Abstract K and J spaces and measure of non-compactness

Math. Nachr. 280, No. 15, 1698 – 1708 (2007) / DOI 10.1002/mana.200510572

Abstract K and J spaces and measure of non-compactness

Fernando Cobos∗1, Luz M. Fernandez–Cabrera∗∗2, and Anton Martınez∗∗∗3

1 Departamento de Analisis Matematico, Facultad de Matematicas, Universidad Complutense de Madrid, 28040 Madrid,Spain

2 Seccion Departamental de Matematica Aplicada, Escuela de Estadıstica, Universidad Complutense de Madrid, 28040Madrid, Spain

3 Departamento de Matematica Aplicada, E.T.S. Ingenieros Industriales, Universidad de Vigo, 36200 Vigo, Spain

Received 4 July 2005, accepted 18 June 2006Published online 8 October 2007

Key words Real interpolation, interpolation with a parameter function, measure of non-compactnessMSC (2000) 46B70, 47B06

We establish a formula for the measure of non-compactness of an operator interpolated by the general realmethod generated by a sequence lattice Γ. The formula is given in terms of the norms of the shift operatorsin Γ.

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

As is well-known, the norm of any operator interpolated by the real or the complex method satisfies the loga-rithmically convex inequality ‖T ‖Aθ,Bθ

≤ ‖T ‖1−θA0,B0

‖T ‖θA1,B1

. This follows easily from the construction of themethods (see [1] or [28]). Similar formulae have been established for other quantities but now some of themrequire a lot of effort to be proved. This is the case of the measure of non-compactness β, a concept that meansmore than only continuity but not so much as compactness.

In 1999, P. Fernandez–Martınez and two of the present authors [9] established the following formula foroperators interpolated by the real method:

βθ,q(T ) ≤ c β0(T )1−θβ1(T )θ. (1.1)

The proof is based on the vector-valued sequence spaces that come up when defining the real interpolation space.Using them and certain families of projections one can decompose and approximate the operator T by otheroperators whose measure of non-compactness can be directly estimated. This technique has its origin in thepapers by Cobos, Edmunds, and Potter [6], Cobos and Fernandez [7], Cobos and Peetre [11] and Cobos, Kuhn,and Schonbek [10].

Previous results on this problem are due to Edmunds and Teixeira [27] (see also the more recent paper [5]).They refer to the special cases when A0 = A1 or B0 = B1 or the couple (B0, B1) satisfies a certain approxima-tion condition. Proofs under these extra assumptions are easier and the results apply also to the complex method.However, it is an open problem if a similar formula to (1.1) holds for the complex method.

The real method (A0, A1)θ,q has two important extensions. The first one is the real method with a functionparameter (A0, A1)ρ,q whose definition is similar to (A0, A1)θ,q but replacing tθ by a more general functionparameter ρ. The second extension is the general real method (A0, A1)Γ. It is defined by replacing the usualweighted Lq norm of (A0, A1)θ,q by a more general lattice norm Γ. It is clear that spaces (A0, A1)Γ includespaces (A0, A1)ρ,q . These two interpolation methods have been studied widely as can be seen in the monographesby Peetre [24] and by Brudnyı and Krugljak [3], or the papers by Gustavsson [17], Janson [19], Cwikel and Peetre

∗ Corresponding author: e-mail: [email protected], Phone: +34 91 394 4453, Fax: +34 91 394 4726∗∗ e-mail: luz [email protected], Phone: +34 91 394 3962, Fax: +34 91 394 4064∗∗∗ e-mail: [email protected], Phone: +34 986 812 153, Fax: +34 986 812 116

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Math. Nachr. 280, No. 15 (2007) 1699

[13], Nilsson [22], [23], Persson [25], Evans, Opic, and Pick [16] and Evans and Opic [15]. To illustrate theirinterest it suffices to consider the couple (L1, L∞). By using the real method we can only obtain Lp and Lp,q

spaces, but interpolating with a function parameter we can get Lorentz–Zygmund spaces Lp,q(log L)γ and otherrelated spaces, and using the general real method we can obtain the majority of symmetric spaces. In fact, anyinterpolation space with respect to (L1, L∞) can be generated by the general real method (see [3] or [23]).

