Section 5, we prove our third main resultTheorem 5.3, which gives a connection between finite-rank operators and differentiability of finitely Lipschitzfunctions (see Definition 5.1). 2. Some Notation and Definitions.
Throughout this paper X , Y and Z always denote normed spaces. By B ( x, r ) and B ( x, r ) we denote theopen and closed balls centered at x and of radius r , respectively, while B X and B X denote the open andclosed unit ball, respectively, and by S X its unit sphere. We write explicitly the underlying space if needed. If A is a subset of X , by A c we denote the complement of A in X , that is X \ A . If V ⊂ X and f : U ⊂ X → Z ,then we denote f | V as the restriction of f to U ∩ V . Lipschitz functions and some variations will be oftenused. Recall the following definitions.
Definition 2.1.
Let X and Z be normed spaces. We say that a function f : U ⊂ X → Z is locally Lipschitz iffor every u ∈ U there exist constants r, L > such that k f ( x ) − f ( y ) k ≤ L k x − y k for every x, y ∈ U ∩ B ( u, r ) .When there exists a constant L > such that the last inequality is valid for every x, y ∈ U , we say that thefunction is Lipschitz. We denote by Lip(
U, Z ) the linear space of Lipschitz functions from U to Z and by Lip x ( U, Z ) the subspaceof functions f ∈ Lip(
U, Z ) which vanish at some fixed point x ∈ U . The space Lip x ( U, Z ) endowed withthe norm k f k Lip := sup x,y ∈ U,x = y k f ( x ) − f ( y ) kk x − y k is a Banach space. In the case Z = R , we simply write Lip( U ) and Lip x ( U ).The set ( C b ( X ) , k · k ∞ ) denotes the Banach space of all real-valued bounded continuous functions on X ,equipped with the sup-norm. By C Gb ( X ) we denote the space of all bounded, Lipschitz, Gˆateaux-differentiablefunctions f from X into R . Let C Fb ( X ) be the space of all bounded, Lipschitz, Fr´echet-differentiable functions f from X into R . These spaces are equipped with the norm k f k = max( k f k ∞ , k f ′ k ∞ ). Recall that by themean value theorem, we have for every f ∈ C Gb ( X ) that k f ′ k ∞ = sup x,x ∈ X ; x = x | f ( x ) − f ( x ) |k x − x k = k f k Lip . The spaces C Gb ( X ) and C Fb ( X ), endowed with the mentioned norm, are Banach spaces (See, [10]).3. Differentiability and compact operators
In this section, we present a result in the line of [4, Theorem 1], which is a nice characterization of limitedoperators in terms of the differentiability of convex functions. In our case, we give a characterization ofcompact operators in terms of differentiability of Lipschitz functions. The main result of this paper reads asfollows.
Theorem 3.1.
Let X and Y be Banach spaces, U ⊂ X an open set and T : Y → X a bounded linearoperator. Then T is compact if and only if for every Banach space Z and every locally Lipschitz function f : U → Z , f ◦ T is Fr´echet differentiable at y ∈ Y whenever f is Gˆateaux differentiable at T y ∈ U . OMPACT AND LIMITED OPERATORS. 3
The ideas for the proofs of Theorem 3.1 (and also of Theorem 5.3) are motivated by the following example,in which we find a Lipschitz function f : ℓ ∞ ( N ) → R which is Gˆateaux-differentiable at 0 and its restrictionto c ( N ) is not Fr´echet-differentiable at 0. This type of function cannot exist if the function f is assumed tobe convex instead of merely Lipschitz, since the canonical isometry of c ( N ) into ℓ ∞ ( N ) is a limited operator(see Theorem 1.3 and [4]).Recall that the core of a set A ⊂ X is the subset core( A ) ⊂ A defined by the points x ∈ A such that forall y ∈ X , there exists t y > x + ty ∈ A for all t ∈ [0 , t y ). It is easily seen that int( A ) ⊂ core( A ).Let f : X → R be a function. By supp( f ) we denote the set { x ∈ X : f ( x ) = 0 } .We need the following elementary result. Proposition 3.2.
Let f : X → R be a function such that ∈ core( X \ supp( f )) . Then the function f isGˆateaux-differentiable at and d G f (0) = 0 .Proof. Take any h ∈ X . Since 0 ∈ core( X \ supp( f )), there exists δ > th ∈ X \ supp( f )for every | t | < δ , which implies f ( th ) = 0 for every | t | < δ . From this we conclude directly that f isGˆateaux-differentiable at 0, with d G f (0) = 0. (cid:3) Example 3.3.
There exists a Lipschitz function f : ℓ ∞ ( N ) → R which is Gˆateaux-differentiable at 0 and itsrestriction to c ( N ) is not Fr´echet-differentiable at 0. Proof.
Consider the sequence ( y n ) n ⊂ ℓ ∞ ( N ) defined by y n = n e n , where e n is the n -th canonical coordinate.Consider the set C n = B ( y n , n ) c and C = ∩ n C n . Consider the function f : ℓ ∞ ( N ) → R defined by f ( x ) = d ( x, C ) = inf {k x − z k : z ∈ C } , which is known to be 1-Lipschitz. Since 0 belongs to the core of C , by Proposition 3.2, we get that f isGˆateaux-differentiable at 0 and its differential is d G f (0) = 0. To show that the restriction of f to c is notFr´echet-differentiable at 0, it suffices to notice that ( y n ) n ⊂ c andlim inf n →∞ f ( y n ) − f (0) k y n k ≥ lim inf n →∞ / n /n = 13 = 0 . (cid:3) In the construction of the function in the above example, we used without further detail that the unitvectors of ℓ ∞ ( N ) are “sufficiently separated” (meaning that they are at distance 1 to each other) and theybelong to the image of the canonical injection ι : c ( N ) → ℓ ∞ ( N ), which is not compact. For a proof in ageneral setting, we need to adapt this idea to obtain a sequence of vectors with the aforementioned propertyand use them in order to obtain a Lipschitz function that has the required property.Before proceeding with the proof of our result, we need the following useful proposition. Proposition 3.4.
