Complex interpolation, minimal methods and compact operators

Math. Nachr. 263–264, 67 – 82 (2004) / DOI 10.1002/mana.200310124

Complex interpolation, minimal methods and compact operators

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 6 March 2002, revised 23 July 2002, accepted 26 July 2002Published online 16 December 2003

Key words Complex interpolation, minimal interpolation methods, maximal interpolation methodsMSC (2000) 46B70

To the memory of Professor Julio R. Bastida

We characterize compact operators between complex interpolation spaces and between spaces obtained by usingcertain minimal methods in the sense of Aronszajn and Gagliardo. Applications to interpolation of compactoperators are also given.

c© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Let A = (A0, A1), B = (B0, B1) be Banach couples and let T : A→ B with T : A0 → B0 compactly (notationis explained in § 2). One of the most important open problems in interpolation theory is to determine whether ornot the interpolated operator by the complex method T : [A0, A1]θ → [B0, B1]θ is compact. A similar problemfor the real interpolation method has been answered in the affirmative by Cwikel [12] (see also the papers byCobos, Edmunds and Potter [6], Cobos and Fernandez [7] and Cobos and Peetre [10]). Positive answers for otherinterpolation methods can be found in the articles by Cobos, Kuhn and Schonbek [9] and by Cwikel and Kalton[13]. In these last two papers affirmative solutions are given for the complex case assuming that the Banachcouples satisfy certain conditions, but in the case of arbitrary couples the problem is still open. See also the morerecent papers by Cwikel, Krugljak and Mastylo [14] and by Schonbek [26].

The complex method shares some properties with the real method. For the problem we are discussing, themore relevant is that both are minimal methods in the sense of Aronszajn and Gagliardo [1]. The real methodis defined by the couple (�1, �1(2−m)) and the intermediate space �q

(2−θm

), while the complex method by

(FL1, FL1(2−m)) and FL1

(2−θm

), where FL1 is the space of Fourier coefficients of Lebesgue integrable

functions (see [19, 23]). But there is an important difference between them: spaces{(�1, �1(2−m)); �q

(2−θm

)}satisfy a certain approximation condition, very useful to work with compact operators, which is not satisfied by{(FL1, FL1(2−m));FL1

(2−θm

)}. Recently, the present authors have opened a new direction to study com-

pactness in [15, 8], where they have characterized compact operators between real interpolation spaces in termsof weaker compactness conditions and convergence of certain sequences of operators involving the K- and theJ-functional. The results apply not only to the classical real method but also to general J- and K-spaces[24, 22, 3].

In the present paper we study this question for the complex method of Calderon and for other minimal methods.In Section 3 we show necessary and sufficient conditions for compactness of operators acting between complexinterpolation spaces. The relevant operators involve now convolutions with the de la Vallee Poussin kernels.

∗ 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


68 Cobos, Fernandez-Cabrera, and Martınez: Interpolation and compactness

Then, in Section 4, we characterize compact operators between spaces generated by the method introduced byPeetre in [25] and the method studied by Gustavsson and Peetre in [18] (called the “± method”). The former isdefined by means of unconditionally convergent series and the latter by using weakly unconditionally convergentseries. These methods produce spaces which are close to those obtained by complex interpolation. For example,on couples of weighted Lp-spaces (p <∞) the three methods give the same space.

Finally, in Sections 5 and 6, we study the more abstract case of minimal methods defined by sequence Banachlattices satisfying the approximation condition, and we apply the new characterizations to improve results ofCobos and Peetre [10], Cobos, Kuhn and Schonbek [9] and Mastylo [21] on interpolation of compact operators.

2 Preliminaries

By a Banach couple A = (A0, A1) we mean two Banach spaces Aj , j = 0, 1, which are continuously embeddedin some Hausdorff topological vector space. Let A0 + A1 be the sum of those spaces and let A0 ∩ A1 be theirintersection. These spaces become Banach spaces when endowed with the norms ‖a‖A0+A1 = K(1, a) and‖a‖A0∩A1 = J(1, a), where for t > 0

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

}, a ∈ A0 +A1 ,

J(t, a) = max{‖a‖A0 , t‖a‖A1

}, a ∈ A0 ∩A1 .

A Banach spaceA is said to be an intermediate space with respect to the couple A ifA0 ∩A1 ↪→ A ↪→ A0 +A1.Here ↪→ means continuous inclusion. The “position” of A within the couple A can be described by using thefunctions

ψ(t, A, A

)= sup{K(t, a) : ‖a‖A = 1} ,

ρ(t, A, A

)= inf

{J(t, a) : a ∈ A0 ∩A1, ‖a‖A = 1

}(see [5]) .

Let B = (B0, B1) be another Banach couple. We define an operator T : A → B to be a linear operator fromA0 +A1 into B0 +B1 whose restriction to each Aj is a bounded operator from Aj into Bj ( j = 0, 1). We put

‖T ‖A,B = max{‖T ‖A0,B0 , ‖T ‖A1,B1

}.