Similar formulae to (1.1) holds for these extensions. The case of the function parameter is due to Cordeiro[12] and the case of the general real method has been established by Szwedek [26] very recently. In both cases,proofs follow step by step the one given in [9]. Brandani da Silva and Fernandez [2] studied also the case ofthe function parameter. They followed a different approach but, as we will show latter, their arguments have aninaccuracy.

In this paper we show a new proof for the corresponding formula to (1.1) in the case of the general real method.The novelty of the approach is the way of handling with the “middle parts” of the operator. These pieces are themore difficult in [9], [12] and [26], but with our new ideas, they can be easily handled. The outcome is an easier,shorter and more transparent proof.

We work with the general real method realized in a discrete way, so Γ will be a lattice of sequences with Z

as index set. It follows from the compactness results of the present authors [8] that some assumptions on Γ areneeded in order that the analogous formula to (1.1) holds. We formulate the assumptions by fixing the behaviourat ∞ and at −∞ of the norms of shift operators in Γ.

The paper is organized as follows. In Section 2, we give some preliminaries on measures of non-compactness.In Section 3, we recall the construction of the general real method. Section 4 deals with norm estimates forinterpolated operators, and Section 5 contains the formula for the measure of non-compactness. We also uncoverthere an inaccuracy in [2], Lemma 4.1.

2 Preliminaries

Let A and B be Banach spaces and let T ∈ L(A, B) be a bounded linear operator. The (ball) measure of non-compactness β(T ) = β(TA,B) = β(T : A → B) of T is defined to be the infimum of the set of all numbersσ > 0 for which there exists a finite subset b1, . . . , bn ⊆ B such that

T (UA) ⊆n⋃

j=1

bj + σUB.

Here, UA (respectively, UB) stands for the closed unit ball of A (respectively, B).It is clear that β(T ) ≤ ‖T ‖, and T is compact if and only if β(T ) = 0. Other properties of β can be found in

[14] and [4].Another functional which measures the non-compactness of T ∈ L(A, B) is γ(T ) defined to be the infimum

of the set of all σ > 0 for which there is a Banach space Z and a compact linear operator R ∈ K(A, Z) such that

‖Tx‖B ≤ σ‖x‖A + ‖Rx‖Z for all x ∈ A.

It can be checked that γ coincides with the seminorm ‖·‖m studied by Lebow and Schechter in [20]. Consequently(see [20, Thm. 3.1])

12

β(T ) ≤ γ(T ) ≤ 2 β(T ) for any T. (2.1)

3 Abstract K and J spaces

Let A = (A0, A1) be a Banach couple, that is, two Banach spaces A0 and A1 which are continuously embeddedin some Hausdorff topological vector space. For t > 0, the K- and the J-functional are defined by

K(t, a) = K(t, a; A0, A1) = inf‖a0‖A0 + t‖a1‖A1 : a = a0 + a1, aj ∈ Aj , a ∈ A0 + A1,

J(t, a) = J(t, a; A0, A1) = max‖a‖A0 , t‖a‖A1 , a ∈ A0 ∩ A1.

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1700 Cobos, Fernandez–Cabrera, and Martınez: Measure of non-compactness

Then K(t, ·)t>0 (respectively, J(t, ·)t>0) is a family of norms on A0 + A1 (respectively, A0 ∩A1), and anytwo of which are equivalent.

We denote by Γ a Banach space of real valued sequences with Z as index set. We assume that Γ contains allsequences with only finitely many non-zero coordinates and that Γ is a lattice, that is, whenever |ξm| ≤ |µm| foreach m ∈ Z and µm ∈ Γ, then ξm ∈ Γ and ‖ξm‖Γ ≤ ‖µm‖Γ.

As Nilsson in [22], we say that Γ is K-non-trivial if

min(1, 2m) ∈ Γ. (3.1)

The lattice Γ is said to be J-non-trivial if

sup

∞∑m=−∞

min(1, 2−m)|ξm| : ‖ξm‖Γ ≤ 1

< ∞. (3.2)

Given a Banach couple A = (A0, A1) and a K-non-trivial sequence space Γ, the K-space AΓ;K = (A0, A1)Γ;K

is formed by all a ∈ A0 + A1 such that K(2m, a) ∈ Γ. We put ‖a‖AΓ;K= ‖K(2m, a)‖Γ.

