On not open linear continuous operators between Banach spaces

Note di Matematica 25, n. 1, 2005/2006, 29–34. On not open linear continuous operators between Banach spaces Angela A. Albanese Dipartimento di Matematica “E. De Giorgi” Università degli Studi di Lecce Via Per Arnesano I–73100 Lecce, Italy [email protected] Abstract. Let X and Y be infinite–dimensional Banach spaces. Let T : X → Y be a linear continuous operator with dense range and T (X) 6= Y . It is proved that, for each ε > 0, there exists a quotient map q : Y → Y1 , such that Y1 is an infinite–dimensional Banach space with a Schauder basis and q ◦ T is a nuclear operator of norm ≤ ε. Thereby, we obtain with respect to the quotient spaces the proper analogue result of Kato concerning the existence of not trivial nuclear restrictions of not open linear continuous operators between Banach spaces. As a consequence, it is derived a result of Ostrovskii concerning Banach spaces which are completions with respect to total nonnorming subspaces. Keywords: not open linear continuous operators, nuclear operators, basic sequence, biorthogonal system, quotient spaces. MSC 2000 classification: 46B20, 47A65, 47B10. Dedicated to the memory of Klaus Floret Introduction A well-known result of Kato is the following (see [6, Proposition 2.c.4]): Let X and Y be infinite–dimensional Banach spaces. Let T : X → Y be a linear continuous operator with T (X) not closed subspace of Y . Then, for every  > 0 there is an infinite–dimensional subspace Z of X so that Z has a Schauder basis and T|Z is a nuclear operator with norm ≤ . This result played a central role in the study of the strictly singular operators and of the perturbation theory of Fredholm operators in the setting of Banach spaces (see [6, Section 2.c]). The aim of this paper is to prove the proper analogue result of Kato with respect to the quotient spaces (see Theorems 1 and 2 of § 2). This type of result has been motivated by the study of some topological invariants in the context of Fréchet spaces (see [1,2]). As a consequence, in §3 we derive a result of Ostrovskii [7] concerning Banach spaces which are completions with respect to total nonnorming subspaces. 30 A.A. Albanese Let us recall some basic definitions. Let (X, || ||) be a Banach space and (X ∗ , || ||∗ ) be its dual topological space. We denote by BX (BX ∗ respectively) the closed unit ball of X (X ∗ respectively). A closed subspace M of X ∗ is said to be total if for every 0 6= x ∈ X there is an f ∈ M such that f (x) 6= 0. A sequence (xn )n∈N in X is said to be a Schauder basis if for x∈X Pevery ∞ there is a unique sequence of scalars (αn )n∈N such that x = α xn . A n n=1 sequence (xn )n∈N of X is said to be a basic sequence if it is a Schauder basis for its closed linear span [xn ]n∈N . A pair of sequences ((xn , x∗n ))n∈N in X × X ∗ is said to be a biorthogonal system if x∗n (xn ) = δnm for every n, m ∈ N. Let T : X → Y be a linear continuous map with (X, || ||) and (Y, | |) Banach spaces. We denote by ||T || its operator norm and by Rg(T ) its range. The map T is said to be a quotient map if T BX = BY . The map T is said to be nuclear if there exist a bounded sequence (x∗n )n∈N in X ∗ , a bounded sequence (yn )n∈N P ∗ in Y , and an element (λn )n∈N ∈ `1 such that T (x) = ∞ λ x n=1 n n (x)yn for every x ∈ X. The notation T|Z means the restriction of T to the subspace Z of X. Moreover, T is called strictly cosingular if there exists no closed subspace N of Y with codim N = ∞ such that q ◦ T : X → Y /N is onto where q denotes the canonical quotient map from Y onto Y /N (see [8]). ◦ For a subset A of X, A, A and A⊥ denote the closure of A in the strong topology, the set {x∗ ∈ X ∗ : ∀x ∈ A |x∗ (x)| ≤ 1} and the set {x∗ ∈ X ∗ : ∀x ∈ w∗ • A x∗ (x) = 0} respectively. For a subset A of X ∗ , A , A and A> denote the closure of A in the weak*–topology, the set {x ∈ X : ∀x∗ ∈ A x∗ (x) = 0} and the set {x ∈ X : ∀x∗ ∈ A x∗ (x) = 0} respectively. Other notation for Banach spaces is standard and we refer the reader, for example, to [6]. 