In the context of Theorem 3.1, the following are equivalent (1)
For every locally Lipschitz function f : U → Z it holds that f ◦ T is Fr´echet-differentiable at y ∈ Y whenever f is Gˆateaux-differentiable at T y ∈ U . (2) For every Lipschitz function f : X → Z it holds that f ◦ T is Fr´echet-differentiable at y ∈ Y whenever f is Gˆateaux-differentiable at T y ∈ X . (3) For every Lipschitz function f : X → Z it holds that f ◦ T is Fr´echet-differentiable at ∈ Y whenever f is Gˆateaux-differentiable at ∈ X . (4) For every Lipschitz function f : X → R it holds that f ◦ T is Fr´echet-differentiable at ∈ Y whenever f is Gˆateaux-differentiable at ∈ X .Proof. It suffices to prove that (4) implies (3) (implications (1) = ⇒ (2) = ⇒ (3) = ⇒ (4) are trivial, while(2) = ⇒ (1) and (3) = ⇒ (2) are simple by noticing that differentiability notions are local and using translations,respectively). M. BACHIR ∗ , G. FLORES † AND S. TAPIA-GARC´IA † Suppose that f : X → Z is a Lipschitz function which is Gˆateaux-differentiable at 0 and f ◦ T is notFr´echet-differentiable at 0. Consider g : X → R given by g ( x ) = k f ( x ) − f (0) − d G f (0) x k (which is triviallyLipschitz). We show that g is Gˆateaux-differentiable at 0 and g ◦ T is not Fr´echet-differentiable at 0, whichleads to a contradiction. For h ∈ S X and t > g ( th ) − g (0) t = k f ( th ) − f (0) − td G f (0) h k t t → −→ , which implies that g is Gˆateaux-differentiable at 0, with d G g (0) = 0. But for t > h ∈ S X | ( g ◦ T )( th ) − ( g ◦ T )(0) − t ( d G g (0) ◦ T ) h | t =sup h ∈ S X k ( f ◦ T )( th ) − ( f ◦ T )(0) − t ( d G f (0) ◦ T ) h k t which does not converge to 0 whenever t goes to 0, since f ◦ T is not Fr´echet-differentiable at 0. (cid:3) To get the results, it suffices to work with statement (4) in Proposition 3.4.3.1.
Characterization of compactness.
We begin this subsection with a definition and two results thatconsist of a central part of the proof of Theorem 3.1. Recall that for a convex set A , the generated conescone { A } is the set { λa : λ > , a ∈ A } . Definition 3.5.
Let ( x n ) n ⊂ X be a bounded sequence. We say that ( x n ) n is β -separated if k x n k = k x m k for every n, m ∈ N and there exists β > such that k x n − x m k ≥ β if n = m . Furthermore, we say that thesequence ( x n ) n is β -cone-separated if the pairwise intersection of the generated cones cone { B ( x n , β ) } is { } . In the case of ℓ ∞ ( N ), the canonical sequence of coordinates ( e n ) n is 1-separated and ( − ε )-cone separated,with ε ∈ (0 , ), and as the next proposition shows, this is not an special behaviour. Proposition 3.6.
Let ( x n ) n ⊂ X be a sequence which is β -separated, then it is β/ -cone-separated.Proof. Let x ∈ X \ { } and 0 < α < k x k . We will first show that the set P α ( x ) := (cid:26) k x kk y k y : y ∈ B ( x, α ) (cid:27) is contained in B ( x, α ). Let y ∈ B ( x, α ), then: (cid:13)(cid:13)(cid:13)(cid:13) x − k x kk y k y (cid:13)(cid:13)(cid:13)(cid:13) ≤ k x − y k + (cid:13)(cid:13)(cid:13)(cid:13) y − k x kk y k y (cid:13)(cid:13)(cid:13)(cid:13) = k x − y k + |k y k − k x k| ≤ k x − y k < α. Now if ( x n ) n is a β -separated sequence, consider for each n ∈ N the sets P β/ ( x n ). With this, P β/ ( x n ) ⊂ B ( x n , β ) for each n ∈ N which by definition of β -separated implies that the family ( P β/ ( x n )) n is pairwisedisjoint. Suppose that there exists n = m such thatcone { B ( x n , β/ } ∩ cone { B ( x m , β/ } 6 = { } and take y = 0 belonging to this intersection. By definition, there exist a, b > y ∈ B ( x n , β/ y ∈ B ( x m , β/
4) such that y = ay = by . Since k x n k = k x m k , we see that k x n kk y k y = k x m kk y k y Equivalently, k x n kk y k y = k x m kk y k y . This implies that P β/ ( x n ) ∩ P β/ ( x m ) = ∅ , which is a contradiction. We conclude that ( x n ) n is β/ (cid:3) OMPACT AND LIMITED OPERATORS. 5
Lemma 3.7.
Let T : Y → X be a bounded operator which is not compact. Then there exists β > and a β -separated sequence ( x n ) n in X such that x n = T y n and ( y n ) n is bounded on Y .Proof. Since
T B Y is not a relatively compact subset of X , there exists a sequence ( z n ) n ⊂ T B Y withoutcluster points. Without lose of generality, we can assume that for some α > n ∈ N , k z n k ≥ α .Let us define the sequence ( x n ) n ⊂ X by x n := αz n / k z n k , for all n ∈ N . Notice that ( x n ) n ⊆ T B Y . Since( z n ) n does not have cluster points, then neither does ( x n ) n . With this, since A = { x n : n ∈ N } ⊂ T B Y ∩ αS X is a not relatively compact, bounded set, it cannot be totally bounded. Thus, there exists β > A cannot be covered by finitely many balls of radius β . Finally, leaving out eventually some points of A , wecan construct a β -separated sequence in T B Y ∩ αS X . (cid:3) Remark 3.8.
