Generalized quasi-Banach sequence spaces and measures of noncompactness

Given 0 < s ≤ 1 and Ã an s-convex function, s − Ã -sequence spaces are introduced. Several quasi-Banach sequence spaces are thus characterized as a particular case of s − Ã-spaces. For these spaces, new measures of noncompactness are also defi ned, related to the Hausdorff measure of noncompactness. As an application, compact sets in s − Ã-interpolation spaces of a quasi-Banach couple are studied.


INTRODUCTION
Lately many researchers have been interested about diverse issues related to quasi-Banach spaces.These spaces arise in a natural way as a generalization of Banach spaces, where the triangular inequality of the norm is changed by a weaker condition.From a geometrical point of view, the convex unitary ball of the Banach space case is replaced in the quasi-Banach case by a nonconvex unitary ball.Besides the classical works by (Aoki 1942), (Rolewicz 1957(Rolewicz , 1985) ) and (Kalton et al. 1985), the study of geometrical aspects is one of the main issues for these spaces, with several results obtained recently, as it may be seen in the works by Albiac and Kalton (2009), Albiac and Leranóz (2010a, b) and Mastylo and Mleczko (2010).Results on quasi-Banach spaces have been applied in related subjects, for example, to obtain important characterizations on H p H p H spaces, for 0 < p ≤ 1, as it may be seen in (Bownik 2005), (Bownik et al. 2010) and (Gomez and Silva 2011).
Another example of quasi-Banach spaces cames from the interpolation theory.In the case of abstract Banach spaces and operators, this theory began with the classical papers by Lions and Peetre (1964), and Calderón (1964), constituting a very active research fi eld.
The study of geometrical aspects of interpolated spaces is one of the main issues.Besides the normed case, the research about the behavior of quasi-Banach spaces under interpolation methods begun with papers by Krée (1967), Holmsted (1970), Peetre (1970) and Sagher (1972).More recently, the subject has attracted a lot of attention, since several properties and issues from the normed case like interpolation of bilinear EDUARDO B. SILVA, DICESAR L. FERNANDEZ and LUDMILA NIKOLOVA operators, geometric aspects, maximal and minimal functors and compactness, have been generalized to the quasi-Banach case.See, for instance, the works of Bergh and Cobos (2000), Cobos et al. (2007), Grafakos and Mastylo (2006), Ghorbani and Modarres (2007), Molina (2009) and Cobos and Persson (1998).
Sequence Banach spaces are also another very active research subject, deeply connected with several defi nitions, characterizations and properties in functional analysis.In a very interesting paper, Mitani and Saito (2007) introduced a class of sequence spaces, called the ℓ Ã sequence spaces, which presents in a unifi ed form, norm and geometric properties of several well-known Banach sequence spaces.Recent papers by Nikolova and Zachariades (2009) and Zachariades (2011) presents an interpolation theory of couples of Banach spaces modeled on ℓÃ spaces and characterizes several geometric properties.These spaces possess good generality and are very workable for a unifi ed application, but the constructions of Mitani and Saito (2007), Nikolova and Zachariades (2009) and Zachariades (2011) are not directly generalized to quasi-Banach sequence spaces.
The notion of measure of noncompactness was introduced by K. Kuratowski.The Kuratowski measure, as well as its variant, called by some authors the Haudorff measure, has a very important role in functional analysis.It is applied to the theories of differential and integral equations as well as to the operator theory.The relation between measures of noncompactness and interpolation theory of linear and non-linear operators is a very active research topic.See, for example Banás and Goebel (1980) and Fernandez and Silva (2010) and the references therein.
In current work, given 0 < s ≤ 1 and an s-convex function Ã, the s − Ã-sequence spaces are introduced.A necessary condition is given which guarantees the existence of these spaces and some properties are proved, including that they are quasi-Banach spaces.This allow us to characterize some quasi-Banach sequence spaces as a particular case of s − Ã-spaces.
New measures of noncompactness related to Hausdorff measure of noncompactness are also introduced to obtain a quantitative version of a classical result by Phillips (1940, Thm. 3.7) [see also Dunford and Schwartz (1967 Lemma IV.5.4, p. 259)] and Brooks and Dinculeanu (1979, Thm. 1).These quantitative results for the quasi-Banach case seems to be new in the literature.
Interpolation spaces on s−Ã-sequence spaces are also defi ned.Compact sets in this interpolation spaces are investigated and a characterization of them is obtained.