In the following sections we deal with interpolation methods, that is, procedures F that associate to each Banachcouple A an intermediate space F

(A)

in such a way that given any other Banach couple B and any T : A→ B,the restriction of T to F

(A)

gives a bounded operator from F(A)

into F(B).

Some interpolation methods are defined by using unconditionally convergent series and weakly uncondition-ally convergent series. Recall that a series

∑∞m=−∞am in the Banach space A is said to be unconditionally

convergent if, for every bounded sequence {λm} of scalars, the series∑∞

m=−∞λmam converges. The series∑∞m=−∞am is called weakly unconditionally convergent if there exists a constant C > 0 such that for every

finite set K ⊆ Z and every sequence {λm} with sup{|λm| : m ∈ Z} ≤ 1, it holds∥∥∑

m∈Kλmam

∥∥A≤ C. See

[19, 20].To characterize compactness of the interpolated operators we shall use the following lemmas. The proofs are

implicit in the arguments of [10, Thms. 1.6 and 1.8].

Lemma 2.1 Let X, A, B, Y be Banach spaces, and assume that B is continuously embedded in Y . LetT : A→ B be a linear, compact operator and for every n ∈ N let Qn : X → A be a linear operator such that

supn∈N

‖Qn‖X,A < ∞ and limn→∞ ‖TQn‖X,Y = 0 .

Then

limn→∞ ‖TQn‖X,B = 0 .

Corollary 2.2 Let X be a Banach space and let A = (A0, A1), B = (B0, B1) be Banach couples. Let A, Bbe intermediate spaces of A, B, respectively. Let T : A → B such that T (A) ⊆ B and T : A → B is compact.


Math. Nachr. 263–264 (2004) / www.interscience.wiley.com 69

For n = 1, 2, . . . let Qn : X → A and assume that

supn∈N

‖Qn‖X,A < ∞ and limn→∞ ‖TQn‖X,A0+A1 = 0 .

Then

limn→∞ ‖TQn‖X,B = 0 .

Lemma 2.3 Let X, Y, Z be Banach spaces, let D be a dense subspace of X and let T : X → Y be a linear,compact operator. Let Sn : Y → Z be linear and bounded for n = 1, 2, . . . . If supn∈N

‖Sn‖Y,Z < ∞ and iflimn→∞‖SnTx‖Z = 0 for all x ∈ D, then

limn→∞‖SnT ‖X,Z = 0 .

Let �q (1 ≤ q ≤ ∞) and c0 be the usual spaces of q-summable [respectively, null] scalar sequences with Z asindex set. For {wm} any positive sequence and E any sequence space, define E(wm) by

E(wm) ={ξ = {ξm} : {wmξm} ∈ E

}.

3 Complex interpolation and compact operators

Given a Banach couple A = (A0, A1), we define F(A)

to be the space of all continuous 2πi-periodic functionsf from the closed strip S = {z ∈ C : 0 ≤ Re z ≤ 1} into A0 + A1, that are analytic on the interior of S, withf(j+ it) ∈ Aj for all t ∈ R, j = 0, 1, and the functions t→ f(j+ it) are continuous from R intoAj . The spaceF(A)

becomes a Banach space when normed by

‖f‖F(A) = max{

supt∈R

‖f(it)‖A0, supt∈R

‖f(1 + it)‖A1

}.

For 0 < θ < 1, the complex interpolation space of Calderon A[θ] = [A0, A1]θ consists of all a ∈ A0 +A1 suchthat a = f(θ) for some f ∈ F

(A). We endow [A0, A1]θ with the norm

‖a‖[θ] = inf{‖f‖F(A) : f(θ) = a, f ∈ F

(A)}.

So, the linear map π : F(A) → [A0, A1]θ defined by π(f) = f(θ), is surjective and [A0, A1]θ coincides with thequotient space of F(A) given by π.

This definition of the complex method is slightly different from those in [4, 2 or 27] where functions of F(A)

are not suppose to be periodic. However, as can be seen in [11], it gives the same spaces up to equivalence ofnorms.

To characterize compact operators between complex interpolation spaces, we shall work with Fourier coeffi-cients of functions of F

(A). Let f ∈ F

(A)

and k ∈ Z. The k-th Fourier coefficient of f is defined by

f(k) =12π

∫ 2π

0

e−kitf(it) dt =e−k

2π

∫ 2π

0

e−kitf(1 + it) dt .

It is clear that f(k) ∈ A0 ∩A1 with∥∥f(k)∥∥

A0≤ ‖f‖F(A) ,

∥∥f(k)∥∥

A1≤ e−k ‖f‖F(A) . (3.1)

The series∑∞

k=−∞f(k)ekθ converges Cesaro 1 to f(θ) in [A0, A1]θ . Moreover,∑∞

k=−∞f(k)ekθ is convergentin A0 +A1 because, by (3.1), we have

∞∑k=−∞

∥∥f(k)∥∥

A0+A1ekθ ≤

∞∑k=0

∥∥f(k)∥∥

A1ekθ +

−∞∑k=−1

∥∥f(k)∥∥

A0ekθ ≤



≤∞∑

k=0

ek(θ−1)‖f‖F(A) +−∞∑

k=−1

ekθ ‖f‖F(A) =(

11 − eθ−1

+e−θ

1 − e−θ

)‖f‖F(A) .

