Generalized quasi-Banach sequence spaces and measures of noncompactness

Abstracts

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

quasi-Banach spaces; quasi-convex functions; measures of noncompactness; interpolation theory

Dado 0 < s ≤ 1 e uma função s-convexa ψ, os espaços de sequencias s – ψ são introduzidos. Vários espaços quase-Banach de sequencias são assim caracterizados como um caso particular dos espaços s – ψ. Para esses espaços novas medidas de não compacidade são também definidas, relacionadas a medida de não compacidade de Hausdorff. Como uma aplicação, conjuntos compactos nos espa, cos de interpolação s – ψ, de um par quase-Banach são estudados.

espaços quase-Banach; funções quase-convexas; medidas de não compacidade; teoria de interpolação

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, 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 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 field.

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 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 definitions, characterizations and properties in functional analysis. In a very interesting paper, Mitani and Saito (2007) introduced a class of sequence spaces, called the ?_ψ sequence spaces, which presents in a unified 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 ?_ψ spaces and characterizes several geometric properties. These spaces possess good generality and are very workable for a unified 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 ψ, the s – ψ -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 – ψ -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–ψ-sequence spaces are also defined. Compact sets in this interpolation spaces are investigated and a characterization of them is obtained.

s-CONVEX FUNCTIONS AND ψ-SEQUENCE SPACES

Let us set

and let ₀₀. Denote |z| = (|z_n |)_{n ∈N} . be the usual basis of c

DEFINITION 1. Given 0 < s ≤ 1, an s-norm on c ₀₀ is a functional satisfying .

(a) ||x|| ≥ 0 for all x ∈ c₀₀ and ||x|| = 0 if, and only if x = 0.

(b) ||λx|| = |λ|||x|| for all ₀₀. and x ∈ c

(c) ||x + y||^s ≤ ||x||^s + ||y||^s for all x, y ∈ c₀₀.

An s-norm on c₀₀is called absolute if ||z|| = || |z| || for every z ∈ c₀₀ and it is called normalized if ||e_i|| = 1, for all .

DEFINITION 2. Given if, and only if |x_i| ≤ |y_i|, An absolute norm on c ₀₀ is monotone if |x| ≤ |y| implies ||x|| ≤ ||y||.. , , then |x| ≤ |y|

DEFINITION 3. Given 0 < s ≤ 1, let ₀₀ such that, for all e_i, there are positive constants c₁ and c₂ satisfying

be the set of all absolute and monotone s-norms on c

Let ₀₀. We denote by Δ_∞ the set

be the set of all absolute, normalized and monotone s-norms on c

LEMMA 1. For all

one has

where ||.||_s and ||.||_∞ are the norms on ? _s and ? _∞, respectively.

PROOF. Given x_n = 0 for n > m and ||x||_∞ = |x₁|. Then,

, let us suppose

Now,

DEFINITION 4. Given 0 < s ≤ 1, a linear space V and a function

, then f is said to be s-convex in the second sense if inequality

holds for all u, v ∈ V and α, β ≥ 0 with α + β = 1. The set of all these functions is denoted by .

If s = 1 the definition means just convexity. Another useful result is

PROPOSITION 1. If then f is non-negative on V. and 0 < s < 1,

Proof of this fact and several another results about s-convex functions appear in (Hudzik and Maligranda 1994).

DEFINITION 5. Given 0 < s ≤ 1, Ψ_{s ,∞} will be the set of all s-convex continuous functions ψ (in the second sense) on Δ_∞, for which ψ(e_n) = 1 and condition (A) is fulfilled, namely

for all λ, whereandwith t_i ≠ 1.,

DEFINITION 6. Given ψ ∈ Ψ_{s ,∞}, the function

is defined by

THEOREM 1. (i). For every

by, we define

Thenψ ∈ Ψ_{s ,∞}.

(ii)For any ψ ∈ Ψ_{s ,∞} and x ∈ c₀₀ we define

Then,and satisfies (2).

Therefore,and Ψ_{s ,∞} are in a one-to-one correspondence under the equation 2.9(2).

