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A spaceability result in the context of hypergroups

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A spaceability result in the context of hypergroups

Seyyed Mohammad Tabatabaie

Department of Mathematics, University of Qom, Qom, Iran sm.tabatabaie@qom.ac.ir

Bentol Hoda Sadathoseyni

Department of Mathematics, University of Qom, Qom, Iran sb.sadathosseini@stu.qom.ac.ir

Received: 14.8.2017; accepted: 4.12.2017.

Abstract. In this paper, by an elementary constractive technique, it is shown that Lr(Z+) − S

q<rLq(Z+) is non-empty, where Z+is the dual of a compact countable hypergroup introduced by Dunkl and Ramirez. Also, we prove that for each r > 1, Lr(Z+)−S

q<rLq(Z+) is spaceable.

Keywords:locally compact hypergroup, spaceability, Lp–space

MSC 2000 classification:primary 46A45, secondary 43A62, 46B45

1 Introduction and Notations

Suppose that X is a topological vector space. A subset S ⊆ X is called spaceable if S ∪ {0} is large enough to contain a closed infinite dimensional subspace of X. This concept was first introduced in [6] and then the term spaceable was used in [1]. There have been several further works on this notion (for example see [4], [7] and [9]). It is well-known that for each 0 < p, there are sequences in lp not belonging to S

q<plq. Usually, this fact is proved by non-constructive techniques, althought a constructive proof, depending on the Principle of Uniform Boundedness can be found in [8]. Actually, in [3] it is proved that for every p > 0, the set lpS

q<plq is even spaceable.

Let Z+ be the set of non-negative integers equipped with discrete topology, M (Z+) be the space of all Radon measures on Z+, and p0 be a fixed prime number. For any k ∈ Z+ and distinct non-zero m, n ∈ Z+ we put

δk∗ δ0= δ0∗ δk:= δk,

δn∗ δn:= 1

pn−10 (p0− 1)δ0+

n−1

X

k=1

pk−10 δk+p0− 2 p0− 1δn, http://siba-ese.unisalento.it/ c 2018 Universit`a del Salento

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and δm ∗ δn := δmax{m,n}. Then, by [5], Z+ equipped with the convolution

∗ : M (Z+) × M (Z+) → M (Z+) defined by µ ∗ ν :=

Z

Z+

Z

Z+

δm∗ δndµ(m)dν(n), (µ, ν ∈ M (Z+),

and the identity mapping on Z+as involution, is a commutative discrete hyper- group. Also, the measure m defined by

m({k}) :=

1, if k = 0,

(p0− 1)pk−10 , if k ≥ 1,

is a Haar measure for Z+. For more details about locally compact hypergroups see [2].

In the sequel, we consider Z+ with above structure, and for each s > 0 we denote Ls(Z+) := Ls(Z+, m).

In this paper, by an elementary technique, we give an algorithm that gen- erates lots of sequences in Lp(Z+) but not in Lq(Z+) for every q < p. This argument can be regarded as a variant of the technique that was used to prove the Banach-Steinhaus Theorem in [8]. Also, we prove that for each r > 1, the set Lr(Z+) −S

q<rLq(Z+) is spaceable in Lr(Z+).

2 Main Results

This is well-known that:

Lemma 1. There is a sequence (an)n=1 of real numbers such that an > 1 for all n, limn→∞an= 1, and P

n=1 1

nan converges.

Lemma 2. If 0 < q < r < ∞, then Lq(Z+) ⊆ Lr(Z+).

Proof. Let f := (xn)n=0∈ Lq(Z+). Since kf kqq= |x0|q+

X

n=1

|xn|q(p0− 1)pn−10 < ∞, (2.1)

there is a positive number N such that for each n ≥ N ,

|xn| (p0− 1)pn−10 1q

< 1.

But for each n ∈ N,

(p0− 1)pn−10 1q

≥ 1.