Whence,

f(θ) =∞∑

k=−∞f(k)ekθ in A0 +A1 . (3.2)

For n ∈ N, let Vn be the de la Vallee Poussin kernel defined by

Vn(t) =∑|k|≤n

eikt +∑

n<|k|≤2n

(2 − |k|

n

)eikt .

Following [13], if f ∈ F(A), we put

Snf(z) = Vn ∗ f(z)

=12π

∫ 2π

0

Vn(−t)f(z + it) dt =∑|k|≤n

f(k)ekz +∑

n<|k|≤2n

(2 − |k|

n

)f(k)ekz .

Since

supn∈N

12π

∫ 2π

0

|Vn(t)| dt = CV < ∞ ,

we have that Snf ∈ F(A) with

‖Snf‖F(A) ≤ CV ‖f‖F(A) . (3.3)

Note also that for each n ∈ N

πSn : F(A) −→ A0 ∩A1 boundedly . (3.4)

Indeed, according to (3.1) we get

‖Snf(θ)‖A0∩A1 ≤∑|k|≤n

∥∥f(k)∥∥

A0∩A1ekθ +

∑n<|k|≤2n

(2 − |k|

n

)∥∥f(k)∥∥

A0∩A1ekθ

≤[ ∑

|k|≤n

max{1, e−k

}ekθ +

∑n<|k|≤2n

(2 − |k|

n

)max

{1, e−k

}ekθ

]‖f‖F(A) .

The following result is contained in [13, Lemma 5].

Lemma 3.1 Let A = (A0, A1) be a Banach couple. There is a constant C depending on θ such that

‖(f − Snf)(θ)‖A0+A1 ≤ C(e−n(1−θ) + e−nθ

) ‖f‖F(A)

for all f ∈ F(A).

P r o o f. According to (3.2) and (3.1), we have

‖(f − Snf)(θ)‖A0+A1

=

∥∥∥∥∥∞∑

m=−∞f(m)emθ −

∑|m|≤n

f(m)emθ −∑

n<|m|≤2n

(2 − |m|

n

)f(m)emθ

∥∥∥∥∥A0+A1

≤∥∥∥∥∥ ∑

n<m≤2n

(mn

− 1)f(m)emθ +

∑2n<m

f(m)emθ

∥∥∥∥∥A1

+



+

∥∥∥∥∥ ∑n<m≤2n

(mn

− 1)f(−m)e−mθ +

∑2n<m

f(−m)e−mθ

∥∥∥∥∥A0

≤∑n<m

∥∥f(m)∥∥

A1emθ +

∑n<m

∥∥f(−m)∥∥

A0e−mθ

≤( ∑

n<m

e−m(1−θ) +∑n<m

e−mθ

)‖f‖F(A)

≤ e−n(1−θ)

1 − eθ−1+

e−nθ

1 − e−θ.

�

We are now able to establish the characterization of compact operators between complex interpolation spaces.

Theorem 3.2 Let A = (A0, A1), B = (B0, B1) be Banach couples and let T : A → B. Then for any0 < θ < 1, the interpolated operator T : [A0, A1]θ → [B0, B1]θ is compact if and only if the followingconditions hold.

(a) T : A0 ∩A1 → [B0, B1]θ is compact.

(b) sup{‖T (f − Snf)(θ)‖B[θ]

: ‖f‖F(A) ≤ 1}→ 0 as n→ ∞.

P r o o f. Suppose that T : [A0, A1]θ → [B0, B1]θ is compact. Since A0 ∩ A1 is continuously embedded in[A0, A1]θ , it is clear that T : A0 ∩ A1 → [B0, B1]θ is compact. Condition (b) follows from Lemma 3.1 andCorollary 2.2.

Conversely, assume that (a) and (b) hold. Condition (a) and (3.4) imply that for any n ∈ N

TπSn : F(A) −→ [B0, B1]θ

is compact. Condition (b) means that the sequence {TπSn}n∈N converges to Tπ in L(F(A), B[θ]

). Then

Tπ : F(A) → [B0, B1]θ is compact and so, since π is a metric surjection, we conclude that T : [A0, A1]θ →

[B0, B1]θ is compact. The proof is complete.

4 The methods of Peetre and Gustavsson-Peetre and compact operators

Let f be a function parameter, that is, let f be a positive function on (0,∞) such that f(t) increases from 0 to∞, f(t)/t decreases from ∞ to 0 and for every t > 0

sf (t) = sup {f(tu)/f(u) : u > 0}is finite, with

sf (t) = o(max{1, t}) as t → 0 and t → ∞ (see [18, 16]) .

The main example of function parameter is f(t) = tθ, where 0 < θ < 1. Then sf = f . In a more generalway, if 0 < θ < 1 and −∞ < γ < ∞, then g(t) = tθ(1 + | log t|)γ is equivalent to a function parameter andsg(t) = tθ(1 + | log t|)|γ|.