PROOF. (i) Given t_i ≠ 1 and λ such that λt_i ≥ (1−λ)+λt_i ≥ 0. Since

implies , then , with , the norm is monotone and one has, for all

and so the condition (A) holds.

(ii) From the definition of ||.||_ψ the properties (a) and (b) of Definition 2.1 are verified. Now, let x ∈ c₀₀. There exists some _i = 0 for all i > n. Then

, where with x

To prove the ”triangle” inequality we first show that ||.||_ψ is monotone.

Given (p_i )_i, (a_i )_i ∈ Δ_∞ such that p_i ≤ a_i for each p_i − a_i = 0 for all i > n. Let us suppose the condition (A) is fulfilled and 0 ≤ p ₁ ≤ a ₁. We denote a − a ₁ + p ₂ + …p_n , p − p ₁ + p ₂ + … p_n and ., there exists some with

Let , i = 2, …, n and i =1, …, n, then

. Note that , , , and after condition (A), one has and . Then

i.e.

which means

Next step is to prove

Since a ₂ ≥ p ₂ ≥ 0, we just act similar to above. In this way, one can get

when a_i ≥ p_i .

Finally, for triangle inequality one has, by the s-convexity of ψ,

The proof of Theorem 1 shows that the condition (A) is really closed with monotonicity of the norm.

DEFINITION 7. Given ψ ∈ Ψ_{s ,∞}, the space ?_ψ is defined by

and c_ψ is the closure of c₀₀ in (?_ψ , || . ||_ψ).

The next result may be proved following the proof of Proposition 2.4 in Mitani and Saito (2007).

PROPOSITION 2. The linear space ?_ψ and c_ψ are s-Banach spaces with the s-norm

EXAMPLE 1. Let ψ_p be the p-convex function obtained from the p-norm || . ||_p of ?_p , 0 < p ≤ 1. Then ψ_p ∈ Ψ_{p ,∞}, and ?_ψ = c_ψ = ?_p .

PROPOSITION 3. Let ψ ∈ Ψ_{s ,∞}. Then, for each

holds

DEFINITION 8. A function ψ ∈ Ψ_{s ,∞} is called regular if c_ψ = ?_ψ .

RELATIVE COMPACTNESS AND ψ-DIRECT SUMS

We have considered spaces ?_ψ as spaces of complex numbers labeled in

and even in be defined by. Therefore, let . An analogous theory may be also developed with labels in

with ψ ∈ Ψ_{s ,∞} we define

. For and , for

To emphasize which space we are considering, we shall use the notation and . The next result follows directly from Proposition 2.,

PROPOSITION 4. Let ψ ∈ Ψ_{s ,∞} and

be a sequence of Banach spaces. Then,

is an s-Banach space when equipped with the norm.

In a similar way, let

be a family of Banach spaces. We define

and . for every

Examples of ψ-direct sums are ?_p direct sums for ψ – ψ_p , 0 < p ≤ 1.

Now, the more general situation of ψ-direct sum of quasi-Banach spaces is considered.

THEOREM 2. If is a sequence of quasi-Banach spaces with quasi-Banach constants C_n , with sup C_n < ∞, then their ψ-direct sum is also a quasi-Banach space.

PROOF. Let C = sup C_n . Then,

where C_ψ is the quasi-Banach constant of ?_ψ .

Using Lemma 1, the completness may be proved as in the Banach case, considered in Proposition 2.4 of Mitani and Saito (2007).

DEFINITION 9. For a bounded set B in X_ψ, the Hausdorff measure of noncompactness of B, χ(B), is defined by

χ_X (B) = inf{ε > 0; there exists a finite set such that }, where U_X is the closed unit ball of X_ψ with center in the origin.,

Now some operators are introduced who will assist in the attainment of the next result.

Let 0 < s ≤ 1, Q_k (x₁, …, x_n , …) = (x₁, …, x_k ) Then, for ψ ∈ Ψ_{s ,∞} we define

, the operator given by a sequence of quasi-Banach spaces as in Theorem 2 and

if(x₁, … , x_k ) ≠ (0, … , 0), and 0 if (x₁, … , x_k ) = (0, … , 0), where ?_ψk is the s–space defined on .