1 Main Result In this section we will prove the proper analogue result of Kato with respect to the quotient spaces. The proof of this result is inspired by the ones given in [3, Theorem 1.2, (2) implies (3)] and [9, Lemma 3]. 1 Theorem. Let (X, || ||) and (Y, | |) be infinite–dimensional Banach spaces. Let T : X → Y be a linear continuous map with dense range and T (X) 6= Y . Then for each ε > 0 there exists a quotient map q : Y → Y1 , such that Y1 is an infinite–dimensional Banach space with a Schauder basis and the map q ◦ T is nuclear with norm ≤ ε. Proof. We first notice that, by Closed Range Theorem, if T has no closed range then T ∗ hasn’t, and also T ∗ restricted to any finite–codimensional sub- 31 On not open linear continuous operators space of Y ∗ hasn’t. Let ε > 0 be fixed. We choose Pa sequence (An )n∈N of positive real numbers with A1 > 1 and An+1 > 1 + nk=1 Ak for all n ∈ N, and we construct by induction a biorthogonal system ((yn , yn∗ ))n∈N in Y × Y ∗ such that |yn | ≤ An , |yn∗ |∗ = 1 and ||T ∗ yn∗ ||∗ ≤ ε −n 2 An (1) for all n ∈ N. Clearly, there exist y1 ∈ Y and y1∗ ∈ Y ∗ such that y1∗ (y1 ) = 1 and (1) holds with n = 1. Turning to our induction step, we assume that (yk )nk=1 ⊂ Y and (yk∗ )nk=1 ⊂ Y ∗ have been chosen in such a way that ((yk , yk∗ ))nk=1 is a biorthogonal P: Y →Y P system and (1) is satisfied for k ∈ {1, . . . , n}. Let define ∗ ∈ Rg(Q∗ ) = by P y = nk=1 yk∗ (y)yk . Then we set Q = I − P and choose yn+1 span{y1 , . . . , yn }⊥ ⊂ Y ∗ so that ∗ ∗ |yn+1 |∗ = 1 and ||T ∗ yn+1 ||∗ ≤ ε An+1 2−(n+1) . ∗ ∗ ), we have Since yn+1 = Q∗ (yn+1 ∗ ∗ 1 = |yn+1 |∗ = sup |yn+1 (Qy)| ≤ |y|≤1 sup z∈Rg(Q), |z|≤||Q|| ∗ |yn+1 (z)|. By construction An+1 > 1 + ||P || ≥ ||Q|| so that we can find yn+1 in Y with the desired properties. Next, let S : X → Y be the map so defined Sx := ∞ X n=1 (T ∗ yn∗ )(x)yn , x ∈ X. (2) Then S ∈ L(X, Y ) is nuclear with norm ≤ ε. Since S ∗ y∗ = ∞ X y ∗ (yn )T ∗ yn∗ n=1 for all y ∗ ∈ Y ∗ , we obtain that T ∗ = S ∗ on span{y1∗ , y2∗ , . . . }. We set Z = {y ∈ Y : ∀n ∈ N yn∗ (y) = 0}. Clearly Y /Z is infinite–dimensional. Since yn∗ (T x − Sx) = ((T ∗ − S ∗ )yn∗ )(x) = 0 for all n ∈ N and x ∈ X, we have that q ◦ T = q ◦ S where q : Y → Y /Z is the canonical quotient map. Consequently, q ◦ T is a nuclear map with norm ≤ ε. In order to obtain a quotient with a Schauder basis, we notice that Rg(q ◦ T ) is dense in Y /Z and Rg(q ◦ S) is separable. Therefore the Banach space Y /Z is separable and then we can apply [6, Theorem 1.b.7] (see [5]) to get 32 A.A. Albanese an infinite–dimensional quotient of Y /Z with a Schauder basis. Finally, the composition of the quotient maps fulfills all required properties and the proof QED is complete. As an immediate consequence, we obtain the following: 2 Theorem. Let (X, || ||) and (Y, | |) be infinite–dimensional Banach spaces. Let T : X → Y be a not open linear continuous operator. Then for each ε > 0 there exists a quotient map q : Y → Y1 , such that Y1 is an infinite–dimensional Banach space and q ◦ T is a nuclear operator of norm ≤ ε. Proof. Let Z = T (X). By assumption, T is a linear continuous map from X into Z with dense range and T (X) 6= Z. Taking any ε > 0, by Theorem 1, there exists a quotient map qZ : Z → Z1 , such that Z1 is an infinite dimensional space with a Schauder basis and qZ ◦ T is a nuclear operator of norm ≤ ε. Put N = ker qZ . Clearly, N ⊂ Z is also a closed subspace of Y and Z1 is isometrically isomorph to a closed subspace of Y1 = Y /N via the canonical map j : Z1 → Y1 defined by j(x + N ) = x + N for all x ∈ Z. Consequently, denoting by q the canonical quotient map from Y onto Y1 , j ◦ qZ = q ◦ i (where i : Z → Y is the canonical inclusion) and hence q ◦ T = j ◦ qZ T is also a nuclear operator QED of norm ≤ ε. This completes the proof. 