Let us set
AC s,∞ AC be the set of all absolute and monotone s-norms on c such that, for all e i , there are positive constants c i , there are positive constants c i 1 and c 2 satisfying Let AN s AN s AN ,∞ be the set of all absolute, normalized and monotone s-norms on c 00 .We denote by Δ ∞ the set LEMMA DEFINITION 4. Given 0 < s ≤ 1, a linear space V and a function f : V → R, then f is said to be s-convex in the second sense if inequality (1) EDUARDO B. SILVA, DICESAR L. FERNANDEZ and LUDMILA NIKOLOVA holds for all u, v 2 V and α, β ≥ 0 with α + β = 1.The set of all these functions is denoted by K 2 s .The set of all these functions is denoted by K s .The set of all these functions is denoted by K (V ) V ) V .
If s = 1 the defi nition means just convexity.Another useful result is , then f is non-negative on V. , then f is non-negative on V Proof of this fact and several another results about s-convex functions appear in (Hudzik and Maligranda 1994).
Therefore, AN s,∞ AN s,∞ AN and ª s,∞ are in a one-to-one correspondence under the equation 2.9(2).

PROOF. (i) Given
AN , the norm is monotone and one has and so the condition (A) holds.
(ii) From the defi nition of ||.|| Ã the properties (a) and (b) of Defi nition 2.1 are verifi ed.Now, let x 2 c 00 .There exists some n 2 N with x i = 0 for all i > n.
To prove the "triangle" inequality we fi rst show that ||.|| Ã is monotone.Given (p Given (p Given ( i ) i , (a i ) i 2 Δ ∞ such that p i ≤ a i for each n 2 N, there exists some n 2 N with p i = a i = 0 for all i > n.Let us suppose the condition (A) is fulfi lled and 0 ≤ p 1 ≤ a 1 .We denote a = a 1 + p 2 + . . .p n , p = p 1 + p 2 + ... p n and λ = and after condition (A), one has Next step is to prove Since a 2 ≥ p 2 ≥ 0, we just act similar to above.In this way, one can get Finally, for triangle inequality one has, by the s-convexity of Ã, The proof of Theorem 1 shows that the condition (A The proof of Theorem 1 shows that the condition (A The proof of Theorem 1 shows that the condition ( ) is really closed with monotonicity of the norm.
The next result may be proved following the proof of Proposition 2.4 in Mitani and Saito (2007).
PROPOSITION 2. The linear space ℓ Ã and c Ã are s-Banach spaces with the s-norm

RELATIVE COMPACTNESS AND Ã-DIRECT SUMS
We have considered spaces ℓ Ã as spaces of complex numbers labeled in N.An analogous theory may be also developed with labels in N 0 = N [ {0} and even in Z.Therefore, let τ : with (z with (z with ( ^n) 2n+1 = z n and z ^2n = z −n , for n 2 N.For Ã 2 ª s,∞ we defi ne To emphasize which space we are considering, we shall use the notation ℓ Ã (N), ℓ Ã (N 0 ) and ℓ Ã (Z).The next result follows directly from Proposition 2. PROPOSITION 4. Let Ã 2 ª s,∞ and ( and ( and X (X ( n X n X ) n2N be a sequence of Banach spaces. Then, In a similar way, let (X In a similar way, let (X In a similar way, let ( n X n X ) n2Z be a family of Banach spaces.We defi ne Examples of Ã-direct sums are ℓ p -direct sums are ℓ p -direct sums are ℓ direct sums for Ã = Ã p Ã p Ã , 0 < p ≤ 1.
Now, the more general situation of Ã-direct sum of quasi-Banach spaces is considered.
For a characterization of the compact sets in Â Ã we need of the following auxiliary result.
K is relatively compact.K In particular, if X is a fi xed quasi-Banach space and X is a fi xed quasi-Banach space and X X n X n X = X, for each X, for each X n 2 N, we have Thus, we obtain from Corollary 1 a similar result to that stated by Brooks and Dinculeanu (1979 Thm.1), now for s-Banach spaces.COROLLARY 2. A set K ½ ℓ p ℓ p ℓ (X (X ( ), 0 < X), 0 < X p ≤ 1, is relatively compact if, and only if: (ii) for each m 2 N, the set K( the set K( the set K m) = {π m π m π (x (x ( ) : x 2 K} is relatively compact in the quasi-norm of X.