It is clear that k = 2, 3, … and . for every

For a sequence P_k ((x_n )) = (x₁, … , x_k , 0, 0, …) and Π_k ({x_n }) = x_k ., we also define

It is clear that Q_n (P_n (x)) = (x₁, … , x_n ), Q_{n +1}(P_n (x)) = (x₁, … , x_n , 0) and since

, then

On the other hand since

, one has

and

i.e.

Hence

i.e. ₁, … , x_n , … , 0), then P_n (x) = x and thus . If x = (x.

THEOREM 3. For a bounded subset

we define

Then if, _ψ, one has

is the Hausdorff measure of noncompactness on X

for all bounded subset B in χ_ψ .

PROOF. For simplicity, let us denote χ_ψ (B). For each bounded subset B

. by , and

Since χ_ψ is s-sub-additive, taking in account the inequality

we get

for all χ_ψ (B) ≤ v_ψ (B).. Therefore,

Conversely, since operators P_n are uniformly bounded, let us define M : = lim sup_{n ⇒∞} ||P_n ||. Then, since ||P_n || = 1, it follows M = 1.

Given a bounded subset B in X_ψ and ε > 0 arbitrary, let B ₀ in X_ψ , such that

and . Thus, there is a finite set

And, since B ₀ is finite, there exist

, such that

for all n ≥ N and x₀ ∈ B ₀. Now, let x be an arbitrary element in B and x₀ ∈ B ₀ chosen such that

. Since

it holds

and, for all x ∈ B and n ≥ N,

Therefore, taking ε ⇒ 0, one has

Finally, since χ_ψ (P_nB) ≤ ||P_n ||χ_ψ (B) ≤ M χ_ψ (B) ≤ χ_ψ (B), one has

The proof is thus complete.

For a characterization of the compact sets in χ_ψ we need of the following auxiliary result.

LEMMA 2. Let Π_n : χ_ψ⇒χ_n be the natural projection on χ_n. Then, for any bounded subset

one has

PROOF. It will be proved that ₁, … , x_n , …), we put y = (0, … , x_n , 0, … , 0, …). Then

. For any x = (x. First we prove

Hence, if ., then

For a given n-th position. Then Π_n (x) = z and , let x = (0, … , z, 0, … , 0, …), with z in the , which implies that . Hence .

Now, given ε > χ_ψ (B), there exist balls B ₁, … , B_M ∈ X_∈ which B_i = B(x_i, ε), such that

Thus

And, since

for each i, there are elements y₁, … , y_M , such that

Therefore, and the result follows.

COROLLARY 1. A set is relatively compact if, and only if:

(i) uniformly on K, for k ⇒ ∞.

(ii) The set K(m) = Π_m (K) = {Π_m (x) : x ∈ K} is relatively compact in the norm of X_m for each .

PROOF. If is relatively compact, then

. Hence and from Theorem 3.4 we obtain

This means that lim sup is the only one point of condensation of

, hence the lim sup is just a lim. This implies that

which means that K. Hence (i) is fulfilled. uniformly on

To prove (ii), using Lemma 3.5, one has

hence .

Now, assuming (i) and (ii), the former means that

It follows from (ii) that K is relatively compact.. Then we get and thus , i.e. and hence

In particular, if X is a fixed quasi-Banach space and X_n = X, for each

, 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 p < 1, is relatively compact if, and only if: , 0 <

(i) k ⇒ ∞, uniformly for x ∈ K, for k ⇒ ∞.,

(ii) for each , the set K(m) = {Π_m (x) : x ∈ K} is relatively compact in the quasi-norm of X.

s – ψ-DIRECT SUMS AND INTERPOLATION SPACES

In Nilsson (1982), the K-interpolation space is defined for a pair of quasi-Banach spaces (E ₀, E ₁) and an A (a quasi-Banach space of real valued sequences with M exists such that |a_n |_A ≤ M||b_n ||_A whenever |a_n | ≤ |b_n | for each ).-lattice as index set and with a monotonicity property:

The K-space consists of all a ∈ E ₀+E ₁ such that {K(2ⁿ, a, E ₀, E ₁)} ∈ 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⁻ⁿ )}_n ∈ A. It will be considered the K interpolation space when

-lattice is the space

with K-nontrivial, i.e. is . Due to Lemma 2.4, it is not difficult to see that .