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Hence, for all n ≥ N , |xn| < 1, and then |xn|r < |xn|q. Therefore, for each n ≥ N we have

|xn|r(p0− 1)pn−10 < |xn|q(p0− 1)pn−10 . By (2.1) we getP

n=1|xn|r(p0− 1)pn−10 < ∞, i.e. f ∈ Lr(Z+). QED

Lemma 3. Let q < r, p0 be a prime number, and (an)n=1 be a sequence as in Lemma 1. If the sequence x = (xn)n=0 is defined by

xn:=

1 if n = 0,

1

(pn−10 (p0−1)nan)1r if n ≥ 1, then x ∈ Lr(Z+) − Lq(Z+).

Proof. From

k x krr=

X

n=0

|xn|rm({n})

= 1 +

X

n=1

1

pn−10 (p0− 1)nan(p0− 1)pn−10

= 1 +

X

n=1

1 nan < ∞

we conclude that x ∈ Lr(Z+). Let 0 < q < r. Since ar

n < r for every n and limn→∞ r

an = r, there is some t0∈ N such that q < art0. Since

n→∞lim an at0

= 1 at0

< 1,

we can choose s ∈ (a1

t0, 1), and there is N0 ∈ N such that aat0n < s < 1 for all n ≥ N0. Hence,

1 ns < 1

n

an at0

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for all n ≥ N0. We have kx k

r at0r at0 =

X

n=0

|xn|

r

at0 m({n})

= 1 +

X

n=1

1

pn−10 (p0− 1)nan1r

r at0

(p0− 1)pn−10

= 1 +

X

n=1

(p0− 1)pn−10 1−at01 1 n

an at0

.

Since P n=1 1

ns = ∞ and

(p0− 1)pn−10 1−

1 at0 ≥ 1, we have

X

n=N0

1 ns

X

n=N0

1 n

an t0

≤ 1 +

X

n=1

(p0− 1)pn−10 1−

1 at0 1

n

an at0

≤ k x k

r at0r at0

,

and so, x /∈ L

r

at0(Z+). Therefore, x /∈ Lq(Z+) because q < ar

t0 and Lq(Z+) ⊆ L

r

at0(Z+). QED

Theorem 1. If 1 ≤ q < r < ∞, then Lq(Z+) is not closed in Lr(Z+).

Proof. First Proof: Let Cc(Z+) be the space of all functions from Z+ into C with finite support. We have Cc(Z+) ⊆ Lq(Z+) ⊆ Lr(Z+), and Cc(Z+) is dense in Lq(Z+) and in Lr(Z+). So, if Lq(Z+) is closed in Lr(Z+), then we have Lq(Z+) = Lr(Z+), a contradiction.

Second Proof: Let T : Lq(Z+) −→ Lr(Z+) be the inclusion mapping. We claim that T is continuous. For this, suppose that (fn)n=0 is a sequence in Lq(Z+) with fn := (xn,i)i=0, and fn → f in Lq(Z+), where f := (xi)i=0 Lq(Z+). For each 0 < ε < 1 there is N ∈ N such that for all n ≥ N ,

kfn− f kqq= |xn,0− x0|q+

X

i=1

|xn,i− xi|q(p0− 1)pi−10 < ε < 1.

Hence, |xn,i− xi|q< 1 for all n ≥ N and i ∈ Z+. Since q < r, we get

|xn,i− xi|r≤ |xn,i− xi|q,

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where n ≥ N and i ∈ Z+, and so kfn−f kr ≤ kfn−f kqfor all n ≥ N . Therefore, fn→ f with respect to the norm k · kr, and T is continuous.

Now, we show that k · kq and k · kr are not equivalent. In contrast, suppose that there exists M > 0 such that for each f ∈ Lq(Z+), kf kq≤ M kf kr. For any n ∈ N, let χn be the characteristic function of the set {n}. Then for all n ∈ N, we have

(p0− 1)pn−10 1q

= kχnkq≤ M kχnkr= M (p0− 1)pn−10 1r , and so,

(p0− 1)pn−10 1q1r

≤ M,

which is a contradiction, since 1q1r > 0. Therefore, the norms k · kq and k · kr

are not equivalent. Finally, suppose that Lq(Z+) is closed in Lr(Z+). Since T is continuous, the induced mapping