In the beginning of the proof of Lemma 3.7 we used that
T B Y is not compact. However, theresult is more general. In fact, in the same way we can prove that if Z is an infinite dimensional subspace of X , then for some β > β -separated sequence in Z . In the case that T is compact, we will finda β -separated sequence in T Y if and only if
T Y is infinite dimensional. However, in this case the associatedsequence ( y n ) n cannot be bounded.Now we are able to present the proof of the main theorem. Proof of Theorem 3.1.
In [6, Lemma 3.1], M. Bachir and G. Lancien proved the necessity, which we includefor the sake of completeness. Suppose that T is compact, Z is a Banach space and f : U ⊂ X → Z is locallyLipschitz and Gˆateaux differentiable at T y . Since f is locally Lipschitz and Gˆateaux differentiable at T y , wededuce that it is Hadamard differentiable at
T y . Then, if d H f ( T y ) denotes the Hadamard differential of f at T y we will have for t > h ∈ B Y k ( f ◦ T )( y + th ) − ( f ◦ T )( y ) − t ( d H f ( T y ) ◦ T ) h k| t | = sup k ∈ T B Y k f ( T y + tk ) − f ( T y ) − td H f ( T y ) k k| t | . From this, since
T B Y is compact, we deduce that if we take limit of t → f ◦ T is Fr´echet differentiable at y , being its Fr´echet differential equal to d H f ( T y ) ◦ T .On the other hand, for the sufficiency we proceed by contradiction. Applying Proposition 3.4 we willconstruct a Lipschitz function f : X → R . Assume that T : Y → X is a noncompact bounded operator. Let( x n ) n := ( T y n ) n ⊂ T B Y be a β -separated sequence given by Lemma 3.7, with ( y n ) n ⊂ B Y . For each n ∈ N ,consider the set C n ⊂ X defined by C n = B ( x n n , β n ) c = X \ B ( x n n , β n ) and C := ∩ n C n . By Proposition 3.6,0 belongs to the core of C since each line passing through 0 intersects at most one set of the sets B ( x n n , β n ).By Proposition 3.2, the function f : X → R defined by f ( x ) = d ( x, C ) (which is 1-Lipschitz) is Gˆateauxdifferentiable at 0 with differential d G f (0) = 0. However, we notice that:lim inf n →∞ ( f ◦ T )( y n /n ) − ( f ◦ T )(0) k y n /n k = lim inf n →∞ β n − k y n /n k ≥ β n {k y n k} > , which shows that f ◦ T is not Fr´echet differentiable at 0, since the sequence ( y n /n ) n goes to 0. (cid:3) As consequence, we obtain that in infinite dimentional Banach space Y , the set of all Lipschitz continuousfunctions that vanish at 0, which are Gˆateaux differentiable but not Fr´echet differentiable at 0, contains asubspace isometric to ℓ ∞ ( N ). More generally, we have the following corollary. Corollary 3.9.
Let T : Y → X be a non compact bounded operator. Let F ⊂ Lip ( X ) be the set defined asfollows: f ∈ F if and only if f is Gˆateaux differentiable at and f ◦ T is not Fr´echet differentiable at or f ≡ . Then, F contains a subspace isometric to ℓ ∞ ( N ) . M. BACHIR ∗ , G. FLORES † AND S. TAPIA-GARC´IA † Proof.
Consider as before a β -separated sequence ( x n ) n ⊂ T B Y and one of its asociated sequences on Y ; ( y n ) n ⊂ B Y such that T y n = x n . For every p ∈ N prime define the sets C p,n = B ( x pn p n , β p n ) c and C p := ∩ n C p,n . Just as before, the functions f p ( x ) = d ( x, C p ) are 1-Lipschitz, Gˆateaux differentiable at 0and such that the compositions f p ◦ T are not Fr´echet differentiable at 0. In the following ( p i ) i stands foran enumeration of the prime numbers. We have that ( f p i ) i ⊂ Lip ( X ) are linearly independent, since theirsupports are disjoint. Moreover, if µ ∈ ℓ ∞ ( N ), the function f µ ( x ) := sup i ∈ I + µ i f p i ( x ) − sup i ∈ I − − µ i f p i ( x ) , where I + = { n ∈ N : µ n ≥ } and I − = N \ I + , is well defined and k µ k ∞ -Lipschitz. That is, the operator L : ℓ ∞ ( N ) → Lip ( X ) given by Lµ = f µ is an isometry. By Proposition 3.6, is easy to see that 0 ∈ core ( ∩ i C p i )and by Proposition 3.2, Lµ is Gˆateaux differentiable at 0. But if µ = 0, f µ is not Fr´echet differentiable at 0,since if µ k = 0, thenlim inf n →∞ ( f µ ◦ T )( y p nk /p nk ) − ( f µ ◦ T )(0) k y p nk /p nk k = lim inf n →∞ f p k ( x p nk /p nk ) − f p k (0) k y p nk /p nk k≥ lim inf n →∞ β p nk − k y p nk /p nk k ≥ β n {k y n k} > . (cid:3) This corollary says that the set F defined there is c -lineable, meaning that it contains an isometric copyof a normed space of dimension c . More on lineability and spaceability can be found in [2], [1], [14] and [9].An interesting case in this framework is whenever the space X admits a smooth bump function. Weunderstand by a bump function a nonnegative continuous function b : X → R different from 0 and withbounded support. The existence of a bump function b ∈ C Gb ( X ) (resp b ∈ C Fb ( X )) is not always trivial andrequires in general some geometrical properties on the underlying Banach space X . For more informationabout smooth bump functions in infinite dimensional Banach spaces, we refer to the book of Deville, Godefroyand Zizler [11] and the survey [13]. Theorem 3.10.
