A note on the geometry of certain classes of linear operators
AA NOTE ON THE GEOMETRY OF CERTAIN CLASSES OF LINEAROPERATORS
DIOGO DINIZ AND ANSELMO RAPOSO JR.
Abstract.
In this note we introduce a new technique to answer an issue posed in [7]concerning geometric properties of the set of non-surjective linear operators. We alsoextend and improve a related result from the same paper. Introduction
In 1872 K. Weierstrass constructed an example af a nowhere differentiable continuousfunction from [0 , on R . This non intuitive result, now known as Weierstrass Mon-ster, was pushed further in 1966, when V. Gurariy constructed an infinite-dimensionalsubspace formed, except for the null vector, by continuous nowhere differentiable func-tions. In 2004, Aron, Gurariy and Seoane [1] investigated similar problems in othersettings, initiating the field of research known as “lineability”: the idea is to look forlinear structures inside exotic subsets of vector spaces. If V is a vector space and α is a cardinal number, a subset A of V is called α -lineable in V if A ∪ { } contains an α -dimensional linear subspace W of V . When V has a topology and the subspace W can be chosen to be closed, we say that A is spaceable. We refer to the book [2] for ageneral panorama of the subject.As a matter of fact, with the development of the theory, it was observed that positiveresults of lineability were quite common, although general techniques are, in general, notavailable. Towards a more demanding notion of linearity, Fávaro, Pellegrino and Tomazintroduced a more involved geometric concept: let α , β and λ be cardinal numbers, with α < β ≤ λ , and let V be a vector space such that dim V = λ . A subset A of V is called ( α, β ) -lineable if, for every subspace W α ⊂ V such that dim W α = α and W α ⊂ A ∪ { } there is a subspace W β ⊂ V with dim W β = β and W α ⊂ W β ⊂ A ∪ { } . When V is atopological vector space, we shall say that A is ( α, β ) -spaceable when the subspace W β can be chosen to be closed .A well-known technique in lineability is known as “mother vector technique”: it con-sists of choosing a vector v in the set A and generating a subspace W ⊂ A ∪ { } containing “copies” of v . However, in general, the vector v does not belong to thegenerated subspace (see, for instance, [8]). Constructing a vector space of prescribeddimension and containing an arbitrary given vector is a rather more involving problem,which is probably another motivation of this more strict approach to lineability. Mathematics Subject Classification.
Key words and phrases.
Spaceability; lineability; sequence spaces.D. Diniz was partially supported by CNPq 301704/2019-8 and Grant 2019/0014 Paraiba StateResearch Foundation (FAPESQ). a r X i v : . [ m a t h . F A ] S e p D. DINIZ AND A. RAPOSO JR.
Under this new perspective, several simple problems, from the point of view of ordi-nary lineability, gain more subtle contours. For instance, it is obvious that the set ofthe continuous linear operators u : (cid:96) p → (cid:96) q that are non-surjective is c -spaceable (hereand henceforth c denotes the continuum). In fact, if π : (cid:96) q → K is the projection atthe first coordinate, just consider the colection of the continuous and non-surjectivelinear operators for which π ◦ u ≡ . Therefore, only ( α, β ) -lineability matters in thisframework.In this note we answer a question posed in [7] on the (1 , c ) -lineability of a certainset of non surjective functions. Our solution uses a technique that, to the best of theauthors knowledge, is new.2. Lineability vs injectivity and surjectivity
Lineability properties of the sets of injective and surjective continuous linear opera-tors between classical sequence spaces were recently investigated by [3] and [5]. In [7,Theorem 3.1] the authors investigated more subtle geometric properties in the settingof non injective continuous linear operators by proving that if p, q ≥ and(2.1) A := { u : (cid:96) p → (cid:96) q : u is linear, continuous and non injective } ,then A is (1 , c ) -lineable. In the same paper the authors pose a question on the (1 , c ) -lineability of the set(2.2) D := { u : (cid:96) p → (cid:96) q : u is linear, continuous and non-surjective } .In this section we shall show that D is (1 , c ) -lineable and, as a matter of fact, ourtechnique works in a more general environment of sequence spaces. We shall say thata Banach sequence space E of X -valued sequences where X is a Banach space is rea-sonable if c ( X ) ⊂ E and, for all x = ( x j ) ∞ j =1 ∈ E and ( α j ) ∞ j =1 ∈ (cid:96) ∞ , we have ( α j x j ) ∞ j =1 ∈ E with(2.3) (cid:13)(cid:13)(cid:13) ( α j x j ) ∞ j =1 (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ( α j ) ∞ j =1 (cid:13)(cid:13)(cid:13) ∞ (cid:13)(cid:13)(cid:13) ( x j ) ∞ j =1 (cid:13)(cid:13)(cid:13) .The class of reasonable sequence spaces includes various classical sequence spaces.For instance, for < p < ∞ , the (cid:96) p ( X ) spaces of p -summable sequences, the (cid:96) wp ( X ) spaces of weakly p -summable sequences and the (cid:96) up ( X ) spaces of unconditionally p -summable sequences are reasonable sequence spaces. The spaces (cid:96) ∞ ( X ) , c ( X ) , ofbounded and null sequences, respectively and the Lorentz spaces (cid:96) ( w,p ) ( X ) are alsoreasonable sequence spaces.Our result reads as follows: Theorem 2.1.