If Γ is J-non-trivial, the J-space AΓ;J = (A0, A1)Γ;J is defined as the collection of all sums a =∑∞

m=−∞um

(convergence in A0 + A1) where um ⊆ A0 ∩ A1 and J(2m, um) ∈ Γ. We set

‖a‖AΓ;J= inf

‖J(2m, um)‖Γ : a =

∞∑m=−∞

um

. (3.3)

The spaces AΓ;K and AΓ;J are Banach spaces. Conditions (3.1) and (3.2) are essential to get meaningfuldefinitions (see [22] and [3]).

Using the fundamental lemma of interpolation theory (see [1, Lemma 3.3.2]), it follows that AΓ;K is continu-ously embedded in AΓ;J . The converse inclusion fails in general. But, as can be seen in [22, Lemma 2.5], if theCalderon transform

Ωξm =

∞∑k=−∞

min(1, 2m−k)ξk

m∈Z

is bounded in Γ, then AΓ;J → AΓ;K and the norm of this embedding is less than or equal to ‖Ω‖Γ,Γ.Note that boundedness of Ω in Γ implies that Γ is K-non-trivial and J-non-trivial. Indeed, let e0 = δ0

mm∈Z

where δ0m is the Kronecker delta. Then

min(1, 2m) = Ωe0 ∈ Γ

and

sup

∞∑m=−∞

min(1, 2−m)|ξm| : ‖ξm‖Γ ≤ 1

≤ ‖Ω‖Γ,Γ/‖e0‖Γ.

4 Norm estimates for interpolated operators

Let B = (B0, B1) be another Banach couple. We write T ∈ L(A, B) to mean that T is a linear operator fromA0 + A1 into B0 + B1 whose restriction to each Aj defines a bounded operator from Aj into Bj for j = 0, 1. Ifthis is the case, it is clear that the restrictions

T : (A0, A1)Γ;K −→ (B0, B1)Γ;K and T : (A0, A1)Γ;J −→ (B0, B1)Γ;J

are bounded with norms less than or equal to max ‖T ‖A0,B0 , ‖T ‖A1,B1. Better estimates can be obtained byusing the norms of the shift operators on Γ (see [8, Lemma 2.6]). We describe next this fact, which is importantfor our later considerations.

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Math. Nachr. 280, No. 15 (2007) 1701

For k ∈ Z, the shift operator τk is defined by

τkξ = ξm+km∈Z for ξ = ξmm∈Z.

Motivated by [8, Thm. 5.4], in what follows we assume that τk is bounded in Γ for all k ∈ Z and the normssatisfy

limn→∞ 2−n‖τn‖Γ,Γ = 0 and lim

n→∞ ‖τ−n‖Γ,Γ = 0. (4.1)

Definition 4.1 For m ∈ Z, we put f(2m) = ‖τm‖Γ,Γ and we define f on the other points of (0,∞) by settingit constant in each interval [2m, 2m+1). That is, we let

f(t) =∥∥τ[log2t]

∥∥Γ,Γ

, t > 0.

Here, the logarithm is taken in base 2 and [·] is the greatest integer function.

We let

M1 = max1 , ‖τ1‖Γ,Γ.Using (4.1) it is not hard to check that the following quantities are finite:

M2 = sup f(t) : 0 < t ≤ 1 = sup ‖τ−n‖Γ,Γ : n ≥ 0 ,

M3 = sup

f(t)t

: 1 ≤ t < ∞

= sup2−n‖τn‖Γ,Γ : n ≥ 0

.

These constants and the fact that

‖τm+k‖Γ,Γ ≤ ‖τm‖Γ,Γ ‖τk‖Γ,Γ , m, k ∈ Z,

yield the following properties of f .

Lemma 4.2 The following holds.

(i) limt→0 f(t) = 0.

(ii) f(st) ≤ M1f(s)f(t) for all t, s > 0, and f(2ms) ≤ f(2m)f(s) for all s > 0 and m ∈ Z.

(iii) If s < t, then f(s) ≤ M1M2f(t).

(iv) limt→∞f(t)

t = 0.

(v) If s < t, then f(t)t ≤ M1M3

f(s)s .

Next we show that norms of the interpolated operators can be estimated by means of f .