Let X and Y be Banach spaces, U ⊂ X an open set and T : Y → X a bounded linearoperator. Assume that X admits a Lipschitz Gˆateaux differentiable bump function b : X → R . Then T iscompact if and only if for every bounded Lipschitz, everywhere Gˆateaux differentiable function f : U → R , f ◦ T is everywhere Fr´echet differentiable.Proof. The necessity is a particular case of Theorem 3.1. On the other hand, we assume that T is a non-compact operator and we will construct a bounded Lipschitz and Gˆateaux differentiable function f : X → R such that f ◦ T fails to be Fr´echet differentiable. Let b : X → R a bump Lipschitz, Gˆateaux differentiablefunction such that b (0) > b ) ⊂ B X . Let ( x n ) n = ( T y n ) n ⊂ T B Y be a β -separated sequencegiven by Lemma 3.7, with ( y n ) n ⊂ Y a bounded sequence. For each n ∈ N , we define b n : X → R by b n ( · ) = b ( n ( · − x n )) /n and f = P ∞ i =0 b n . Since the functions { b n : n ∈ N } have pairwise disjoint supportand have uniformly bounded Lipschitz constant, f is Lipschitz. Moreover, by construction f is Gˆateauxdifferentiable on X . Now, the proof follows in a similar way as Theorem 3.1, leading to that f is not Fr´echetdifferentiability at 0. We leave the details to the reader. (cid:3) Application to a Banach Stone Like Theorem.
The following definition was introduced in [5] (seealso [3]) and the following axioms in [3].
Definition 3.11. (The property P F ) Let ( X, d ) be a complete metric space and ( A, k · k ) be a closedsubspace of C b ( X ) (the space of all real-valued bounded continuous functions on X ). We say that A has theproperty P F if, for each sequence ( x n ) n ⊂ X , the two following assertions are equivalent: (1) The sequence ( x n ) n converges in ( X, d ) . OMPACT AND LIMITED OPERATORS. 7 (2)
The associated sequence of the Dirac masses ( δ x n ) n converges in ( A ∗ , k · k ∗ ) , where the Dirac massassociated to a point x ∈ X , is the linear continuous form δ x : ϕ ϕ ( x ) for each ϕ ∈ A . Axioms.
Let (
X, d ) be a complete metric space and A be a space of functions included in C b ( X ). We saythat the space A satisfies the axioms ( A )-( A F ) if the space A satisfies:( A ) The space ( A, k · k ) is a Banach space such that k · k ≥ k · k ∞ ,( A ) The space A contain the constants,( A ) For each n ∈ N there exists a positive constant M n such that for each x ∈ X there exists a function h n : X → [0 ,
1] such that h n ∈ A , k h n k ≤ M n , h n ( x ) = 1 and diam(supp( h n )) < n +1 . This axiom implies inparticular that the space A separate the points of X ,( A F ) the space A has the property P F .A simple adaptation of the proof in [5, Proposition 2.5.], shows that the spaces C Gb ( X ) and C Fb ( X ) havethe property P F for every Banach space X . In addition, we assume that these spaces contain a bump functionrespectively, then they will satisfy the axiom ( A ). Thus, the spaces C Gb ( X ) and C Fb ( X ) satisfy the axioms( A )-( A F ) whenever they contain a bump function respectively, and so we can apply the extension of theBanach-Stone theorem established in [3, Corollary 1.3.]. This is what we are going to do in Theorem 3.12,using our previous Theorem 3.10. Theorem 3.12.
Let X and Y be Banach spaces having a bump function in C Gb ( X ) and C Fb ( Y ) respectively(see [10] ). Then, the following assertions are equivalent. (1) There exists an isomorphism
Φ : C Gb ( X ) → C Fb ( Y ) such that k Φ( f ) k ∞ = k f k ∞ and k (Φ( f )) ′ k ∞ = k f ′ k ∞ for all f ∈ C Gb ( X )(2) X and Y are isometrically isomorphic and of finite dimension. The proof will be given after the following lemma.
Lemma 3.13.
For every a, b ∈ X , we have k a − b k = sup f ∈ C Fb ( X ) \{ } , k f ′ k ∞ > | f ( a ) − f ( b ) |k f ′ k ∞ = sup f ∈ C Gb ( X ) \{ } , k f ′ k ∞ > | f ( a ) − f ( b ) |k f ′ k ∞ Proof.
By the Hanh-Banach theorem, there exists p a,b ∈ B X ∗ such that be k a − b k = p a,b ( a − b ). For each ω >
0, let α ω : R → R such that α ω ∈ C Fb ( R ), 1-Lipschitz and α ω ( t ) = t if | t | ≤ ωω + 1 if t ≥ ω + 2 − ω − t ≤ − ω − f ω ( x ) = α ω ◦ p a,b ( x ), for all x ∈ X . We have that f ω ∈ C Fb ( X ) and is 1-Lipschitzfor every ω >
0. By choosing ω ≥ k a k , k b k ) | f ω ( a ) − f ω ( b ) | = | α ω ◦ p a,b ( a ) − α ω ◦ p a,b ( b ) | = | p a,b ( a ) − p a,b ( b ) | = | p a,b ( a − b ) | = k a − b k . M. BACHIR ∗ , G. FLORES † AND S. TAPIA-GARC´IA † Therefore, k f ′ ω k ∞ = 1. It follows that k a − b k ≥ sup f ∈ C Gb ( X ) \{ } , k f ′ k ∞ > | f ( a ) − f ( b ) |k f ′ k ∞ ≥ sup f ∈ C Fb ( X ) \{ } , k f ′ k ∞ > | f ( a ) − f ( b ) |k f ′ k ∞ ≥ | f ω ( a ) − f ω ( b ) | = k a − b k (cid:3) Proof of Theorem 3.12.