Given a Banach couple A = (A0, A1), let UCf

(A)

be the Banach space of all sequences {am}m∈Z with{am} ⊆ A0 ∩A1 and such that

∑∞m=−∞am/f(2m) and

∑∞m=−∞2mam/f(2m) are unconditionally convergent

in A0 and A1, respectively. The norm in UCf

(A)

is given by

‖{am}‖UCf(A) = maxj=0,1

sup

{∥∥∥∥∥∞∑

m=−∞

2jmλmam

f(2m)

∥∥∥∥∥Aj

: ‖{λm}‖�∞ ≤ 1

} .



The interpolation space 〈A0, A1〉fconsists of all a ∈ A0 + A1 which are sums of the form a =

∑∞m=−∞am

with {am} ∈ UCf

(A). The norm in 〈A0, A1〉f

is defined by

‖a‖〈A0,A1〉f= inf

{‖{am}‖UCf(A) : a =

∞∑m=−∞

am

}.

For f(t) = tθ , this interpolation method was introduced by Peetre in [25]. The case of a general functionparameter was studied by Janson [19] and Ovchinnikov [23]. When f(t) = tθ , we write 〈A0, A1〉θ. The space〈A0, A1〉θ is contained in the complex interpolation space [A0, A1]θ. If A = (A0, A1) is a Banach couple ofcomplexified Banach lattices of measurable functions on some measure space, then 〈A0, A1〉θ = [A0, A1]θ .Details may be found in [19, 23].

To characterize compact operators between spaces obtained by interpolation using the 〈·, ·〉f

-method, we de-note now by π the operator from UCf

(A)

into 〈A0, A1〉fassigning to every sequence {am} its sum π{am} =∑∞

m=−∞am. It is clear that π is a metric surjection. We shall also need operators {Pn}n∈N defined by

Pn{am} = {. . . , 0, 0, a−n, a−n+1, . . . , an−1, an, 0, 0, . . .} .For any n ∈ N, the operator Pn belongs to L

(UCf

(A), UCf

(A))

and its norm is 1. Note also that πPn :UCf

(A)→ A0 ∩A1 is bounded because

‖πPn{am}‖A0∩A1 = max

{∥∥∥∥∥n∑

m=−n

f(2m)am

f(2m)

∥∥∥∥∥A0

,

∥∥∥∥∥n∑

m=−n

f(2m)2m

2mam

f(2m)

∥∥∥∥∥A1

}

≤ max{

max−n≤m≤n

{f(2m)} , max−n≤m≤n

{f(2m)

2m

}}‖{am}‖UCf(A)

≤ max{f(2n), 2nf(2−n)

} ‖{am}‖UCf(A) .

Theorem 4.1 Let A = (A0, A1), B = (B0, B1) be Banach couples and let T : A → B. Then for anyfunction parameter f , the interpolated operator T : 〈A0, A1〉f

→ 〈B0, B1〉fis compact if and only if the

following conditions hold.(a) T : A0 ∩A1 → 〈B0, B1〉f

is compact.

(b) sup{∥∥T (∑|m|>nam)

∥∥〈B0,B1〉f

: ‖{am}‖UCf(A) ≤ 1}

−→ 0 as n → ∞.

P r o o f. If conditions (a) and (b) are satisfied, then for each n ∈ N, the operator TπPn : UCf

(A) →

〈B0, B1〉fis compact because it can be factorized in the form

UCf

(A) 〈B0, B1〉f

A0 ∩A1�

��

��

�

��

��

πPn T

Furthermore, by (b), the sequence{‖Tπ − TπPn‖UCf (A),〈B0,B1〉f

}converges to 0. ThereforeTπ : UCf

(A)→

〈B0, B1〉fis compact. Since π is a metric surjection, this implies that T : 〈A0, A1〉f

→ 〈B0, B1〉fis also com-

pact.Conversely, if T : 〈A0, A1〉f

→ 〈B0, B1〉fis compact, then (a) follows from the continuous inclusion

A0 ∩A1 ↪→ 〈A0, A1〉f.

It is easy to see that

supn∈N

‖π − πPn‖UCf (A),〈A0,A1〉f≤ 1 . (4.1)



On the other hand, given any {am} ∈ UCf

(A), we have

‖(π − πPn){am}‖A0+A1 =

∥∥∥∥∥ ∑|m|>n

am

∥∥∥∥∥A0+A1

≤∥∥∥∥∥ ∑

m>n

f(2m)2m

2mam

f(2m)

∥∥∥∥∥A1

+

∥∥∥∥∥ ∑m<−n

f(2m)am

f(2m)

∥∥∥∥∥A0

≤ max{f(2n)

2n, f(2−n)

}‖{am}‖UCf(A) .

Therefore

‖π − πPn‖UCf (A),A0+A1≤ max

{sf (2n)

2n, sf (2−n)

}f(1) −→ 0 as n → ∞ , (4.2)

and (b) follows from (4.1), (4.2) and Corollary 2.2.