Lemma 4.3 Let A = (A0, A1) and B = (B0, B1) be Banach couples, let T ∈ L(A, B

)and let Γ be a

K-non-trivial sequence space satisfying (4.1). Then

‖T ‖AΓ;K,BΓ;K= 0 if ‖T ‖Aj,Bj = 0 for j = 0 or 1

and

‖T ‖AΓ;K,BΓ;K≤ c ‖T ‖A0,B0f

(‖T ‖A1,B1/‖T ‖A0,B0

)otherwise.

Here, c = f(2).If, in addition, Γ is J-non-trivial and the Calderon transform Ω is bounded in Γ with ‖Ω‖Γ,Γ = δ, then

‖T ‖AΓ;J ,BΓ;K= 0 if ‖T ‖Aj,Bj = 0 for j = 0 or 1

and

‖T ‖AΓ;J ,BΓ;K≤ c δ ‖T ‖A0,B0f

(‖T ‖A1,B1/‖T ‖A0,B0

)otherwise.



P r o o f. Let σj > ‖T ‖Aj,Bj for j = 0, 1. Find r ∈ Z satisfying 2r−1 ≤ σ1/σ0 < 2r. For any m ∈ Z, wehave

K(2m, T a) ≤ σ0K(2mσ1/σ0, a) ≤ σ0K(2m+r, a).

Then

‖Ta‖BΓ;K≤ σ0‖K(2m+r, a)‖Γ ≤ σ0‖τr‖Γ,Γ‖a‖AΓ;K

= σ0f(2σ1/σ0)‖a‖AΓ;K≤ f(2)σ0f(σ1/σ0)‖a‖AΓ;K

.

Now, if ‖T ‖A1,B1 = 0, letting σ1 → 0 and using Lemma 4.2 (i), we see that ‖T ‖AΓ;K,BΓ;K= 0. If ‖T ‖A0,B0 =

0, the argument is similar but letting σ0 → 0 and using Lemma 4.2 (iv). If ‖T ‖Aj,Bj > 0 for j = 0, 1, take anyε > 0 and choose σj = (1 + ε)‖T ‖Aj,Bj . Since

‖T ‖AΓ;K,BΓ;K≤ c (1 + ε)‖T ‖A0,B0f

(‖T ‖A1,B1/‖T ‖A0,B0

)the wanted inequality follows by letting ε → 0.

Finally, in the additional assumption on Γ, we know that the norm of the embedding AΓ;J → AΓ;K is lessthan or equal to δ. Whence, the result follows from the first part.

We end this section with some important examples. For 1 ≤ q ≤ ∞ we let q be the usual space of q-summablescalar sequences with Z as index set. Given any sequence ωm of positive numbers, we put

q(ωm) = ξ = ξm : ωmξm ∈ q .

Example 4.4 For Γ = q

(2−θm

)with 1 ≤ q ≤ ∞ and 0 < θ < 1, K- and J-spaces coincide and they are

equal to the classical real interpolation space

(A0, A1)q(2−θm);K = (A0, A1)q(2−θm);J = (A0, A1)θ,q

(see [1], [3] or [28]). It is not difficult to check that the norm of the Calderon transform Ω in q

(2−θm

)is less

than or equal to(3 − 2θ − 21−θ

)−1.

Example 4.5 Let ρ : (0,∞) → (0,∞) be a function parameter, that is to say, a positive function such that ρ(t)increases from 0 to ∞, ρ(t)/t decreases from ∞ to 0 and, for every t > 0, sρ(t) = sup ρ(ts)/ρ(s) : s > 0 isfinite and sρ(t) = o(max1, t) as t → 0 and t → ∞ (see [17], [18]). Taking 1 ≤ q ≤ ∞ and Γ = q(1/ρ(2m)),K- and J-spaces agree again, being now equal to the real interpolation space with function parameter ρ

(A0, A1)q(1/ρ(2m));K = (A0, A1)q(1/ρ(2m));J = (A0, A1)ρ,q

(see [24], [19], [17], [25]). The norm of Ω in q(1/ρ(2m)) is bounded by the series∑∞

m=−∞ min1, 2−msρ(2m)which converges because sρ(t) = o(max1, t). When 0 < θ < 1 and ρ(t) = tθ, we recover the spaces(A0, A1)θ,q of the previous example.