s − Ã-DIRECT SUMS AND INTERPOLATION SPACES
In Nilsson (1982), the K-interpolation space is defi ned for a pair of quasi-Banach spaces ( K-interpolation space is defi ned for a pair of quasi-Banach spaces ( K E -interpolation space is defi ned for a pair of quasi-Banach spaces (E -interpolation space is defi ned for a pair of quasi-Banach spaces ( 0 , E 1 ) and an Z-lattice A (a quasi-Banach space of real valued sequences with Z as index set and with a monotonicity property: The K-space consists of all K-space consists of all K a 2 E 0 +E +E + 1 such that {K(2 K(2 K n , a, E 0 , E 1 )} 2 A. One more condition has been put on A, namely to be K-nontrivial, which equivalently may be written as the condition {min(1, 2 K-nontrivial, which equivalently may be written as the condition {min(1, 2 K − n )} n 2 A. It will be considered the K interpolation space when K interpolation space when K Z-lattice is the space Ã .Now, the following characterization may be proved.THEOREM 4. Let E i be quasi-Banach spaces with quasi-Banach constants C i be quasi-Banach spaces with quasi-Banach constants C i be quasi-Banach spaces with quasi-Banach constants C , i i , i i = 0, 1, and let M be the quasi-Banach constant of c Ã .Then, the interpolation space (E (E ( 0 , E 1 ) θ,Ã is a quasi-Banach space and its quasi-Banach constant does not exceed M max( its quasi-Banach constant does not exceed M max( its quasi-Banach constant does not exceed M C 0 C 0 C ,C 1 ).
From the inequality K( K( K t, a) ≤ max(1, t/ t/ t s /s / )K( K( K s, a) we get and the same for b in the place of a. Thus, In the case of the second part of the theorem, from (3) The results for this measure are analogous to those given in Theorem 3 for the measure ν Ã (B (B ( ).Thus the following theorem for the compactness of bounded sets in the Ã-interpolation spaces may be obtained: THEOREM 5. Given quasi-Banach spaces E 0 Given quasi-Banach spaces E 0 Given quasi-Banach spaces E and E 1 , a bounded set K is relatively compact in (E (E ( 0 E 0 E , E 1 ) θ,Ã , for 0 < θ < 1 and Ã regular if, and only if, Here we use the fact that if K is relatively compact in K is relatively compact in K E 0 + E 1 then, it is also relatively compact in X n X n X = 2 −θn E 0 + 2 −(θ−1)n E 1 .
When Ã = Ã p Ã p Ã , is obtained for the quasi-Banach case the Theorem 3.2 from Fernandez and Silva (2006), which for the Banach case is a result originally given in Peetre (1968) and also proved in Fernandez-Cabrera (2002).Similar results for general real interpolation methods appear in Cobos, et al. (2005).
EDUARDO B. SILVA, DICESAR L. FERNANDEZ and LUDMILA NIKOLOVA Now some operators are introduced who will assist in the attainment of the next result.
If the norm || .||Ã is translation invariant, this constant may be estimated by M2 constant may be estimated by M2 constant may be estimated by M θ C 1−θ 0 C θ 1 .EDUARDO B. SILVA, DICESAR L. FERNANDEZ and LUDMILA NIKOLOVAPROOF.To prove the fi rst part we will use Lemma 3.11.1 fromBergh and Löfstron (1976).Let E i be quasi-normed with constants C i