2 A Consequence Let (X, || ||) be an infinite–dimensional Banach space. Let M be a total subspace of X ∗ . Define the completion of X with respect to M as the completion of X under the norm ||x||M = sup{|x∗ (x)| : x∗ ∈ M, ||x∗ ||∗ ≤ 1}. Denote by XM this completion. If the norm || ||M is equivalent to the initial norm of X, then the subspace M is said to be norming. It is clear that if M is norming, then XM = X. If M ∗ ∗ = span B σ(X ,X) ⊃ M . is a total nonnorming subspace of X, then XM M Total nonnorming subspaces were studied by many authors. In particular, in [4] Davis and Lindenstrauss proved that a Banach space X has a total nonnorming subspace in its dual if and only if X has infinite codimension in its second dual. In [7] Ostrovskii considered the problem to characterize what kind of Banach spaces are completions of some other Banach spaces with respect to a total nonnorming subspace. In particular, he showed that: 3 Theorem. If a Banach space Z is the completion of some other Banach space with respect to a total nonnorming subspace, then Z ∗ contains a norming 33 On not open linear continuous operators subspace M and a σ(Z ∗ , Z)–closed infinite–dimensional subspace N such that δ(M, N ) > 0 and the quotient Z/N > is separable. Recall that if U and V are subspaces of a Banach space (X, || ||) the number δ(U, V ) = inf{||u − v|| : u ∈ U and ||u|| = 1, v ∈ V } is called the inclination of U to V . Now, this result follows from Theorem 1 as a consequence. Indeed: Proof. Let Z = XM for some Banach space (X, || ||) and a total nonnorming subspace M of X ∗ . Every element of M is a linear functional on XM with the same norm. Further, M considered as a subspace of Z ∗ is clearly norming. Since the inclusion iM : (X, || ||) ,→ (XM , || ||M ) is a linear continuous map with dense range and not open, taking e.g. ε = 1/2, as it was proved in Theo∗ satisfying the rem 1, there is a biorthogonal system ((xn , x∗n ))n∈N in XM × XM ∗ conditions in (1) such that, if we set M1 = {x ∈ XM : xn (x) = 0 for all n ∈ N}, the quotient space XM /M1 is infinite–dimensional and separable. ∗ . Clearly, N is σ(X ∗ , X )–closed. Since X /M is Let N = M1⊥ ⊂ XM 1 M M M separable as already observed, it remains only to show that δ(M, N ) > 0. Let z ∗ ∈ M with ||z ∗ ||∗M = 1 = ||z ∗ ||∗ . Then, taking P any  > 0, there is x ∈ X ∗ with ||x|| = 1 such that z ∗ (x) ≥ 1 − . Put w = x − ∞ n=1 xn (x)xn = x − Sx (cf. (2)). Then w ∈ M1 = N > and ||w − x|| ≤ 1/2. It follows that z ∗ (w) ≥ z ∗ (x) + z 0 (w − x) ≥ 1 −  − 1 1 = − ; 2 2 hence, for each y ∗ ∈ N , 2 2 2 1 1 2 ||z ∗ − y ∗ ||∗M ≥ |(z ∗ − y ∗ )(w)| = |z ∗ (w)| ≥ ( − ) = − . 3 3 3 2 3 3 By the arbitrarity of , it follows that ||z ∗ − y ∗ ||∗M ≥ 1/3 for all y ∗ ∈ N , thereby QED implying that δ(M, N ) ≥ 1/3 > 0. References [1] A.A. Albanese: The density condition in quotients of quasinormable Fréchet spaces, Studia Math. 125 (2) (1997), 131–141. [2] A.A. Albanese: The density condition in quotients of quasinormable Fréchet spaces, II, Rev. Mat. Complut. Madrid 12 (1999), 73–84. [3] J.W. Brace, R.R. Kneece: Approximation of strictly singular and strictly cosingular operators using nonstandard analysis, Trans. Amer. Math. Soc. 168 (1972), 483–496. [4] W.J. Davis, J. Lindenstrauss: On total nonnormin subspaces, Proc. Amer. Math. Soc. 31 (1972), 109–111. 34 A.A. Albanese [5] W.B. Johnson, H.P. Rosenthal: On w*–basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77–92. [6] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces I, Springer–Verlag, Berlin 1977. [7] M.I. Ostrovskii: Characterization of Banach spaces which are completions with respect to total nonnorming subspaces, Arch. Math. 60 (1993), 349–358. [8] A. Pelczynski: On strictly singular and strictly cosingular operators, I and II, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 31–41. [9] V. Wrobel: Strictly cosingular operators on (DF)–spaces, Math. Z. 174 (1980), 281–287.