Let V (cid:54) = { } be a normed vector space and X (cid:54) = { } be a Banach space.Let E be a reasonable sequence space of X -valued sequences. The set D V,E = { u : V → E : u is linear, continuous and non-surjective } is (1 , c ) -lineable. NOTE ON THE GEOMETRY OF CERTAIN CLASSES OF LINEAR OPERATORS 3 The proof
Fixed v ∈ D V,E \ { } , let N v = { k ∈ N : π k ◦ v (cid:54)≡ } ,where π k : E → X ( x j ) ∞ j =1 (cid:55)→ x k is the k -th projection over X . If N v is a proper subset of N , the proof is simple. In fact,if j ∈ N \ N v , since c ( X ) ⊂ E , it is obvious that the subspace N := { u : V → E : u is linear, continuous and π j ◦ u ≡ } is contained in D V,E and it is also plain that v ∈ N . We will prove that dim N ≥ c .Since v is not identically zero, there exists x ∈ V such that v ( x ) = w (cid:54) = 0 . By theHahn-Banach Theorem, there is a continuous linear functional ϕ : E → K , such that ϕ ( w ) = 1 . Fixing a ∈ X \ { } , for each k ∈ N , let us define w k = (0 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) j + k − , a, , , . . . ) ∈ E and consider the linear operators T k : E → E given by T k ( w ) = ϕ ( w ) w k (cid:107) w k (cid:107) .Obviously, the operators T k are continuous and (cid:107) T k (cid:107) = sup (cid:107) w (cid:107)≤ (cid:13)(cid:13)(cid:13)(cid:13) ϕ ( w ) w k (cid:107) w k (cid:107) (cid:13)(cid:13)(cid:13)(cid:13) = sup (cid:107) w (cid:107)≤ | ϕ ( w ) | = (cid:107) ϕ (cid:107) .Hence, the operators S k = T k ◦ v are continuous too and (cid:107) S k (cid:107) = (cid:107) T k ◦ v (cid:107) ≤ (cid:107) T k (cid:107) (cid:107) v (cid:107) = (cid:107) ϕ (cid:107) (cid:107) v (cid:107) .Notice that S k ( x ) = w k / (cid:107) w k (cid:107) and, consequently, π j + k ◦ S k (cid:54)≡ .Hence, S k ∈ N for each k ∈ N . It is obvious that the set { S k : k ∈ N } ⊂ N is linearly independent. Define Ψ : (cid:96) → L ( V ; E )( a n ) ∞ n =1 (cid:55)→ (cid:80) ∞ n =1 a n S n .Since Ψ is well-defined, linear and injective we have dim Ψ( (cid:96) ) = c and since Ψ( (cid:96) ) ⊂ N the proof of the case N v (cid:54) = N is done. D. DINIZ AND A. RAPOSO JR.
Now, let us suppose that N v = N . By (2.3) we know that, for each ( α n ) ∞ n =1 ∈ (cid:96) ∞ , S v ( α n ) ∞ n =1 : V → Ex (cid:55)→ ( α n ( v ( x )) n ) ∞ n =1 is a well-defined continuous linear operator. It is plain that S v ( α n ) ∞ n =1 ∈ D V,E whenever ( α n ) ∞ n =1 is a sequence in (cid:96) ∞ having some null entry (because, if α i = 0 , then the i -thcoordinate of S v ( α n ) ∞ n =1 ( x ) is zero for all x ∈ V ). Let us consider, therefore, ( α n ) ∞ n =1 ∈ (cid:96) ∞ such that α n (cid:54) = 0 for all n ∈ N and fix w = ( w n ) ∞ n =1 ∈ E \ v ( V ) . Since ( α n w n ) ∞ n =1 ∈ E ,we have ( α n w n ) ∞ n =1 ∈ E \ S v ( α n ) ∞ n =1 ( V ) .In fact, if there was x ∈ V such that S v ( α n ) ∞ n =1 ( x ) = ( α n w n ) ∞ n =1 , we would have ( α n w n ) ∞ n =1 = ( α n ( v ( x )) n ) ∞ n =1 and, since α n (cid:54) = 0 for all n ∈ N , we would have v ( x ) = w , which is impossible. Hence, S v ( α n ) ∞ n =1 ∈ D V,E .Now consider the linear map