Let θ ∈ (0, 1) and X_n = 2^−θn E ₀ + 2^−(θ−1)n E ₁. The K-interpolation space (E ₀, E ₁)_θ,ψ may be defined as the subspace of all constant sequences in a ∈ E ₀ + E ₁, such that ||a||_θ,ψ ≤ ∞, where

be a Banach couple, . Since , this is the space of all

Now, the following characterization may be proved.

THEOREM 4. Let E_i be quasi-Banach spaces with quasi-Banach constants C_i , i = 0, 1, and let M be the quasi-Banach constant of c_ψ. Then, the interpolation space (E ₀, E ₁)_θ,ψ is a quasi-Banach space and its quasi-Banach constant does not exceed M max(C ₀,C ₁). If the norm || . ||_ψ is translation invariant, this constant may be estimated by M2^θ .

PROOF. To prove the first part we will use Lemma 3.11.1 from Bergh and Löfstron (1976). Let E_i be quasi-normed with constants C_i , i = 0, 1. Then

From the inequality K(t, a) ≤ max(1, t/s)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)

Now, choose ^m ≤ C ₁/C ₀ ≤ 2^{m +1}. Then, since

, such that 2 is translation invariant, we have

The same is obtained to

. Therefore,

Now, we may get a result about relative compactness and s – ψ-interpolation spaces using a modification of results from the previous section.

For a sequence τ operator from Section 3, , let us consider the

and

Then, it is appropriate to consider a modification of the measure of noncompactness v, namely

The results for this measure are analogous to those given in Theorem 3 for the measure v_ψ (B). Thus the following theorem for the compactness of bounded sets in the ψ-interpolation spaces may be obtained:

THEOREM 5. Given quasi-Banach spaces E ₀ and E ₁, a bounded set K is relatively compact in (E ₀, E ₁)_θ,ψ, for 0 < θ < 1 and ψ regular if, and only if,

(i) uniformly in x ∈ K, where c(k, n) equals 2^−θk if |k| ≥ n and 0, if k = − n,− n + 1, … , n − 1, n.

(ii) The set K is relatively compact in E ₀ + E ₁.

Here we use the fact that if K is relatively compact in E ₀ + E ₁ then, it is also relatively compact in X_n = 2^−θn E ₀ + 2^−(θ−1)n E ₁.

When ψ = ψ_p , 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).

Research of L. Nikolova was partially supported by Sofia University SRF under contract #133/2012. The authors would like to thank to the anonymous referees for helpful corrections and suggestions.

