The history of a general criterium on spaceability

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1 Open Math. 27; 5: Open Mathematics Open Access Review Article Víctor M. Sánchez* The history of a general criterium on spaceability DOI.55/math-27-9 Received January 2, 27; accepted February 9, 27. Abstract: There are just a few general criteria on spaceability. This survey paper is the history of one of the first ones. Let I and I 2 be arbitrary operator ideals and E and F be Banach spaces. The spaceability of the set of operators I.E; F / n I 2.E; F / is studied. Before stating the criterium, the paper summarizes the main results about lineability and spaceability of differences between particular operator ideals obtained in recent years. They are the seed of the ideas contained in the general criterium. Keywords: Spaceability, Operator ideal MSC: 47L2 Introduction During the last two decades many authors have been searching for large linear structures of mathematical objects enjoying certain special properties. If X is a topological vector space, a subset A of X is said to be spaceable if A [ fg contains an infinite-dimensional closed subspace. The subset A is called lineable if A [ fg contains an infinite-dimensional subspace (not necessarily closed). After these definitions were introduced by Aron, Gurariy and Seoane-Sepúlveda in [], many authors showed that many non-linear special sets can often have these properties. See, for instance, the recent works [2 8], just to cite some. However, before the publication of [] some authors already found large linear structures enjoying special properties even though they did not explicitly used terms like lineability or spaceability (see [9, ]). We refer the interested reader to a very recent book on the topic ([]) where many examples can be found and techniques are developed in several different frameworks. Surprisingly, there exist very few general criteria on spaceability. In fact, most results on concrete properties have been proved directly and constructively. Perhaps, the first general criterium on spaceability appeared in [2] where Wilansky proved that if Y is a closed vector subspace of a Banach space X, then X n Y is spaceable if and only if Y has infinite codimension. An improvement of this result, where X is a Fréchet space, is ascribed by Kitson and Timoney to Kalton (see [3, Theorem 2.2]). Kitson and Timoney used it to obtain the following theorem: Theorem. ([3]). Let.E n / n2n be a sequence of Banach spaces, F a Fréchet space and T n W E n! F bounded linear operators. Let G be the span of S nd T n.e n /. If G is not closed in F, then the set F n G is spaceable. Three years before, lineability of the difference between a couple of particular operator ideals was studied for the first time. Thus, in [4], Puglisi and Seoane-Sepúlveda showed that if E and F are Banach spaces where E has the two series property, then the set L.E; F / n.e; F / is lineable, where denotes the ideal of -summing operators. *Corresponding Author: Víctor M. Sánchez: Department of Mathematical Analysis, Faculty of Mathematics, Complutense University of Madrid, 284 Madrid, Spain, victorms@ucm.es 27 Sánchez, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs Brought 3. to License. you by Universidad Complutense Madrid

2 The history of a general criterium on spaceability 253 The following year, and partially answering a question posed in [4], Botelho, Diniz and Pellegrino proved in [5] that if E is a superreflexive Banach space containing a complemented infinite-dimensional subspace with unconditional basis, or F is a Banach space having an infinite unconditional basic sequence, then the set K.E; F / n p.e; F / is lineable for every p, where K denotes the ideal of all compact operators. More generally, in [3], the spaceability of the set of operators K.E; F / n S p p.e; F / was obtained as a consequence of Theorem. when E is a superreflexive Banach space. Finally, and recently, in [6], the authors continued this research considering general operator ideals in the sense of Pietsch I and I 2 and Banach spaces E and F, such that the set I.E; F / n I 2.E; F / is non-empty. They introduced the new notion of -reproducible space for obtaining the spaceability of this set when E or F belongs to this class of spaces and I.E; F / is complete with the usual operator norm or with another complete ideal norm. Many classical Banach spaces are -reproducible, in particular, rearrangement invariant spaces and spaces of continuous functions. For completeness, most of the original proofs have been included. 