Since Φ is an isomorphism for the norm k · k ∞ , then from [3, Corollary 1.3.], thereexists an homeomorphism T : Y → X and a continuous function ǫ : Y → {± } such that Φ( f )( y ) = ǫ ( y ) f ◦ T ( y ) for all f ∈ C Gb ( X ) and all y ∈ Y . Since the space Y is a connected space, we have that ǫ = 1 or ǫ = −
1. Replacing Φ by − Φ if necessary, we can assume without loss of generality that Φ( f ) = f ◦ T for all f ∈ C Gb ( X ). We are going to prove that T is an isometry. Let y , y ∈ Y . Using Lemma 3.13 and the factthat k (Φ( f )) ′ k ∞ = k f ′ k ∞ for all f ∈ C Gb ( X ), we have k T ( y ) − T ( y ) k = sup f ∈ C Gb ( X ) \{ } , k f ′ k ∞ > | f ( T ( y )) − f ( T ( y )) |k f ′ k ∞ = sup f ∈ C Gb ( X ) \{ } , k f ′ k ∞ > | f ◦ T ( y ) − f ◦ T ( y ) |k f ′ k ∞ = sup f ∈ C Gb ( X ) \{ } , k f ′ k ∞ > | Φ( f )( y ) − Φ( f )( y ) |k (Φ( f )) ′ k ∞ = sup g ∈ C Fb ( Y ) \{ } , k g ′ k ∞ > | g ( y ) − g ( y ) |k g ′ k ∞ = k y − y k Thus, T : Y → X is a surjective isometry. From Mazur-Ulam theorem [22], T is an affine map, equivalently T − T (0) is linear. Finally, T − T (0) is a linear surjective isometry from Y onto X . So X and Y are isometricallyisomorphic. On the other hand, since f ◦ T ∈ C Fb ( Y ), whenever f ∈ C Gb ( X ), then T − T (0) is a compactoperator by Theorem 3.10, which implies thanks to Riesz lemma that X and Y are finite dimensional. Thus, X and Y are finite dimensional and isometrically isomorphic. The converse is clear. Indeed, since Gˆateauxand Fr´echet differentiability coincides for Lipschitz functions in finite dimensional Banach space, we have that C Gb ( X ) = C Fb ( X ). On the other hand, if T : Y → X is an isometric isomorphism, then the operator givenby Φ( f ) = f ◦ T is an isomorphism between C Fb ( X ) and C Fb ( Y ) satisfying the two desired conditions. (cid:3) Proposition 3.14.
Let X , Y be Banach spaces and T : Y → X be a bounded linear operator. Then, thefollowing assertions are equivalent. (1) T is compact operator with dense range (2) The operator
Φ : C Gb ( X ) → C Fb ( Y ) defined by Φ( f ) = f ◦ T is a well-defined injective bounded linearoperator.Proof. Suppose that T is compact, then f ◦ T ∈ C Fb ( Y ) whenever f ∈ C Gb ( X ) by [6, Lemma 3.1.], so Φmaps C Gb ( X ) into C Fb ( Y ). By the density of the range of T , Φ is injective. Then, it is clear that Φ is abounded linear operator satisfying k Φ( f ) k ∞ ≤ k f k ∞ and k (Φ( f )) ′ k ∞ ≤ k f ′ k ∞ k T k for all f ∈ C Gb ( X ). Forthe converse, since Φ maps C Gb ( X ) into C Fb ( Y ), then by Theorem 3.10 the operator T is compact. Supposeby contradiction that T ( Y ) = X . There exists x ∈ X such that x T ( Y ). By the Hanh-Banach theorem,there exists a linear continuous form p ∈ X ∗ such that p ( x ) = 1 and p T ( Y ) = 0. Let α : R → R be such that OMPACT AND LIMITED OPERATORS. 9 α ∈ C Fb ( R ) and α ( t ) = t − ≤ t ≤
24 if t ≥
40 if t ≤ f ( x ) = α ◦ p . Thus, f ∈ C Gb ( X ) and we have f ◦ T = 0. Thus, Φ( f ) = Φ(0) = 0 but f = 0since f ( x ) = 1. This contradict the injectivity of Φ. (cid:3) Limited operators and co-Lipschitz delta-convex mappings
We recall the following definition introduced in [7] and studied in several papers (see for instance [17] and[20]).
Definition 4.1.
Let X and E be two metric spaces and f : X → E be a mapping. We say that f isco-Lipschitz, if there exists a constant c > such that for all x ∈ X and all r > we have B ( f ( x ) , cr ) ⊂ f ( B ( x, r )) . The co-Lipschitz property is intimately related with the metric regularity of functions, since if a function f : X → E is co-Lipschitz, then is metrically regular near ( x, f ( x )) for every x ∈ X . The definition (andmore) of metric regularity of multivalued functions can be found in [16], [21] and references therein. We alsorecall the definition of delta-convex mapping introduced by Vesel´y and Zaj´ıˇcek in [23]. Definition 4.2.
Let X and E be two Banach spaces and h : X → E be a map. We say that h is d.cmapping, that is delta-convex mapping if and only if, there exists a convex continuous function, called acontrol function, g : X → R , such that e ∗ ◦ h + g is convex continuous for every e ∗ ∈ E ∗ . Theorem 4.3. (see [12, Theorem 4.] ) Let X and E be two Banach spaces and h : X → E be a d.c mappingwith a control function g . If g is Fr´echet-differentiable at x then, h is Fr´echet-differentiable at x . Example of Fr´echet-differentiable not linear d.c mapping.