If we replace “unconditionally convergent series” by “weakly unconditionally convergent series” in the defini-tion of 〈A0, A1〉f

, then we get the interpolation space 〈A0, A1, f〉 introduced by Gustavsson and Peetre in [18].The norm in 〈A0, A1, f〉 is defined by

‖a‖〈A0,A1,f〉= infa=

∞∑m=−∞

am

maxj=0,1

sup

∥∥∥∥∥∑

m∈K

2jmλmam

f(2m)

∥∥∥∥∥Aj

: K ⊆ Z, #(K) < ∞, ‖{λm}‖�∞ ≤ 1

.

It turns out that 〈A0, A1〉fis the closure of A0 ∩ A1 in 〈A0, A1, f〉 [19, Thm. 8]. Moreover, 〈A0, A1〉f

=〈A0, A1, f〉 provided that A1 does not have a subspace isomorphic to c0 [19, Thm. 7]. Additional information onthis interpolation method can be found in [17, 23].

Repeating the proof of Theorem 4.1 with only minor modifications we get the following characterization ofcompact operators between Gustavsson-Peetre’s spaces.

Theorem 4.2 Let A = (A0, A1), B = (B0, B1) be Banach couples and let T : A → B. Then for anyfunction parameter f , the interpolated operator T : 〈A0, A1, f〉 → 〈B0, B1, f〉 is compact if and only if thefollowing conditions hold:

(a) T : A0 ∩A1 → 〈B0, B1, f〉 is compact.

(b) sup{∥∥T (∑|m|>nam)

∥∥〈B0,B1,f〉 : ‖{am}‖WUCf (A) ≤ 1

}→ 0 as n→ ∞.

5 Minimal methods and compact operators

We first recall the definition of Aronszajn-Gagliardo minimal method in the way presented in [10]. Let X =(X0, X1) be a fixed Banach couple and let X be a fixed intermediate space with respect to X . Given any Banachcouple A = (A0, A1), let U = U

(A)

be the collection of all S : X → A such that ‖S‖X,A ≤ 1. If W is anyBanach space, we write �1[W ] = �1

[U(A),W]

to mean the Banach space of all absolutely summable familiesw = {wS} of elements of W with U as index set, normed by

‖w‖�1[W ] =∑S∈U

‖wS‖W .

For W = X0 +X1 and x = {xS} ∈ �1[X0 +X1], we put

πx =∑S∈U

SxS .

Note that π : �1[X0 +X1] → A0 +A1 is bounded with norm less than or equal to 1. Furthermore, the restrictionof π to �1

[Xj

]gives a bounded operator from �1

[Xj

]into Aj (j = 0, 1) with

∥∥π : �1[Xj

]→ Aj

∥∥ ≤ 1.



The Aronszajn-Gagliardo minimal method G(A)

= G[X;X

](A)

consists of all those a ∈ A0 + A1, suchthat there exists x ∈ �1[X ] with a = πx. The space G

(A)

is a Banach space in the natural quotient norm

‖a‖G(A) = inf

{∑S∈U

‖xS‖X : a = πx

}.

Interpolation methods that we have worked with in the previous sections, can be described as minimal meth-ods. To be exact, given any function parameter f , we have

〈A0, A1〉f= G

[(c0, c0(2−m)); c0(1/f(2m))

](A0, A1) , (5.1)

〈A0, A1, f〉 = G[(c0, c0(2−m)); �∞(1/f(2m))

](A0, A1) (5.2)

(see [19, 23]). On the other hand, if 0 < θ < 1 and FL1 is the space of Fourier coefficients of Lebesgueintegrable functions

FL1 ={{f(m)

}m∈Z

: f ∈ L1[0, 2π]}

with the norm∥∥{f(m)

}∥∥FL1

= ‖f‖L1[0,2π], then

[A0, A1]θ = G[(FL1, FL1(2−m));FL1

(2−θm

)](A0, A1) (see [19]) . (5.3)

The real method (A0, A1)θ,q [2, 27] and the real method with a function parameter (A0, A1)f,q [24, 16] areother important interpolation methods that can be also realized as minimal methods. We have

(A0, A1)θ,q = G[(�1, �1(2−m)); �q

(2−θm

)](A0, A1) , (5.4)

(A0, A1)f,q = G[(�1, �1(2−m)); �q(1/f(2m))

](A0, A1) . (5.5)

Let E �= {0} be a Banach space of scalar sequences over Z. We assume that E is a lattice, that is, whenever|ξm| ≤ |µm| for each m ∈ Z and {µm} ∈ E then {ξm} ∈ E and ‖{ξm}‖E ≤ ‖{µm}‖E . We also assume thatE is symmetric, meaning that for any k ∈ Z, the shift operator τk{ξm}m∈Z = {ξm+k}m∈Z satisfies that

‖τkξ‖E = ‖ξ‖E for any ξ = {ξm}m∈Z ∈ E .

In what follows, we suppose that (X0, X1) = (E,E(2−m)) with E as before. Thus

‖τk‖X0,X0 = 1 , ‖τk‖X1,X1 = 2k , k ∈ Z . (5.6)

Referring to X we suppose that it is a Banach lattice of sequences over Z, intermediate with respect to X =(X0, X1) and such that for each k ∈ Z the operator τk is bounded in X . Note that this kind of minimal methodscomprises examples (5.1), (5.2), (5.4) and (5.5).