2 Preliminaries An operator ideal in the sense of Pietsch (see [7, 8]) is defined as follows: Definition 2.. Let B denote the class of all Banach spaces and let L denote the class of all bounded linear operators between Banach spaces. An operator ideal I is a mapping" I W B B! 2 L such that. For each pair of Banach spaces E and F the collection of operators I.E; F / (or I.E/ if E D F ) is a subspace of the space L.E; F / (or L.E/ if E D F ) of bounded linear operators from E to F containing all finite-rank operators. 2. If in a scheme of bounded linear operators E S! E T! F S 2! F we have T 2 I.E; F /, then S 2 ı T ı S 2 I.E ; F /. Definition 2.2. An ideal norm defined on an ideal I is a rule k k I that assigns to every operator T 2 I a nonnegative number kt k I satisfying the following conditions:. kx yk I D kx k E kyk F for x 2 E ; y 2 F where.x y/.x/ D x.x/y for x 2 E. 2. ks C T k I ksk I C kt k I for S; T 2 I.E; F /. 3. ks 2 ı T ı S k I ks 2 kkt k I ks k for S 2 2 L.F; F /, T 2 I.E; F / and S 2 L.E ; E/. The last condition implies that kt k I D jjkt k I for T 2 I and 2 K. Thus, we have indeed a norm. Moreover, kt k kt k I where k k denotes the usual operator norm of L.E; F /, which is an example of ideal norm. A classical closed operator ideal endowed with the canonical operator norm of L.E; F / is the ideal K of all compact operators. A classical non-closed operator ideal with respect to the operator norm is the ideal q;p of.q; p/-summing operators. Recall that if p q <, an operator T 2 L.E; F / is called.q; p/-summing (or p-summing if p D q) if there is a constant C so that, for every choice of an integer n and vectors.x i / n in E, we have nx kt.x i /k q! =q C sup kx k! =p nx jx.x i /j p : The smallest possible constant C defines a complete ideal norm on this operator ideal, denoted by q;p./. If =p =q =r =s and p r, one has q;p s;r (see [7, page 459]). Let us recall now the definition of a rearrangement invariant space. Given a measure space.; /, where is the interval Œ; or Œ; / and is the Lebesgue measure, or D N and is the counting measure, the distribution function x associated to a scalar measurable function x on is defined by x.s/ D ft 2 W jx.t/j > sg. The decreasing rearrangement function x of x is defined by x.t/ D inffs 2 Œ; / W x.s/ tg. A Banach space

3 254 V.M. Sánchez.E; k k E / of measurable functions defined on is said to be a rearrangement invariant space if the following conditions are satisfied:. If y 2 E and jx.t/j jy.t/j -a.e. on, then x 2 E and kxk E kyk E. 2. If y 2 E and x D y, then x 2 E and kxk E D kyk E. Important examples of rearrangement invariant spaces are L p, Lorentz, Marcinkiewicz and Orlicz spaces. For properties of rearrangement invariant spaces we refer to [9 2]. 3 Results The first result about lineability of the difference between a couple of particular operator ideals appeared in [4]. In this paper the authors showed that if E and F are Banach spaces where E has the two series property, then L.E; F / n.e; F / is lineable, where denotes the ideal of -summing operators. Definition 3.. A Banach space E is said to have the two series property provided there exist unconditionally convergent series P f i in E and P x i in E such that X jf j.x i /j 2 A kf j k j D =2 D : For instance, every L p -space has the two series property for < p <. Lemma 3.2 ([4]). Let E be a Banach space satisfying the two series property. Let P f i in E and P x i in E such that P P jf j.x i /j 2 =2 j D kf j k D. Then, there exists a sequence of countable pairwise disjoint subsets of N,.A n / n2n, such that X =2 jf j.x i /j2 A D kf j k j 2A n for each n 2 N. Theorem 3.3 ([4]). Let E be a Banach space satisfying the two series property. Then the set L.E; `2/ n.e; `2/ is lineable. Proof. Since E satisfies the two series property, there exist unconditionally convergent series P f i in E and P x i in E such that =2 X jf j.x i /j 2 A D : kf j k j D Let.A n / n2n be the sequence given by Lemma 3.2. For each n 2 N, let us define the operator T n W E T n.x/ D X k2a n f k.x/ kf k k =2 e k:! `2 by If kxk E, then kt n.x/k X jf k.x/j2 A kf k k k2a n =2! =2 X jf k.x/j 2 sup kf k k kd kxkd kd =2 X jf k.x/j! < : Thus, T n is well-defined and T n 2 L.E; `2/ for every n 2 N.