Every linear bounded mapping is trivially aFr´echet-differentiable d.c mapping, but the class of d.c mappings with Fr´echet-differentiable control functionis larger. We know, for example, from [23, Proposition 1.11.] that if H is a Hilbert space and Y is anynormed space, then every C , mapping h : H → Y is d.c with a control function Lip( h ′ ) k · k (which isan everywhere Fr´echet-differentiable control function). For interesting results about d.c Lipschitz isomorphicnormed spaces, we refer to [12]. We need the following proposition. Proposition 4.4.
Let
X, Y be a Banach spaces and h : X → Y be a continuous d.c mapping with a controlfunction g . Let f : Y → R be a convex continuous function. Then, f ◦ h + g is convex continuous.Proof. Since h is d.c mapping , there exists a convex continuous function g : X −→ R , such that e ∗ ◦ h + g isconvex continuous for every e ∗ ∈ Y ∗ . By the Fenchel theorem, there exists a family of continuous linear maps( e ∗ i ) i ∈ I and real numbers ( c i ) i ∈ I , such that f ( y ) = sup i ∈ I e ∗ i ( y ) + c i for all y ∈ Y . Thus, f ( h ( x )) + g ( x ) =sup i ∈ I { e ∗ i ◦ h ( x ) + c i + g ( x ) } for all x ∈ X . It follows that f ◦ h + g is convex, since e ∗ i ◦ h + c i + g is convexfor each i ∈ I . On the other hand, since f, h and g are continuous, then f ◦ h + g is continuous. (cid:3) We recall that the Bourgain-Diestel theorem established in [8] says that: if a bounded linear operator T : Y → X is limited then it is strictly cosingular, that is, the only Banach spaces E for which we can find alinear bounded operator q X : X → E such that q X ◦ T is surjective, are finite dimensional. This result appliedwith the identity map, gives the well known Josefson-Nissenzweig theorem [15], [18], with another proof. Wegive below an extension of the Bourgain-Diestel theorem, where the assumption of bounded linearity of q X is replaced by the more general condition of co-Lipschitz d.c mapping. Our proof is based on Theorem 1.3and the existence of convex Gˆateaux not Fr´echet differentiable function at some point in infinite dimensional,which is a result based on the Josefson-Nissenzweig theorem. For a canonical contruction of such functions, werefeer to [4]. Thus, Theorem 1.3 together with Josefson-Nissenzweig theorem implies the following extensionof Bourgain-Diestel theorem. ∗ , G. FLORES † AND S. TAPIA-GARC´IA † Theorem 4.5. ( Extension of Bourgain-Diestel theorem ) Let T : Y → X be a bounded linear operator.Then, we have that (1) = ⇒ (2) = ⇒ (3) = ⇒ (4) . (1) T is limited operator. (2) The only Banach spaces E for which we can find a d.c mapping q X : X → E with a control function g Fr´echet-differentiable at and such that q X (0) = 0 and B E (0 , t ) ⊂ q X ◦ T ( B Y (0 , t )) for all t > ,are finite dimensional. (3) The only Banach spaces E for which we can find a d.c mapping q X : X → E with a control function g Fr´echet-differentiable at some point x and such that q X ◦ T is co-Lipschitz, are finite dimensional. (4) The only Banach spaces E for which we can find a linear bounded operator q X : X → E such that q X ◦ T is surjective, are finite dimensional.Proof. The implications (2) = ⇒ (3) = ⇒ (4) are trivial. Let us prove (1) = ⇒ (2). In fact, we are going toprove ¬ (2) = ⇒ ¬ (1). Indeed, let E be an infinite dimensional Banach space. We can find a convex Lipschitzcontinuous function f : E → R which is Gˆateaux-differentiable but not Fr´echet-differentiable at 0, with f (0) = 0 and f ′ (0) = 0 (see a construction of such a function in [4]). Since f is not Fr´echet-differentiable at0, then ( ∃ ε > ∀ δ > ∀ t, < t ≤ δ ) sup h ∈ B E (0 , t − | f ( th ) | ≥ ε. Suppose that q X : X → E is a d.c mapping with a control function g Fr´echet-differentiable at 0, such that q X (0) = 0 and B E (0 , t ) ⊂ q X ◦ T ( B Y (0 , t )), for all t >
0. Since f ′ (0) = 0, the only candidate for Fr´echet-derivative of f ◦ q X ◦ T at 0 is 0. But since q X (0) = 0 and B E (0 , t ) ⊂ q X ◦ T ( B Y (0 , t )) for all t >
0, we havefor every 0 < t ≤ δ , sup h ∈ B Y (0 , t − | f ◦ q X ◦ T ( th ) | ≥ sup h ∈ B E (0 , t − | f ( th ) | ≥ ε. This shows that f ◦ q X ◦ T is not Fr´echet-differentiable at 0. By Proposition 4.4, f ◦ q X + g is convex continuousand g is Fr´echet-differentiable at 0. Using Theorem 4.3, q X is Fr´echet-differentiable at 0, since its controlfunction g has that property. It follows that f ◦ q X + g is a convex continuous function Gˆateaux-differentiableat 0 = T (0). Suppose by contradiction that T is limited. Then, f ◦ q X ◦ T + g ◦ T is Fr´echet-differentiable at 0by [4, Theorem 1.] which implies that f ◦ q X ◦ T is Fr´echet-differentiable at 0 (since g ◦ T has that property)which is a contradiction. Hence, T is not limited. (cid:3) Corollary 4.6.