Shift operators τk in q(1/ρ(2m)) satisfy ‖τk‖q(1/ρ(2m)),q(1/ρ(2m)) ≤ sρ

(2k

). This allows us to work with

sρ instead of f in this case. Note that sρ is a submultiplicative function which satisfies all statements of Lemma4.2 with M1 = M2 = M3 = 1.

Example 4.6 Consider now Γ = q(1/g(2m)), where g : (0,∞) → (0,∞) is a measurable function equiva-lent to a function parameter ρ, meaning that there are positive constants c1, c2 such that

c1 g(t) ≤ ρ(t) ≤ c2 g(t) for all t > 0.

This is the case of the functions

g(θ,γ)(t) = tθ(1 + | log t|)γ for 0 < θ < 1 and γ ∈ R,


Math. Nachr. 280, No. 15 (2007) 1703

and

g(θ;α0,α∞)(t) =

tθ(1 − log t)α0 if 0 < t ≤ 1,

tθ(1 + log t)α∞ if 1 ≤ t < ∞,

where 0 < θ < 1 and α0, α∞ ∈ R. Again we have Aq(1/g(2m));J = Aq(1/g(2m));K . This kind of interpolationspaces has been studied in [21], [15] and [16]. For Γ = q(1/g(θ,γ)(2m)), instead of the function f we can workwith sg(θ,γ)(t) = tθ(1 + | log t|)|γ|. In the case Γ = q(1/g(θ;α0,α∞)(2m)) we can use tθ(1 + | log t|)|α0|+|α∞|.

Examples where K- and J-spaces do not coincide can be found in [8].

5 Interpolation of the measure of non-compactness

Let Γ be a K- and J-non-trivial sequence space with Ω bounded in Γ and with shift operators satisfying (4.1).Let A = (A0, A1), B = (B0, B1) be Banach couples and let T ∈ L

(A,B

). It follows from [8, Thm. 5.4], that

T : (A0, A1)Γ;J → (B0, B1)Γ;K is compact provided T : Aj → Bj compactly for j = 0 or 1. Next we showhow far can be the interpolated operator from being compact when T : Aj → Bj fails to be compact for j = 0, 1.

Given any sequence Wm of Banach spaces and any sequence λm of positive numbers, we denote byΓ(λmWm) the vector-valued sequence space

Γ(λmWm) =u = um : um ∈ Wm and ‖u‖Γ(λmWm) = ‖ λm‖um‖Wm ‖Γ < ∞

.

When λm = 1 for all m ∈ Z, we write simply Γ(Wm).In what follows, Gm denotes the Banach space A0 ∩ A1 endowed with the norm J(2m, ·; A0, A1), and Fm

stands for the Banach space B0 +B1 normed by K(2m, ·; B0, B1). Then the J-space (A0, A1)Γ;J is the quotientspace of Γ(Gm) given by the surjective map π : Γ(Gm) → (A0, A1)Γ;J defined by πum =

∑∞m=−∞um. On

the other hand, the map j : (B0, B1)Γ;K → Γ(Fm) which associates to each element b ∈ B0 + B1 the constantsequence jb = . . . , b, b, b, . . . is a metric injection.

We shall decompose and approximate the operator T by using the maps j, π and the families of projections thatwe introduce next. Given any n ∈ N, we define projections Pn , Q+

n , Q−n on the couple (1(Gm) , 1(2−mGm))

by

Pnum = . . . , 0, 0, u−n, u−n+1, . . . , un−1, un, 0, 0, . . .,Q+

n um = . . . , 0, 0, un+1, un+2, . . . ,Q−

n um = . . . , u−n−2, u−n−1, 0, 0, . . ..Note that, for each n ∈ N, the identity operator I can be written as I = Pn + Q+

n + Q−n . The operators

Pn , Q+n , Q−

n have norm 1 on 1(Gm), 1(2−mGm) and Γ(Gm). Moreover, for each n ∈ N, it holds

‖Q+n ‖1(Gm),1(2−mGm) = 2−(n+1) = ‖Q−

n ‖1(2−mGm),1(Gm)

and

‖Pn‖1(Gm),1(2−mGm) = 2n = ‖Pn‖1(2−mGm),1(Gm). (5.1)

We shall also use these families of projections acting on the couple (∞(Fm) , ∞(2−mFm)) . We call themRn , S+

n , S−n . Note that they have analogous properties.