4 The history of a general criterium on spaceability 255 But P kt n.x i /k D for every n 2 N. Then T n 62.E; `2/ for each n 2 N. Because of the pairwise disjointness of the sets A n, we have that the sequence.t n / n2n is linearly independent in L.E; `2/. Finally, le us show that every nonzero bounded linear operator in the linear span of the sequence.t n / n2n does not belong to the ideal of -summing operators. It is enough to consider the linear combination of two elements because the general case follows similarly. Thus, if ; 2 2 K and n ; n 2 2 N, then (assuming, without loss of generality, that ) we have that X xi T n xi C 2 T n2 D X B X jf k.x /j2 C kf k k k2a n X k2a n2 Since P x i is an unconditional convergent series in E, we obtain the result. ˇ ˇˇˇ2 ˇf 2 k x i C A kf k k =2 D : Now, given two Banach spaces E and F, where E enjoys the two series property, a linearly independent sequence.ft n / n2n can be constructed from the sequence.t n / n2n made in Theorem 3.3 whose linear span is contained in L.E; F / n.e; F /. Thus, we have Corollary 3.4 ([4]). Let E; F be Banach spaces, where E has the two series property. Then the set L.E; F / n.e; F / is lineable. In [22], Davis and Johnson proved that the set K.E; F / n p.e; F / is non-empty whenever E is a superreflexive Banach space and F is any Banach space. The question about the lineability of this set was posed in [4] and partially answered in [5, Theorem 2.]: Theorem 3.5 ([5]). Let E be a superreflexive Banach space. If either E contains a complemented infinitedimensional subspace with unconditional basis or F is a Banach space having an infinite unconditional basic sequence, then K.E; F / n p.e; F / is lineable for every p. Proof. First, let us suppose that E contains a complemented infinite-dimensional subspace E with unconditional basis.e n / n2n. We consider a decomposition of N into infinitely many infinite pairwise disjoint subsets.a k / k2n. Since.e n / n2n is an unconditional basis, it is well known that.e n / n2ak is an unconditional basic sequence for every k 2 N. Let us denote by E k the closed span of.e n / n2ak. As a subspace of a superreflexive space, E k is superreflexive as well, and there exists T k 2 K.E k ; F / n p.e k ; F / for each k 2 N. If C is the unconditional basis constant of.e n / n2n, then X X n a n e n C a n e n nd for every n D and scalars a n. We denote by P k W E! E k the canonical projection onto E k for each k 2 N. For x D P nd a ne n 2 E we have for a suitable choice of signs n and n. Thus, nd 2P k.x/ D X X X 2a n e n D n a n e n C n a ne n n2a k nd nd kp k.x/k C kxk: So each P k is continuous, kp k k C and E k is complemented in E. If W E! E is the projection onto E, for each k 2 N we define ft k D T k ı P k ı. Since.P k ı /.x/ D x for every x 2 E k, then ft k 2 K.E; F / n p.e; F /. Given scalars ; : : : ; n with at least one k, there is a weakly p-summable sequence.x j / j 2N E k such that P j D kt k.x j /k p D C. The sequence.x j / j 2N is

5 256 V.M. Sánchez weakly p-summable in E, ft k.x j / D T k.x j / and et i.x j / D if i k for every j 2 N. Thus, X X k ft.x j / C C n ft n.x j /k p D k k T k.x j /k p D C; j D proving that the span of.ft k / k2n is contained in K.E; F / n p.e; F /. Let us prove now that the set of operators.ft k / k2n is linearly independent. Let ; : : : ; n be scalars such that ft C C n ft n D. Choosing x k 2 E k such that ft k.x k / for each k 2 f; : : : ; ng, we have that j D D ft.x k / C C n ft n.x k / D k ft k.x k /: It follows that k D for every k 2 f; : : : ; ng. Now, let us suppose that F contains a subspace F with unconditional basis.e n / n2n having unconditional basis constant C. Considering again the subsets.a k / k2n as above, we define F k as the closed span of.e n / n2ak and let P k W F! F k be the corresponding projections. We also obtain as above that kp k k C. For each k 2 N there exists T k 2 K.E; F k / n p.e; F k /. If y i 2 F i and y j 2 F j with i j, then ky i k D kp i.y i C y j /k C ky i C y j k: We define now the operator ft k by the composition of T k with the inclusion from F k to F. It is clear that ft k 2 K.E; F / n p.e; F / for each k 2 N. Since k et i.x/ C ft j.x/k C k et i.x/k for every x 2 E, then et i C ft j 2 K.E; F / n p.e; F / for all i; j 2 N. It is easy to deduce that the span of.ft k / k2n is contained in K.E; F / n p.e; F /. The linear independence is obtained as in the first case. In [3], Theorem 3.5 was