Let E be any infinite dimensional Banach space. Then, there exists no d.c mapping q : ℓ ∞ ( N ) → E with a control function g Fr´echet-differentiable at such that the restriction q | c ( N ) (to c ( N ) ) isco-Lipschitz.Proof. The proof follows from Theorem 4.5, since the canonical embedding from c ( N ) into ℓ ∞ ( N ) is a limitedoperator (see [8]). (cid:3) Remark 4.7.
Every convex Lipschitz continuous function from f : c ( N ) → R has a convex Lipschitzcontinuous extension to ℓ ∞ ( N ), it suffices to consider the function˜ f ( x ) := inf y ∈ c ( N ) { f ( y ) + L ( f ) k x − y k ∞ } , where L ( f ) denotes the Lipschitz constant of f . However, if a function f : c ( N ) → R is Gˆateaux-differentiablebut not Fr´echet-differentiable at some point a ∈ c ( N ), then f cannot have an extension to ℓ ∞ ( N ) whichis convex Lipschitz continuous and Gˆateaux-differentiable at a . This is due to the fact that the identitymapping i : c ( N ) −→ ℓ ∞ ( N ) is a limited operator, and so by [4] every restriction to c ( N ) of convexLipschitz continuous and Gˆateaux-differentiable function at a , must be Fr´echet-differentiable at a . OMPACT AND LIMITED OPERATORS. 11 Finite Rank Operators
Our final aim is to shown how the idea of the proof of Theorem 3.1 can be used to characterize finiterank operators. The forthcoming result involves the bornology generated by the bounded sets of X whichare contained in finite dimensional subspaces. To get this result, we introduce a new class of functions whichcontains all the Lipschitz functions. Definition 5.1.
We say that a function f : U ⊂ X → Z is finitely locally Lipschitz if for every finitedimensional affine subspace Y of X such that U ∩ Y = ∅ , f | ( Y ∩U ) is locally Lipschitz. If every restrictionis Lipschitz, we simply say that f is finitely Lipschitz. We denote by FLip( U , Z ) the linear space of finitelyLipschitz functions from U ⊂ X to Z and by FLip x ( U , Z ) the subspace of functions f ∈ FLip(
X, Z ) whichvanish at some fixed point x ∈ U . In the case Z = R , we simply write FLip( U ) and FLip x ( U ) . Remark 5.2.
Let X and Z be two Banach spaces, U ⊂ X be any open set and x ∈ U . In case that X isfinite dimensional, it is clear that FLip( U ) = Lip( U ) (the same holds for the local concept), but in generalthe inclusion Lip( U ) ⊂ FLip( U ) is strict (the same holds for the local concept). In fact, whenever U = X ,the equality holds if and only if X is finite dimensional. This is easily seen by noticing that if ϕ is any notbounded linear functional, then ϕ belongs to FLip( X ) \ Lip( X ). As we can see, in infinite dimension a finitelylocally Lipschitz functions does not even need to be continuous. Theorem 5.3.
Let X and Y be Banach spaces, U ⊂ X be an open set and T : Y → X be a bounded linearoperator. Then T has finite rank if and only if for every Banach space Z and every finitely locally Lipschitzfunction f : U → Z , f ◦ T is Fr´echet differentiable at y ∈ Y whenever f is Gˆateaux differentiable at T y ∈ U . Remark 5.4.
By similar arguments, we deduce a simplification of the statement of Theorem 5.3 as we did inTheorem 3.1 (using Proposition 3.4). This means that for the sufficiency, we will show the proof for U = X , Z = R and y = 0. Proof.
The necessity part goes alongs the lines of the necessity of Theorem 3.1. Let T be a finite rankoperator and let f : U ⊂ X → Z be a locally finitely Lipschitz and Gˆateaux differentiable at T y ∈ U . Since
T Y is a finite dimensional subspace of X , the function g := f | T Y is Lipschitz and Fr´echet differentiable at
T y . Then, if d F g ( T y ) denotes the Fr´echet differential of g at T y , we have thatsup h ∈ B Y k ( f ◦ T )( y + th ) − ( f ◦ T )( y ) − t ( d F g ( T y ) ◦ T )( h ) k| t | = sup u ∈ T B Y k f ( T y + tu ) − f ( T y ) − td F g ( T y )( u ) k| t | = sup u ∈ T B Y k g ( T y + tu ) − g ( T y ) − td F g ( T y )( u ) k| t | From the last line, since g is Fr´echet differentiable at T y , we deduce that the first supremum goes to 0 when t →
0. Then f ◦ T is Fr´echet differentiable at y , being its Fr´echet differential equal to d G g ( T y ) ◦ T .To prove the sufficiency we will proceed by contradiction. Suppose that T : Y → X is a bounded operatorsuch that T Y is infinite dimensional. By Remark 3.8 we have that there exists a β -separated sequence ( x n ) n in T Y . Since x n ∈ T Y , for each n there exists y n ∈ Y such that T y n = x n . Now, define the sequence ofsubsets C n = B (cid:18) x n n k y n k , β n k y n k (cid:19) c . Since ( x n ) n is a β -separated sequence, by Proposition 3.6 we deduce that the family ( C cn ) n is pairwise disjoint.With this, the functions f n : X → R given by f n ( x ) = k y n k d ( x, C n ) n ∈ N ∗ , G. FLORES † AND S. TAPIA-GARC´IA † are k y n k -Lipschitz and have pairwise disjoint support. Consider now f : X → R given by f ( x ) = sup n f n ( x ) . It is easy to see that this function is well defined.We claim that f ∈ FLip( X ). Let V be a finite dimensional affine subspace of X and suppose that V intersects infinitely many of the sets ( C cn ) n , namely ( C cn k ) k . Take any sequence ( v k ) k such that v k ∈ V ∩ C cn k .If we consider v ′ k := n k k y k k v k , we see that for k, j ∈ N β ≤ k x n k − x n j k ≤ k x n k − v ′ k k + k v ′ k − v ′ j k + k v ′ j − x n j k ≤ β k v ′ k − v ′ j k . which implies that ( v ′ k ) k ⊂ V does not have accumulation points, which is impossible because V is finitedimensional and k v ′ k k ≤ k v ′ k − x n k k + k x n k k ≤ β M, where M = k x n k (for all n ). Then, V intersects only finitely many of the sets ( C cn ) n , namely ( C cn k ) Nk =1 . Sincethe functions f n have pairwise disjoint support it follows that Lip( f | V ) ≤ max {k y n k k : k = 1 , ..., N } , whichproves the claim. Similar to the Theorem 3.1, we have that 0 belongs to the core of the complement of thesupport of f . Thus, by Proposition 3.2 we deduce that f is Gˆateaux differentiable at 0, and d G f (0) = 0.However, we notice that:lim inf n →∞ f ◦ T ( y n /n k y n k ) − f ◦ T (0) k y n /n k y n kk = lim inf n →∞ k y n k β n k y n k − n = β > , which shows that f ◦ T is not Fr´echet differentiable at 0 since the sequence ( y n /n k y n k ) n goes to 0. (cid:3) Corollary 5.5.