Given any T : A → B, it is easy to check that ‖T ‖G(A),G(B) ≤ ‖T ‖A,B . Next, guided by [8, 21], we showthat a more useful estimate holds. Namely, the norm of the interpolated operator can be controlled by the normsof shift operators.

Lemma 5.1 Let (X0, X1) and X be as before. Let A = (A0, A1), B = (B0, B1) be Banach couples and letT : A→ B. Then for any n ∈ N:

(i) if ‖T ‖A0,B0 ≤ 2−n and ‖T ‖A1,B1 ≤ 1, then ‖T ‖G(A),G(B) ≤ 2−n‖τn‖X,X;(ii) if ‖T ‖A0,B0 ≤ 1 and ‖T ‖A1,B1 ≤ 2−n, then ‖T ‖G(A),G(B) ≤ ‖τ−n‖X,X .

P r o o f. Suppose ‖T ‖A0,B0 ≤ 2−n and ‖T ‖A1,B1 ≤ 1. The proof of the other case can be carried out in thesame way.

Let a ∈ G(A)

with ‖a‖G(A) ≤ 1. Given any ε > 0, we can find a summable family x = {xS} ∈ �1[X ] witha = πx and

∑S∈U‖xS‖X ≤ 1 + ε. Thus

Ta =∑S∈U

TSxS =∑S∈U

2nTSτ−nτn(2−nxS) .



Operators 2nTSτ−n belong to L(X,B) and, by (5.6), we have

‖2nTSτ−n‖X0,B0 ≤ 2n ‖T ‖A0,B0 ‖τ−n‖X0,X0 ≤ 2n ‖T ‖A0,B0 ≤ 1 ,

‖2nTSτ−n‖X1,B1 ≤ 2n ‖T ‖A1,B1 ‖τ−n‖X1,X1 ≤ 2n ‖τ−n‖X1,X1 ≤ 1 .

Therefore

‖Ta‖G(B) ≤∑S∈U

‖τn(2−nxS)‖X ≤ (1 + ε)2−n ‖τn‖X,X .

Taking the infimum over all ε > 0, the result follows.

Remark 5.2 It follows from the lemma that if the norms of the shift operators in X satisfy

2−n ‖τn‖X,X −→ 0 and ‖τ−n‖X,X −→ 0 as n → ∞ ,

and if Tn : A → B for n = 1, 2, . . . , with supn∈N ‖Tn‖A,B < ∞ and either{‖Tn‖A0,B0

}or{‖Tn‖A1,B1

}converges to 0, then

{‖Tn‖G(A),G(B)

}converges to 0.

Functions ψ(t, G(A), A)

and ρ(t, G(A), A)

can be also estimated by means of the norms of shift operators:

Lemma 5.3 Let (X0, X1) and X be as before, and let A = (A0, A1) be a Banach couple. Then there areconstants C1, C2 > 0 such that for each n ∈ N:

(i) ψ(2n, G

(A), A) ≤ C1 ‖τn‖X,X ;

(ii) ρ(2−n, G(A), A

) ≥ 1/C2 ‖τn‖X,X .

P r o o f. The result is an easy consequence of Lemma 5.1 by using the same argument as in [8, Lemma 2.9].

For ξ = {ξm}m∈Z ∈ X0 +X1 and n ∈ N, we let

Pnξ = {. . . , 0, 0, ξ−n, ξ−n+1, . . . , ξn−1, ξn, 0, 0, . . .} ,Q+

n ξ = {. . . , 0, 0, ξn+1, ξn+2, . . . } ,Q−

n ξ = {. . . , ξ−n−2, ξ−n−1, 0, 0, . . . } .

Given any Banach couple A = (A0, A1), the operators Pn, Q+n , Q

−n extend in a natural way to the operators

Pn, Q+n , Q

−n from �1[X0 +X1] into �1[X0 +X1]. Indeed, for x = {xS} ∈ �1[X0 +X1] we set

Pnx = {PnxS}S∈U

and we define Q+n and Q−

n similarly.The fundamental properties of these operators areE1: The identity operator I on �1[X0 +X1] can be decomposed as I = Pn + Q+

n + Q−n for each n ∈ N .

E2: They are uniformly bounded in (�1[X0], �1[X1]),

maxj=0,1

{∥∥Pn

∥∥�1[Xj ],�1[Xj ]

,∥∥Q+

n

∥∥�1[Xj ],�1[Xj ]

,∥∥Q−

n

∥∥�1[Xj ],�1[Xj ]

}≤ 1 , n = 1 , 2 , . . .

E3: For each n ∈ N, max{∥∥Q+

n

∥∥�1[X0],�1[X1]

,∥∥Q−

n

∥∥�1[X1],�1[X0]

}≤ 2−(n+1) and∥∥Pn

∥∥�1[X0]+�1[X1],�1[X0]∩�1[X1]

≤ 2n .

The properties (E1), (E2) and (E3) follow from the corresponding properties of Pn, Q+n andQ−

n in (X0, X1).Let us check, for example, that∥∥Pn

∥∥�1[X0]+�1[X1],�1[X0]∩�1[X1]

≤ 2n .