improved by establishing spaceability, and indeed a single infinite dimensional closed subspace valid for all p. In order to obtain this improvement we will also need the following result: Proposition 3.6 ([3]). Let E n (n 2 N) and F be Fréchet spaces and T n W E n! F bounded linear operators. Let G be the span of S nd T n.e n /. If G is closed in F, then there exists k 2 N such that G is the span of S k nd T n.e n /. Theorem 3.7 ([3]). Let E and F be infinite dimensional Banach spaces. If E is superreflexive, then [ K.E; F / n p.e; F / is spaceable. p< Proof. Since K.E; F / is closed in the operator norm, it follows that p.e; F / \ K.E; F / is closed in p.e; F /, hence a Banach space in the norm p./ for every p. In the proof of [22, Theorem], it is shown that the norm induced by p.e; F / on the finite-rank operators is not equivalent to the operator norm for p < when E is superreflexive. Thus, p.e; F / \ K.E; F / is not closed in K.E; F /. Due to Proposition 3.6 the union [ p.e; F / \ K.E; F / p< p.e; F / \ K.E; F / D [ p2n is not closed and the result then follows from Theorem.. Finally, in [6], the authors attained a general criterium. They introduced the notion of -reproducible space in order to obtain the spaceability of the set I.E; F / n I 2.E; F /, where I and I 2 are general operator ideals in the sense of Pietsch, E or F belongs to that class of spaces and I.E; F / is complete with the usual operator norm or with another complete ideal norm.

6 The history of a general criterium on spaceability 257 Definition 3.8. A Banach space E is said to be -reproducible if there exists a sequence.e n / n2n of complemented subspaces, where P n W E! E n is a bounded projection, such that each E n is isomorphic to E, P i ı P j D if i j, and for all k 2 N the projections fp k D P k nd P n W E! L k nd E n are uniformly bounded. Notice that this notion is an isomorphic property. Also, if E and F are -reproducible Banach spaces, then E F and the dual E are also -reproducible (see [6, Proposition 3.2]). Every Banach space E with a Schauder decomposition.e n / n2n (see [23, page 47]) such that each subspace E n is isomorphic to E, is -reproducible. However, any indecomposable space is not -reproducible (recall that a Banach space E is indecomposable if there do not exist infinite-dimensional closed subspaces F and G of E with E D F G). Proposition 3.9 ([6]). Every rearrangement invariant space E is -reproducible. Proof. First, let E be a rearrangement invariant space on Œ;. For every a 2 Œ; / and r 2.; a we consider the complemented subspace E a;r D fx 2 E W supp x Œa; a C r g and the bounded projection P a;r W E! E a;r given by P a;r.x/ D x Œa;aCr for x 2 E. For a measurable function x we define the linear operators t a T a;r.x/.t/ D x.a;acr.t/ r and S a;r.x/.t/ D x.. t/a C t.a C r// which are bounded from L to L and from L to L. Thus, using the Calderón-Mitjagin interpolation theorem ([2, Theorem 2.a.]), T a;r and S a;r are bounded from E to E. These operators also have the following properties:..s a;r ı T a;r /.x/ D x for every x 2 E. 2..T a;r ı S a;r /.x/ D x for every x 2 E a;r. 3. T a;r W E! E a;r is an isomorphism. 4. S a;r W E a;r! E is an isomorphism. Let us show only that T a;r is injective (it is easy to prove the rest of the properties). Indeed, if x 2 E n fg, then there exists n 2 N such that.a n / > where A n is defined as A n D ft 2 Œ; W jx.t/j > =ng. Thus.A n / > with A n D f. t/a C t.a C r/ W t 2 A ng. And jt a;r.x/.s/j > =n for every s 2 A n. For every n 2 N we consider a n D and r 2 n n D. Let E 2 n n D E an ;r n and P n D P an ;r n. Since fp k D for all k 2 N, we conclude that E is -reproducible. We consider now a rearrangement invariant space E on Œ; /. Let fa n W n 2 Ng be a disjoint sequence of subsets of Œ; / where A n D S kd.a n;k; a n;k C for an increasing sequence.a n;k / k2n N, and the complemented subspaces E n D fx 2 E W supp x A n g: Given a measurable function x, we define the linear operators and T n.x/.t/ D S n.x/.t/ D X x.t C k a n;k /.an;k ;a n;k C.t/ kd X x.t C a n;k k /.k ;k.t/: kd Since.T n.x// D x,.s n.x// x and.s n ı T n /.x/ D x, we have that T n W E! E n is an isometry and S n W E n! E is an isomorphism. Now, reasoning as in the Œ; case we obtain the result. Finally, we consider a symmetric sequence space. Let fa k W k 2 Ng be a disjoint partition of N where the subset A k is the range of an injective map ' k W N! N for every k 2 N.