Let T : Y → X be a bounded operator with infinite rank. Then the set F ⊂ FLip( X ) suchthat f ∈ F if and only if f is Gˆateaux differentiable at and f ◦ T is not Fr´echet differentiable at or f ≡ ,contains a subspace algebraically isomorphic to ℓ ∞ ( N ) .Proof. This proof goes similar to Corollary 3.9 but with suitable modifications. Consider as before a β -separated sequence ( x n ) n := ( T y n ) n ⊂ T Y . Let p ∈ N be a prime number and, for n ∈ N , define the set C p,n and the functions f p n , g p : X → R by C p,n = B (cid:18) x p n n k y p n k , β n k y p n k (cid:19) c , f p n ( x ) = k y n k dist( x, C p,n ) , and g p ( x ) = sup n ∈ N f p n ( x ) . Thanks to Theorem 5.3, we know that g p belongs to FLip( X ), is Gˆateaux differentiable at 0 and g p ◦ T isnot Fr´echet differentiable at 0. Let ( p n ) n be an enumeration of prime numbers. Since ( x n ) n is β -separated,the sets { supp( g p n ) n : n ∈ N } are pairwise disjoint. In a similar way as in the proof of Corollary 3.9, we candeduce that the linear operator L : ℓ ∞ ( N ) → FLip( X ) given by Lµ ( x ) := sup i ∈ I + µ i g p i ( x ) − sup i ∈ I − − µ i g p i ( x ) , where I + = { n ∈ N : µ n ≥ } and I − = N \ I + , is well defined, injective and satisfies that for every µ ∈ ℓ ∞ ( N ) µ = 0, Lµ is Gˆateaux differentiable at 0, but Lµ ◦ T is not Fr´echet differentiable at 0. (cid:3) Remark 5.6.
By construction, none of the functions f ∈ F evoked in Corollary 5.5 are Lispchitz, butinstead, they are finitely Lipschitz. Thus, if T : Y → X is a noncompact operator, then we can find twosubspaces of FLip( X ), infinite dimensionals with uncountable Hammel basis, F and F (given by Corollary3.9 and Corollary 5.5 respectively) such that satisfy for each f ∈ F ∪ F , f is Gˆateaux differentiable at 0, f ◦ T is not Fr´echet differentiable at 0. Since we can ensure that in F there are no Lipschitz function exceptfrom 0, F ∩ F = { } . OMPACT AND LIMITED OPERATORS. 13
Open question 5.7.
In view of [4, Theorem 1], Theorem 3.1 and Theorem 5.3, it seems interesting to studythe case of weak-Hadamard differentiability, and furthermore the abstract framework of β -differentiability,with β a given bornology on X . That is, since for a bornology β on X we can define a notion of differentiability:Is it true that there exists a class of functions F β such that an operator T : Y → X sends the unit ball B Y to T B Y ∈ β iff for each function f ∈ F β , f ◦ T is Fr´echet differentiable at y ∈ Y , whenever f is Gˆateauxdifferentiable at T y ? In [4] is shown that this is true whenever β is the bornology of limited sets, by using theconvex functions. We have shown the same result but for the Hadamard bornology by using the Lipschitzfunctions and moreover, we have defined the finitely Lipschitz functions for the bornology generated bybounded convex sets with finite dimensional span, which more or less corresponds to the convex version ofthe Gˆateaux bornology. Acknowledgment
The authors would like to thank A. Daniilidis for fruitful discussions. The second author was sup-ported by the grants CONICYT-PFCHA/Doctorado Nacional/2017-21170003, FONDECYT 1171854 andCMM Grant AFB170001. The third author was supported by the grants CONICYT-PFCHA/DoctoradoNacional/2018-21181905, Monge Invitation Programe of ´Ecole Polytechnique, FONDECYT 1171854 andCMM Grant AFB170001.
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Dissertationes Math. (RozprawyMat.) 289 (1989). ∗ , G. FLORES † AND S. TAPIA-GARC´IA † ∗ Laboratoire SAMM 4543, Universit´e Paris 1 Panth´eon-Sorbonne, Centre P.M.F. 90 rue Tolbiac, 75634 Pariscedex 13, France
E-mail address : [email protected] † Departamento de Ingenier´ıa Matem´atica, CMM (CNRS UMI 2807) Universidad de Chile, Beauchef 851, Chile
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