First note that Pn : X0 + X1 → X0 ∩ X1 is bounded with ‖Pn‖X0+X1,X0∩X1 ≤ 2n. Indeed, given anyξ ∈ X0 +X1, if ξ = ξ0 + ξ1 with ξj ∈ Xj we have

‖Pnξ‖X0∩X1 ≤ max{‖Pnξ0‖X0 + ‖Pnξ1‖X0 , ‖Pnξ0‖X1 + ‖Pnξ1‖X1

}≤ max

{‖ξ0‖E + 2n ‖ξ1‖E(2−m), 2n ‖ξ0‖E + ‖ξ1‖E(2−m)

}≤ 2n

(‖ξ0‖X0 + ‖ξ1‖X1

).

Whence ‖Pnξ‖X0∩X1 ≤ 2n‖ξ‖X0+X1 . Now, if x = {xS} ∈ �1[X0 +X1], we get

∥∥Pnx∥∥

�1[X0]∩�1[X1]= max

{∑S∈U

‖PnxS‖X0 ,∑S∈U

‖PnxS‖X1

}

≤ max

{2n∑S∈U

‖xS‖X0+X1 , 2n∑S∈U

‖xS‖X0+X1

}= 2n ‖x‖�1[X0+X1] .

We shall also need the following property that follows from (E3) and Lemma 5.1.

E4 : There exists a constant C such that for each n ∈ N∥∥Q+n

∥∥�1[X],�1[X1]

≤ C 2−n ‖τn‖X,X ,∥∥Q−

n

∥∥�1[X],�1[X0]

≤ C ‖τ−n‖X,X .

Next we show that ideas of the proof of Theorem 4.1 work in this more general context, giving a characteriza-tion of compact operators between minimal spaces.

Theorem 5.4 Let (X0, X1) and X be as before. Assume further that

2−n ‖τn‖X,X −→ 0 and ‖τ−n‖X,X −→ 0 as n → ∞ . (5.7)

Let A = (A0, A1), B = (B0, B1) be Banach couples, let B be an intermediate space with respect to B and letT : A → B such that T

(G(A)) ⊆ B. Then T : G

(A) → B is compact if and only if the following conditions

hold.(a) T : A0 ∩A1 → B is compact.

(b)∥∥Tπ − TπPn

∥∥�1[X],B

→ 0 as n→ ∞.

P r o o f. Suppose first that (a) and (b) are satisfied. Factorization

�1[X ] ↪→ �1[X0] + �1[X1]Pn−−→ �1[X0] ∩ �1[X1]

π−−→ A0 ∩A1T−−→ B

and condition (a) yield that for each n ∈ N the operator TπPn : �1[X ] → B is compact. Using (b), we concludethat Tπ : �1[X ] → B is compact. This implies compactness of T : G

(A)→ B because π is a metric surjection.

Conversely, if T : G(A) → B is compact, then (a) is a consequence of the inclusion A0 ∩ A1 ↪→ G

(A).

To establish (b), note that the sequence{∥∥π − πPn

∥∥�1[X],G(A)

}is bounded so that by the Corollary 2.2 and the

compactness of T : G(A)→ B, it suffices to prove that{∥∥π − πPn

∥∥�1[X],A0+A1

}={∥∥πQ+

n + πQ−n

∥∥�1[X],A0+A1

}converges to 0. By (E4),∥∥πQ+

n

∥∥�1[X],A1

≤ ∥∥Q+n

∥∥�1[X],�1[X1]

≤ C 2−n ‖τn‖X,X ,∥∥πQ−n

∥∥�1[X],A0

≤ ∥∥Q−n

∥∥�1[X],�1[X0]

≤ C ‖τ−n‖X,X .

The result follows from (5.7).



If f is any function parameter and X = c0(1/f(2m)), �∞(1/f(2m)) or �q(1/f(2m)) then (5.7) is satisfiedbecause

‖τn‖X,X ≤ sf (2n) = o(max{1, 2n}) .Hence the theorem applies to examples (5.1), (5.2), (5.4) and (5.5). However the theorem does not apply to thecomplex method (example (5.3)). The reason is that, on one hand, FL1 is not a lattice and, on the other hand, wedo not know operators with similar properties to those of Q+

n and Q−n .

Next we use Theorem 5.4 to obtain new information on interpolation of compact operators by minimalmethods.

Theorem 5.5 We make the same assumptions on (X0, X1) and X as in Theorem 5.4. Let A = (A0, A1),B = (B0, B1) be Banach couples and let T : A → B. Then T : G

(A) → G

(B)

is compact in any of thefollowing three cases:

(i) T : A0 → B0 and T : A1 → B1 are compact.(ii) T : A0 → B0 is compact and B0 ↪→ B1.

(iii) T : A0 → B0 is compact and A0 ↪→ A1.

P r o o f. According to Theorem 5.4, it suffices to check that (a) and (b) hold. To prove (a) note that the operatorT : A0 ∩A1 → B0 is compact in any of the three cases. Moreover, by (5.7) and Lemma 5.3,

limt→0

t

ρ(t, G(B), B) = lim

n→∞2−n

ρ(2−n, G

(B), B) ≤ lim

n→∞C22−n ‖τn‖X,X = 0 .