T : Le q(Z+) −→ T (Lq(Z+)) = (Lq(Z+), k · kr), T (f ) := T (f ) = f,e is a bijective continuous linear operator between Banach spaces. So, by the Open Mapping Theorem, eT is an isomorphism, and in particular the norms k · kq and k · kr are equivalent on Lq(Z+), a contradiction. Therefore, Lq(Z+) is not closed

in Lr(Z+). QED

By [9, Theorem 2.4], we can conclude:

Corollary 1. If 0 < q < r < ∞, then Lr(Z+) − Lq(Z+) is spaceable.

Here, we recall the following result (see [9, Theorem 3.3]).

Theorem 2. Suppose that X is a Frechet space, and for each n = 1, 2, . . ., Zn is a Banach space. Let for each n ∈ N, Tn : Zn −→ X be bounded lin- ear operators. If Y := span(S

n=1Tn(Zn)) is not closed in X, then X − Y is spaceable.

Theorem 3. If r > 1, then

Lr(Z+) −[

q<r

Lq(Z+)

is spaceable.

Proof. The sequence x ∈ Lr(Z+) constructed in Lemma 3 does not belong to Lq(Z+) for all q < r, i.e. Lr(Z+) −S

q<rLq(Z+) is non-empty.

Let q < r. Since Cc(Z+) is dense in Lr(Z+), and Cc(Z+) ⊆ Lq(Z+) ⊆ Lr(Z+),

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Lq(Z+) is dense in Lr(Z+), and so S

q<rLq(Z+) is not closed in Lr(Z+). By Lemma 2, the setS

q<rLq(Z+) is actually a countable union:

[

q<r

Lq(Z+) =

[

n=1

Lqn(Z+), (2.2)

where q1< q2 < · · · < r, and moreover, the sequence (qn)nconverges to r. Easily, one can see that span (S

n=1Lqn(Z+)) =S

n=1Lqn(Z+). So, applying Theorem 2 (put Zn:= (Lqn(Z+), k · kqn) and Tn: Zn,→ Lr(Z+)), Lr(Z+) −S

q<rLq(Z+)

is spaceable. QED

Acknowledgements. We would like to thank the referee of this paper for a careful reading and helpful comments. Also, we are specially grateful to Professor G. Botelho for his effective comments and suggestions on this paper.

References

[1] R. M. Aron, V. I. Gurariy, J. B. Seoane-Sep´ulveda: Lineability and spaceability of sets of functions on R, Proc. Amer. Math. Soc., 133, n. 3 (2005), 795–803.

[2] W. R. Bloom, H. Heyer: Harmonic analysis of probability measures on hypergroups, De GruYte, Berlin, 1995.

[3] G. Botelho, D. Diniz, V. V. F´avaro, D. Pellegrino: Spaceability in Banach and quasi-Banach sequence spaces, Linear Algebra Appl., 434, n. 5 (2011), 1255–1260.

[4] G. Botelho, V. V. F´avaro, D. Pellegrino, J. B. Seoane-Sep´ulveda:

Lp[0, 1]\S

q>pLq[0, 1] is spaceable for every p > 0, Linear Algebra Appl., 436 (2012), 2963–2965.

[5] C. F. Dunkl, D. E. Ramirez: A family of countably compact P-hypergroups, Trans.

Amer. Math. Soc., 202 (1975), 339–356.

[6] V. P. Fonf, V. I. Gurariy, M. I. Kadets: An infinite dimensional subspace of C[0, 1]

consisting of nowhere differentiable functions, C. R. Acad. Bulgare Sci., 52 (1999), 13–16.

[7] V. I. Gurariy, L. Quarta: On lineability of sets of continuous functions, J. Math. Anal.

Appl., 294 (2004), 62–72.

[8] H. Hahn: Uber folgen linearer operationen, Monatsh. Math. Phys., 32 (1922), 3–88.

[9] D. Kitson, R. M. Timoney: Operator ranges and spaceability, J. Math. Anal. Appl., 378 (2011), 680–686.

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