Λ : (cid:96) ∞ → L ( V ; E )Λ (( µ n ) ∞ n =1 ) = S v ( µ n ) ∞ n =1 .We have just proved that S v ( α n ) ∞ n =1 ∈ D V,E for all ( α n ) ∞ n =1 ∈ (cid:96) ∞ ; thus Λ ( (cid:96) ∞ ) ⊂ D V,E . Note that Λ is injective. In fact, if ( µ n ) ∞ n =1 ∈ (cid:96) ∞ and Λ (( µ n ) ∞ n =1 ) = 0 , since N v = N , itfollows that, for all k ∈ N , there is x ( k ) ∈ V such that (cid:0) v (cid:0) x ( k ) (cid:1)(cid:1) k (cid:54) = 0 . However, the k -th coordinate of S v ( µ n ) ∞ n =1 (cid:0) x ( k ) (cid:1) is µ k (cid:0) v (cid:0) x ( k ) (cid:1)(cid:1) k , which must be null and, consequently, µ k = 0 for all k and ( µ n ) ∞ n =1 = 0 .Observe that, choosing ( λ n ) ∞ n =1 = (1 , , , . . . ) ∈ (cid:96) ∞ , then v = Λ (( λ n ) ∞ n =1 ) ∈ Λ ( (cid:96) ∞ ) .Since Λ is injective, we have dim (Λ ( (cid:96) ∞ )) = dim ( (cid:96) ∞ ) = c ,and the proof is done. 4. Final remarks
We finish this note by showing that the previous technique gives us the followingimprovement of [7, Theorem 3.1]:
Theorem 4.1.
Let V (cid:54) = { } be a normed vector space and X (cid:54) = { } be a Banach space.Let E be a Banach sequence space such that c ( X ) ⊂ E . If A V,E := { u : V → E : u is linear, continuous and non-injective } (cid:54) = { } then A V,E is (1 , β ) -spaceable, where β = max { c , dim X } . NOTE ON THE GEOMETRY OF CERTAIN CLASSES OF LINEAR OPERATORS 5
Proof.
Fixed v ∈ A V,E \ { } , let x , y ∈ V , with x (cid:54) = y , be such that v ( x ) = (( v ( x )) n ) ∞ n =1 = (( v ( y )) n ) ∞ n =1 = v ( y ) .It is obvious that the subspace M := { u : V → E : u is linear, continuous and u ( x ) = u ( y ) } is contained in A V,E and v ∈ M . It is sufficient to show that M is closed and dim M ≥ β .Since v is not identically zero, there exists ξ ∈ V such that v ( ξ ) = w (cid:54) = 0 . By theHahn-Banach Theorem, there is a continuous linear functional ϕ : E → K , such that ϕ ( w ) = 1 . Let { a γ : γ ∈ Γ } be a Hamel basis of V . For each γ ∈ Γ and each k ∈ N ,let us define w γk = (0 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) k − , a γ , , , . . . ) ∈ E and consider the linear operators T γk : E → E given by T γk ( w ) = ϕ ( w ) w γk (cid:107) w γk (cid:107) .Obviously, the operators T γk are continuous and, thus, the operators R γk = T γk ◦ v arecontinuous too. Notice that R γk ( ξ ) = w γk / (cid:107) w γk (cid:107) and, consequently, π k ◦ R γk (cid:54)≡ and thus, R γk ∈ M for each k ∈ N . Let us see that { R γk : k ∈ N , γ ∈ Γ } ⊂ M is linearly independent. In fact, let ( k , γ ) , . . . , ( k n , γ n ) ∈ N × Γ be pairwise distinctand let λ , . . . , λ n ∈ K such that λ R γ k + · · · + λ n R γ n k n = 0 .Then λ R γ k ( ξ ) + · · · + λ n R γ n k n ( ξ ) = λ w γ k (cid:13)(cid:13) w γ k (cid:13)(cid:13) + · · · + λ n w γ n k n (cid:13)(cid:13) w γ n k n (cid:13)(cid:13) and it is plain that λ = · · · = λ n = 0 if k i (cid:54) = k j whenever i (cid:54) = j , with i, j ∈ { , . . . , n } .With no lost of generality, assuming that k = k = · · · = k p , p ≤ n , and k i (cid:54) = k if i > p , we have that the k -th coordinate of λ w γ k (cid:13)(cid:13) w γ k (cid:13)(cid:13) + · · · + λ n w γ n k n (cid:13)(cid:13) w γ n k n (cid:13)(cid:13) is λ (cid:13)(cid:13) w γ k (cid:13)(cid:13) a γ + · · · + λ p (cid:13)(cid:13)(cid:13) w γ p k p (cid:13)(cid:13)(cid:13) a γ p .By hypothesis, k = · · · = k p implies γ , . . . , γ p pairwise distinct and, therefore, a γ , . . . , a γ p are linearly independent. Hence, λ = · · · = λ p = 0 and λ p +1 w γ p k p (cid:13)(cid:13)(cid:13) w γ p k p (cid:13)(cid:13)(cid:13) + · · · + λ n w γ n k n (cid:13)(cid:13) w γ n k n (cid:13)(cid:13) . D. DINIZ AND A. RAPOSO JR.