7 258 V.M. Sánchez For x D.x n / n2n we define the linear operators T k.x/ D.a n / n2n with ( x m if ' a n D k.m/ D n if n 62 A k and S k.x/ D.x 'k.n// n2n. If E k D fx 2 E W supp x A k g, we have that T k W E! E k is an isometry and S k W E k! E is an isomorphism. And reasoning again as in the Œ; case we obtain the result. Also, the space C Œ; is -reproducible (see [6, Proposition 3.4]). As a consequence, the space C.K/ is - reproducible for any uncountable compact metric space K (see [24, Theorem 4.4.8]). The following lemma will be useful in the proof of the general criterium: Lemma 3. ([6]). Let I be an operator ideal.. If E and E 2 are isomorphic Banach spaces, then there exists a bijection between I.E ; F / and I.E 2 ; F / for every Banach space F. 2. If G is a closed subspace of the Banach space E and T 2 I.E; F /, then the restriction T j G 2 I.G; F / for every Banach space F. Theorem 3. ([6]). Let I and I 2 be operator ideals such that I.E; F / n I 2.E; F / is non-empty for a couple of Banach spaces E and F. If E or F is -reproducible and I.E; F / is complete for an ideal norm, then I.E; F / n I 2.E; F / is spaceable. Proof. Let T 2 I.E; F /ni 2.E; F /. If E is a -reproducible Banach space with isomorphisms n W E n! E and bounded projections P n W E! E n, for every n 2 N we consider the operator T n D T ı n ı P n which belongs to I.E; F / n I 2.E; F /. Indeed, using Lemma 3., if T n 2 I 2.E; F /, then we have that T n j En D T ı n 2 I 2.E n ; F /, but this is not true. The sequence.t n / n2n is formed by linearly independent operators. To show this, if P k nd a nt n D, restricting to E j we obtain a j D with j k. In the same way, it can be showed that P k nd a nt n cannot belong to I 2.E; F /. Thus, I.E; F / n I 2.E; F / is lineable. Furthermore,.T n / n2n is a basic sequence in I.E; F /. Indeed, for any integers k < m and any choice of scalars. n / n2n we have kx mx mx n T n D n T n ı fp k n T n fp k : nd I nd I nd I P Let S 2 ŒT n W n 2 N I.E; F / with S D n T n. Then there exists n 2 N such that n. We have nd that Sj En D n T ı n 62 I 2.E n ; F /. Thus, S 62 I 2.E; F / and ŒT n W n 2 N I.E; F / n I 2.E; F /. If F is -reproducible with isomorphisms. n / n2n, for each n 2 N we consider the operator T n D n ı T which belongs to I.E; F / n I 2.E; F /. The sequence.t n / n2n is formed by linearly independent operators. Thus, we obtain that I.E; F / n I 2.E; F / is lineable. And.T n / n2n is a basic sequence. Indeed, for any integers k < m and any choice of scalars. n / n2n we have kx mx n T n D P f k ı n T n X m fp k n T n : nd I nd I nd I Let S 2 ŒT n W n 2 N I.E; F / with S D P nd n T n. There exists n 2 N such that n. If S 2 I 2.E; F /, then P n ıs 2 I 2.E; F /, but this is not true because P n ıs D n T n. Then ŒT n W n 2 N I.E; F /n I 2.E; F /. The general criterium can be extended in the following way: Theorem 3.2 ([6]). If E or F is a -reproducible Banach space, I is an operator ideal such that I.E; F / is complete for an ideal norm, and.i n / n2n is a sequence of operator ideals such that the set I.E; F / n I n.e; F / is non-empty for every n 2 N, then the set I.E; F / n S nd I n.e; F / is spaceable.