Theorem 5.1 Let Γ be a K- and J-non-trivial sequence space with the Calderon transform Ω bounded on Γ.Assume that shift operators satisfy

2−n‖τn‖Γ,Γ −→ 0 and ‖τ−n‖Γ,Γ −→ 0 as n −→ ∞and let f be the function introduced in Definition 4.1. Let A = (A0, A1), B = (B0, B1) be Banach couples andlet T ∈ L

(A, B

). Then

β(TAΓ;J ,BΓ;K

) ≤ C β(TA0,B0)f(β(TA1,B1)/β(TA0,B0)

)where C = 12 δM2

1M2M3f(2), with δ = ‖Ω‖Γ,Γ and M1, M2, M3 are the constants in Lemma 4.2.



P r o o f. We write β0 = β(TA0,B0) and β1 = β(TA1,B1). We have the following diagram of bounded opera-tors

1(Gm) π−−−−→ A0T−−−−→ B0

j−−−−→ ∞(Fm),

1(2−mGm) π−−−−→ A1T−−−−→ B1

j−−−−→ ∞(2−mFm),

———————————————————————————-———————————————————————————-

Γ(Gm) π−−−−→ (A0, A1)Γ;JT−−−−→ (B0, B1)Γ;K

j−−−−→ Γ(Fm).

Since π is a metric surjection and ‖I‖BΓ;J ,BΓ;K≤ δ, we have for each n ∈ N

β(TAΓ;J ,BΓ;K

)= β

(Tπ : Γ(Gm) −→ BΓ;K

)≤ β

(TπP2n : Γ(Gm) −→ BΓ;K

)+ β

(Tπ(Q+

2n + Q−2n) : Γ(Gm) −→ BΓ;K

)≤ δβ

(TπP2n : Γ(Gm) −→ BΓ;J

)+ β

(Tπ(Q+

2n + Q−2n) : Γ(Gm) −→ BΓ;K

).

In order to estimate the first term, take any ε > 0 and put σj = (1 + ε)βj for j = 0, 1. By (2.1), there areBanach spaces Z0, Z1 and compact linear operators S0 ∈ K(A0, Z0), S1 ∈ K(A1, Z1) such that

‖Tx‖Bj ≤ 2σj‖x‖Aj + ‖Sjx‖Zj , for all x ∈ Aj , j = 0, 1. (5.2)

Let (Z0 ⊕Z1)1 be the direct sum of Z0 and Z1, normed by ‖(x, y)‖ = ‖x‖Z0 +‖x‖Z1 , and for each m ∈ Z, putWm = (Z0 ⊕ Z1)1 . Find r ∈ Z such that 2r−1 ≤ σ1/σ0 < 2r and let S : Γ(Gm) → Γ(Wm) be the operatordefined by Sum = wm with

wm =

‖τr‖Γ,Γ(S0um, 2m−rS1um) if − 2n ≤ m ≤ 2n,

0 otherwise.

This operator is a finite sum of compact linear operators, so S ∈ K(Γ(Gm), Γ(Wm)).For any u = um ∈ Γ(Gm), it follows from ‖um‖A0 ≤ J(2m, um), ‖um‖A1 ≤ (

2m−r σ1σ0

)−1J(2m, um)

and (5.2) that

‖Tum‖Bj ≤ 2σj

(2m−r σ1

σ0

)−j

J(2m, um) + ‖Sjum‖Zj , j = 0, 1. (5.3)

Let

um =

um if − 2n ≤ m ≤ 2n,

0 otherwise,

and define J(2m, um) similarly. Then, (5.3) and (3.3) yield

‖TπP2nu‖BΓ;J≤ ‖J(2m, T um+r)m‖Γ

≤ ‖τr‖Γ,Γ‖max(‖T um‖B0 , 2m−r‖T um‖B1)‖Γ

≤ ‖τr‖Γ,Γ

∥∥max

(2σ0J(2m, um) + ‖S0um‖Z0 , 2σ0J(2m, um) + 2m−r‖S1um‖Z1

)∥∥Γ

≤ ‖τr‖Γ,Γ

∥∥2σ0J(2m, um) + ‖S0um‖Z0 + 2m−r‖S1um‖Z1

∥∥Γ

≤ ‖τr‖Γ,Γ2σ0‖J(2m, um)‖Γ + ‖Su‖Γ(Wm)

≤ 2f(2)σ0f(σ1/σ0)‖J(2m, um)‖Γ + ‖Su‖Γ(Wm),

where we have used Lemma 4.2 in the last inequality. It follows from (2.1) that

β(TπP2n : Γ(Gm) −→ BΓ;J

) ≤ 4f(2)σ0f(σ1/σ0)


Math. Nachr. 280, No. 15 (2007) 1705

and therefore, for any n ∈ N,

β(TπP2n : Γ(Gm) −→ BΓ;J

) ≤ 4f(2)(1 + ε)β0f(β1/β0).