REFERENCES

• Albiac F and Kalton NJ. 2009. Lipschitz structure of quasi-Banach spaces. Israel J Math 170: 317-335.
• Albiac F and Leránoz C. 2010a. Drops in quasi-Banach spaces. J Geom Anal 20: 525-537.
• Albiac F and Leránoz C. 2010b. Uniqueness of unconditional basis in quasi-Banach spaces which are not sufficiently Euclidean. Positivity 14: 579-584.
• Aoki T. 1942. Locally bounded linear topological spaces. Proc Imp Acad Tokyo 18: 588-594.
• Banás J and Goebel K. 1980. Measures of Noncompactness in Banach Spaces, Marcel Dekker, 97 p.
• Bergh J and Cobos F. 2000. A maximal description for the real interpolation method in the quasi-Banach case. Math Scand 87: 22-26.
• Bergh J and Löfstron J. 1976. Interpolation Spaces: An Introduction. Springer-Verlag, 207 p.
• Bownik M. 2005. Boundedness of operators on Hardy spaces via atomic decompositions. Proc Amer Math Soc 133(12): 3535-3542.
• Bownik M, Li B, Yang D and Zhou Y. 2010. Weighted anisotropic product Hardy spaces and boundedness of sublinear operators. Math Nachr 283(3): 392-442.
• Brooks JK and Dinculeanu N. 1979. Conditional expectation and weak and strong compactness in spaces of Bochner integrable functions. J Mult Anal 9: 420-427.
• Calderón AP. 1964. Intermediate spaces and interpolation, the complex method. Studia Math 24: 113-190.
• Cobos F, Fernández-Cabrera LM and Martínez A. 2005. Compact operators between K- and J-spaces. Studia Math 166: 199-220.
• Cobos F, Fernández-Cabrera LM, Manzano A and Martínez A. 2007. Logarithmic interpolation spaces between quasi-Banach spaces. Zeit Fur Anal Anwendungen 26(1): 65-86.
• Cobos F and Persson LE. 1998. Real interpolation of compact operators between quasi-Banach spaces. Math Scand 82: 138-160.
• Dunford N and Schwartz J. 1967. Linear Operators - Part I. Interscience Pub. Inc., New York, 858 p.
• Fernandez DL and Silva EB. 2006. Hausdorff measures of noncompactness and interpolation spaces. Serdica Math J 32(2-3): 179-184.
• Fernandez DL and Silva EB. 2010. Generalized measures of noncompactness of sets and operators in Banach spaces. Acta Math Hungarica 129: 227-244.
• Fernández-Cabrera LM. 2002. Compact operators between real interpolation spaces. Math Ineq Appl 5: 283-289.
• Gomez Lap and Silva EB. 2011. Vector-valued singular integral operators on product Hardy space Hp. Indian J Math 53(2): 339-369.
• Ghorbani Z and Modarres SMS. 2007. Minimal and maximal description for the real interpolation methods in the case of quasi-Banach n-tuples. Int. J Math Analysis 1(14): 689-696.
• Grafakos L and Mastylo M. 2006. Interpolation of bilinear operators between quasi-Banach spaces. Positivity 10: 409-429.
• Holmsted T. 1970. Interpolation of quasi-normed spaces. Math Scand 26: 177-199.
• Hudzik H and Maligranda L. 1994. Some remarks on s-convex functions. Aequationes Math 48: 100-111.
• Kalton NJ, Peck NT and Rogers JW. 1985. An F-Space Sampler. London Math. Lecture Notes, vol. 89. Cambridge Univ. Press, Cambridge.
• Krée P. 1967. Interpolation d'espaces qui ne sont ni normés ni complets. Applications. Ann Inst Fourier 17: 137-174.
• Lions JL and Peetre J. 1964. Sur une classe d'espaces d'interpolation. Pub Math de l'I H E S 19: 5-68.
• Mastylo M and Mleczko P. 2010. Solid hulls of quasi-Banach spaces of analytic functions and interpolation. Nonl Anal 73: 84-98.
• Mitani K and Saito KS. 2007. On genezalized ?p-spaces. Hiroshima Math J 37: 1-12.
• Molina JAL. 2009. Existence of complemented subspaces isomorphic to ?q in quasi Banach interpolation spaces. Rocky Mt J Math 39(3): 899-926.
• Nikolova L and Zachariades T. 2009. On ψ interpolation spaces. Math Ineq Appl 12(4): 827-838.
• Nilsson P. 1982. Reiteration theorems for real interpolation and approximation spaces. Ann Math Pura Appl 132: 291-330.
• Peetre J. 1968. A theory of interpolation of normed spaces. Lecture notes, Brasilia 1963 [Notas Mat. 39].
• Peetre J. 1970. A new approach in interpolations spaces. Studia Math 34: 23-42.
• Phillips RS. 1940. On linear transformations. Trans Amer Math Soc 48: 516-541.
• Rolewicz S. 1957. On a certain class of linear metric spaces. Bull Acad Pol Sci Cl III(5): 471-473.
• Rolewicz S. 1985. Metric Linear Spaces. In: Mathematics and Its Applications, 2nd ed., East European Series, vol. 20. Reidel, Dordrecht.
• Sagher Y. 1972. Interpolation of r-Banach spaces. Stud Math 41: 45-70.
• Zachariades T. 2011. On ?ψ spaces and infinite ψ-direct sums of Banach spaces. Rocky Mt J Math 41(3): 971-997.

Publication Dates

• Publication in this collection
  June 2013

History

• Received
  28 Nov 2011
• Accepted
  17 July 2012