Again, with no lost of generality, assuming that k = k p +1 = · · · = k p , p ≤ n , and k i (cid:54) = k if i > p , we have that the k -th coordinate of λ p +1 w γ p k p (cid:13)(cid:13)(cid:13) w γ p k p (cid:13)(cid:13)(cid:13) + · · · + λ n w γ n k n (cid:13)(cid:13) w γ n k n (cid:13)(cid:13) is λ p +1 (cid:13)(cid:13)(cid:13) w γp +1 k p (cid:13)(cid:13)(cid:13) a γ p + · · · + λ p (cid:13)(cid:13)(cid:13) w γ p k p (cid:13)(cid:13)(cid:13) a γ p .By hypothesis, k p +1 = · · · = k p implies γ p +1 , . . . , γ p pairwise distinct and, therefore, a γ p , . . . , a γ p are linearly independent. Hence, λ p +1 = · · · = λ p = 0 . Proceeding inthis way, after finitely many steps, or we get λ = · · · = λ n = 0 or we obtain m < n such that λ = · · · = λ m = 0 and k m +1 , . . . , k n are pairwise distinct. So we have λ m +1 w γ m +1 k m +1 (cid:13)(cid:13)(cid:13) w γ m +1 k m +1 (cid:13)(cid:13)(cid:13) + · · · + λ n w γ n k n (cid:13)(cid:13) w γ n k n (cid:13)(cid:13) and, as we know, this implies λ m +1 = · · · = λ n = 0 . Notice that, at this moment, wehave shown that(4.1) dim M ≥ max {ℵ , dim X } .Supposing that u lies in the closure of M , let ( u n ) ∞ n =1 be a sequence in M such that lim n →∞ u n = u (we are considering, as usual, the canonical sup norm in the set of contin-uous linear operators L ( V ; E ) from V to E ). Since u ( x ) = lim n →∞ u n ( x ) = lim n →∞ u n ( y ) = u ( y ) ,we conclude that u ∈ M . Thus, M is closed in L ( V ; E ) and dim M ≥ c . Since c > ℵ ,from here and (4.1) we get dim M ≥ β . (cid:3) We recall that, as commented in [7] it is not true that (1 , c ) -lineability is inheritedby inclusions. So, the following result, which is proved by combinations of the previoustechniques, shall be noticed. Theorem 4.2.
Let V , X and E be as in Theorem 2.1. The set C V,E := { u : V → E : u is linear, continuous, non-surjective and non-injective } is (1 , c ) -lineable. References [1] R.M. Aron, V.I. Gurariy, J.B. Seoane-Sepúlveda, Lineability and spaceability of sets of functionson R , Proc. Am. Math. Soc. 133 (2005), 795–803.[2] R.M. Aron, L. Bernal-González, D. Pellegrino, J.B. Seoane-Sepúlveda, Lineability: The Searchfor Linearity in Mathematics. Monographs and Research Notes in Mathematics. CRC Press, BocaRaton (2016).[3] R. M. Aron, L. Bernal-González, P. Jiménez-Rodríguez, G. Muñoz-Fernández, J. B. Seoane-Sepúlveda, On the size of special families of linear operators, Linear Algebra Appl. 544 (2018),186–205. NOTE ON THE GEOMETRY OF CERTAIN CLASSES OF LINEAR OPERATORS 7 [4] L. Bernal-González, D. Pellegrino, J.B. Seoane-Sepúlveda, Linear subsets of nonlinear sets of topo-logical vector spaces, Bull. Amer. Math. Soc. 51 (2014) 71–130.[5] D. Diniz, V.V. Fávaro, D. Pellegrino, A. Raposo Jr, Spaceability of the sets of surjective andinjective operators between sequence spaces, to appear in RACSAM.[6] V.V. Fávaro, D. Pellegrino, P. Rueda, On the size of the set of unbounded multilinear operatorsbetween Banach sequence spaces, Linear Algebra Appl. 606 (2020), 144–158.[7] V.V. Fávaro, D. Pellegrino, D. Tomaz, Lineability and spaceability: a new approach, Bull. Braz.Math. Soc. 51 (2020), 27–46.[8] D. Pellegrino, E. Teixeira, Norm optimization problem for linear operators in classical Banachspaces. Bull. Braz. Math. Soc. 40 (2009), 417–431.
Unidade Acadêmica de Matemática e Estatística, Universidade Federal de CampinaGrande, 58109-970 - Campina Grande, Brazil.
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