We proceed now with β(Tπ

(Q+

2n + Q−2n

)). Since j is a metric injection and I = Rn + S+

n + S−n , we have

β(Tπ(Q+

2n + Q−2n) : Γ(Gm) −→ BΓ;K

)≤ 2β

(jTπ

(Q+

2n + Q−2n

): Γ(Gm) −→ Γ(Fm)

)≤ 2

[β(RnjTπQ+

2n

)+ β

(RnjTπQ−

2n

)+ β

(S+

n jTπQ−2n

)+ β

(S−

n jTπQ+2n

)+ β

(S+

n jTπQ+2n

)+ β

(S−

n jTπQ−2n

)],

where the last six operators act from Γ(Gm) into Γ(Fm). We shall estimate each of these terms by the norm ofthe corresponding operator. Moreover, taking into account that the norms of the embeddings ([8, Lemmatas 3.4and 4.5])

Γ(Gm) → (1(Gm), 1(2−mGm)

)Γ;J

= 1(G)Γ;J

and

∞(F )Γ;K =(∞(Fm), ∞(2−mFm)

)Γ;K

→ Γ(Fm)

are less than or equal to 1 and using the norm estimates for interpolated operators given in Lemma 4.3, the problemreduces to estimating the norms of the restrictions of the operators to the spaces in the couples(1(Gm), 1(2−mGm)) and (∞(Fm), ∞(2−mFm)). Subsequently, given any S ∈ L

(1(G), ∞(F )

)we put

‖S‖0 = ‖S‖1(Gm),∞(Fm), ‖S‖1 = ‖S‖1(2−mGm),∞(2−mFm).

By the factorization

1(Gm)

1(2−mGm)

∞(Fm)

∞(2−mFm)

Rn

jTπ

Q+2n

and properties in (5.1), we have∥∥RnjTπQ+2n

∥∥0≤ 2−(2n+1)‖T ‖A1,B12

n −→ 0 as n −→ ∞.

On the other hand, it is clear that∥∥RnjTπQ+2n

∥∥1≤ ‖T ‖A1,B1 for all n ∈ N.

Therefore, applying Lemmas 4.3 and 4.2, we get

β(RnjTπQ+

2n

) ≤ ∥∥RnjTπQ+2n

∥∥1(G)Γ;J ,∞(F )Γ;K

≤ δf(2)∥∥RnjTπQ+

2n

∥∥0f(∥∥RnjTπQ+

2n

∥∥1/∥∥RnjTπQ+

2n

∥∥0

)≤ δf(2)M2

1 M2

∥∥RnjTπQ+2n

∥∥0f (‖T ‖A1,B1) f

(1/‖RnjTπQ+

2n‖0

) −→ 0as n −→ ∞.

The term β(RnjTπQ−

2n

)can be treated similarly.

For S+n jTπQ−

2n, using the factorization



1(2−mGm)

1(Gm)

∞(2−mFm)

∞(Fm)

S+n

jTπ

Q−2n

we derive ∥∥S+n jTπQ−

2n

∥∥1≤ 2−(3n+2)‖T ‖A0,B0 −→ 0 as n −→ ∞.

Since ∥∥S+n jTπQ−

2n

∥∥0≤ ‖T ‖A0,B0 ,

Lemmas 4.3 and 4.2 yield

β(S+

n jTπQ−2n

) ≤ ∥∥S+n jTπQ−

2n

∥∥1(G)Γ;J ,∞(F )Γ;K

≤ δf(2)∥∥S+

n jTπQ−2n

∥∥0f(∥∥S+

n jTπQ−2n

∥∥1/∥∥S+

n jTπQ−2n

∥∥0

)≤ δf(2)M2

1 M3‖T ‖A0,B0f (1/‖T ‖A0,B0) f(∥∥S+

n jTπQ−2n

∥∥1

) −→ 0as n −→ ∞.