8 The history of a general criterium on spaceability 259 Proof. Let S n 2 I.E; F / n I n.e; F / for every n 2 N. If E is a -reproducible Banach space with isomorphisms. n / n2n and bounded projections.p n / n2n, let us consider the operators S n ı n ı P n 2 I.E; F / n I n.e; F / for every n 2 N. Then T D X nd S n ı n ı P n 2 n ks n ı n ı P n k I belongs to I.E; F / n I n.e; F / for every n 2 N. Now, reasoning as in the proof of Theorem 3. we can construct a sequence.t k / k2n such that ŒT k W k 2 N I.E; F / n I n.e; F / for every n 2 N. If F is a -reproducible Banach space with isomorphisms. n / n2n, let us consider the operators n ı S n 2 I.E; F / n I n.e; F / for every n 2 N. Then belongs to I.E; F / n S nd I n.e; F /. T D X nd n 2 n k n ı S n ı S nk I Corollary 3.3 ([6]). Let E and F be Banach spaces, and fi p W p 2 Œa; b g be a family of operator ideals such that I p.e; F / I q.e; F / if p < q with continuous inclusion. If E or F is a -reproducible Banach space and I b.e; F / is complete for an ideal norm, then the set I b.e; F / n S p<b I p.e; F / is spaceable. In general, Theorem 3. does not hold for arbitrary Banach spaces. Consider for instance the spaces with few operators given in [25, 26]. They are hereditary indecomposable Banach spaces on which every bounded linear operator is a compact perturbation of a scalar multiple of the identity. Finally, let us remark that in the special case of considering Banach operator ideals I and I 2 with proper continuous inclusion I 2 I where the ideal I 2 is not closed in I, the set of operators I.E; F / n I 2.E; F / is always spaceable. This follows from Theorem.. Many applications of the general criterium can be found in [6] where a good number of particular operator ideals are considered: compact operators, strictly singular operators, strictly co-singular operators, finitely strictly singular operators,.q; p/-summing operators and Schatten operator classes. References [] Aron R.M., Gurariy V.I., Seoane-Sepúlveda J.B., Lineability and spaceability of sets of functions on R, Proc. Amer. Math. Soc., 25, 33, [2] Aron R.M., García-Pacheco F.J., Pérez-García D., Seoane-Sepúlveda J.B., On dense-lineability of sets of functions on R, Topology, 29, 48, [3] Azagra D., Muńoz-Fernández G.A., Sánchez V.M., Seoane-Sepúlveda J.B., Riemann integrability and Lebesgue measurability of the composite function, J. Math. Anal. Appl., 29, 354, [4] Bernal-González L., Ordońez M., Lineability criteria, with applications, J. Funct. Anal., 24, 266, [5] Cariello D., Seoane-Sepúlveda J.B., Basic sequences and spaceability in `p spaces, J. Funct. Anal., 24, 266, [6] Enflo P.H., Gurariy V.I., Seoane-Sepúlveda J.B., Some results and open questions on spaceability in function spaces, Trans. Amer. Math. Soc., 24, 366, [7] Gámez-Merino J.L., Muńoz-Fernández G.A., Sánchez V.M., Seoane-Sepúlveda J.B., Sierpiński-Zygmund functions and other problems on lineability, Proc. Amer. Math. Soc., 2, 38, [8] Ruiz C., Sánchez V.M., Nonlinear subsets of function spaces and spaceability, Linear Algebra Appl., 24, 463, [9] Gurariy V.I., Subspaces and bases in spaces of continuous functions, Dokl. Akad. Nauk SSSR, 966, 67, , in Russian [] Levin B., Milman D., On linear sets in the space C consisting of functions of bounded variation, Zapiski Inst. Mat. Mekh. Kharkov, 94, 6, 2-5, in Russian, English summary [] Aron R.M., Bernal-González L., Pellegrino D., Seoane-Sepúlveda J.B., Lineability: the search for linearity in mathematics, CRC Press, 26 [2] Wilansky A., Semi-Fredholm maps in FK spaces, Math. Z., 975, 44, 9-2 [3] Kitson D., Timoney R.M., Operator ranges and spaceability, J. Math. Anal. Appl., 2, 378,

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