Analogously,

β(S−

n jTπQ+2n

) −→ 0 as n −→ ∞.

Consider now S+n jTπQ+

2n. Repeating the arguments in [9], pages 32–33, with only minor modifications weobtain that there is N ∈ N such that for all n ≥ N we have∥∥S+

n jTπQ+2n

∥∥0≤ ∥∥jTπQ+

2n

∥∥0≤ 2(1 + ε)β0

and ∥∥S+n jTπQ+

2n

∥∥1≤ 2(1 + ε)β1.

Whence,

β(S+

n jTπQ+2n

) ≤ ∥∥S+n jTπQ+

2n

∥∥1(G)Γ;J ,∞(F )Γ;K

≤ δf(2)∥∥S+

n jTπQ+2n

∥∥0f(∥∥S+

n jTπQ+2n

∥∥1/∥∥S+

n jTπQ+2n

∥∥0

)≤ 2δf(2)M2

1M2M3(1 + ε)β0f (β1/β0) .

The remaining term β(S−

n jTπQ−2n

)can be estimated similarly.

Collecting all estimates and keeping in mind that Mj ≥ 1 for j = 1, 2, 3, we obtain that for all ε > 0,

β(TAΓ;J ,BΓ;K

) ≤ 12δM21M2M3f(2)(1 + 2ε)β0f

(β1/β0

).

Finally, letting ε → 0, the formula of the theorem follows.

Next we write down the special case of the real method with a function parameter (Example 4.5). We replacef by sρ and so we can take Mj = 1 for j = 1, 2, 3.


Math. Nachr. 280, No. 15 (2007) 1707

Corollary 5.2 Let ρ : (0,∞) → (0,∞) be a function parameter and 1 ≤ q ≤ ∞. Let A = (A0, A1),B = (B0, B1) be Banach couples and let T ∈ L

(A, B

). Then

β(TAρ,q,Bρ,q

) ≤ M β(TA0,B0)sρ

(β(TA1,B1)/β(TA0,B0)

)where M = 12 sρ(2)

∑∞m=−∞ min1, 2−msρ(2m).

A similar result holds for spaces considered in Example 4.6 with sρ replaced by sg.

Remark 5.3 As we have said at the Introduction, the case of the function parameter has been studied in [2]by a different method, based on the fact that given any ε > 0 there is N ∈ N such that

‖Tπ − TπPN‖1(2−jmGm),Bj≤ 4β(TAj ,Bj ) + ε , j = 0, 1 (5.4)

(see [2, Lemma 4.2]). Unfortunately, (5.4) is not true in general as the following example shows. We work withthe sequence spaces ∞ and ∞(n) with N as index set.

Let A = (∞(n), ∞), B = (∞, ∞), choose T ∈ L(A, B

)as the identity operator T ξn = ξn and

take ε = 1/2. Since I : ∞(n) → ∞ is compact, we have β(TA0,B0) = 0. However, for any N ∈ N, it holds‖Tπ − TπPN‖1(Gm),B0 > 1/2. Indeed, let xN =

xN

m

m∈Z

be the vector-valued sequence defined by

xNm =

⎧⎪⎨⎪⎩0, 0, 0, · · · if m > −N − 1,

2m+N ,2m+N

2,2m+N

3, · · ·

if m ≤ −N − 1.

We have ∥∥xN∥∥

1(Gm)=

∞∑m=−∞

max∥∥xN

m

∥∥∞(n)

, 2m∥∥xN

m

∥∥∞

=

∑m≤−N−1

2m+N = 1.

Consequently,

‖Tπ − TπPN‖1(Gm),B0 ≥ ∥∥(Tπ − TπPN)xN∥∥

B0

=∥∥TπxN

∥∥B0

=

∥∥∥∥∥ ∑

m≤−N−1

2m+N ,∑

m≤−N−1

2m+N

2,

∑m≤−N−1

2m+N

3, . . .

∥∥∥∥∥∞

= 1 > 1/2.

Acknowledgements The authors have been supported in part by the Spanish Ministerio de Educacion y Ciencia (MTM2004-01888).

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Abstract K and J spaces and measure of non-compactness