Whence, compactness of T : A0 ∩A1 → G(B)

follows from [5, Thm. 3.2/(a)].For the proof of (b) let �1 = (�1[X0], �1[X1]). By [10, Lemma 2.1],

�1[X ] ↪→ G(�1[X0], �1[X1]) = G(�1).

Then ∥∥Tπ − TπPn

∥∥�1[X],G(B)

≤ ∥∥Tπ(I − Pn

)∥∥G(�1),G(B)

≤ ∥∥TπQ+n

∥∥G(�1),G(B)

+∥∥TπQ−

n

∥∥G(�1),G(B)

.

We shall show that each one of these two terms goes to 0 when n → ∞. We start with∥∥TπQ+

n

∥∥G(�1),G(B)

. By(5.7) and Lemma 5.1, it is enough to prove that∥∥TπQ+

n

∥∥�1[X0],B0

−→ 0 as n → ∞ . (5.8)

This is immediate from Lemma 2.1, due to the compactness of T : A0 → B0, since{∥∥πQ+

n

∥∥�1[X0],A0

}is

bounded and∥∥πQ+n

∥∥�1[X0],A0+A1

≤ ∥∥πQ+n

∥∥�1[X0],A1

≤ ∥∥Q+n

∥∥�1[X0],�1[X1]

≤ 2−n −→ 0 as n → ∞ .

Let us consider now{∥∥TπQ−

n

∥∥G(�1),G(B)

}. By (5.7) and Lemma 5.1, this sequence converges to 0 if∥∥TπQ−

n

∥∥�1[X1],B1

−→ 0 as n → ∞ . (5.9)

In the case (i), condition (5.9) follows from compactness of T : A1 → B1 with a similar reasoning to (5.8). Inthe cases (ii) and (iii) where we cannot use the compactness of T : A1 → B1, we factorize TπQ−

n by means ofthe diagrams

�1[X1]Q−

n−−→ �1[X0]Tπ−−→ B0 ↪→ B1 (case (ii)) ,

�1[X1]Q−

n−−→ �1[X0]π−−→ A0 ↪→ A1

T−−→ B1 (case (iii)) .



We derive ∥∥TπQ−n

∥∥�1[X1],B1

≤ ∥∥Q−n

∥∥�1[X1],�1[X0]

‖Tπ‖�1[X0],B1

≤ 2−n ‖Tπ‖�1[X0],B1 −→ 0 as n → ∞ .

The proof is complete.

Theorem 5.5 improves a previous result of Cobos and Peetre [10, §2]. Writing down Theorem 5.5 for the caseof Peetre’s method we get:

Corollary 5.6 Let f be a function parameter, let A = (A0, A1), B = (B0, B1) be Banach couples and letT : A→ B. Then T : 〈A0, A1〉f

→ 〈B0, B1〉fis compact in any of the following three cases:

(1) T : A0 → B0 and T : A1 → B1 are compact.(2) T : A0 → B0 is compact and B0 ↪→ B1.(3) T : A0 → B0 is compact and A0 ↪→ A1.

For f(t) = tθ, Corollary 5.6 was proved in [10]. The case (1) for any function parameters was established byMastylo in [21] by using duality and compactness results for maximal methods.

We may also particularize Theorem 5.5 for the case of the Gustavsson-Peetre’s method. The outcome is aresult analogous to Corollary 5.6.

6 Minimal and maximal methods

Let E be a symmetric Banach lattice of sequences as in Section 5, put Y0 = E, Y1 = E(2−m) and let Y be aBanach lattice of sequences intermediate with respect to Y = (Y0, Y1) and such that for each m ∈ Z the shiftoperator τm is bounded in Y . Given any Banach couple B = (B0, B1), let V = V

(B)

be the collection of allS : B → Y such that ‖S‖B,Y ≤ 1. For W an arbitrary Banach space, �∞[W ] = �∞

[V(B),W]

denotes theBanach space of all bounded W -valued families w = {wS} with V as index set, normed by

‖w‖�∞[W ] = sup {‖wS‖W : S ∈ V} .If W = Y0 + Y1 and b ∈ B0 +B1, we put

νb = {Sb}S∈V .

It is clear that ν : B0 +B1 → �∞[Y0 + Y1] is bounded with norm less than or equal to 1. Moreover, for j = 0, 1,ν : Bj → �∞

[Yj

]has also norm less than or equal to 1.

The Aronszajn-Gagliardo maximal method H(B)

= H[Y ;Y

](B)

is the Banach space of all those b ∈B0 +B1, such that νb ∈ �∞[Y ]; the norm in H

(B)

is

‖b‖H(B) = ‖νb‖�∞[Y ] = sup{‖Sb‖Y : S ∈ V

}.

The real method and the real method with the function parameter f can be realized in this way. Namely (see[19]),

(B0, B1)θ,q = H[(�∞, �∞(2−m)); �q

(2−θm

)](B0, B1) , (6.1)

(B0, B1)f,q = H[(�∞, �∞(2−m)); �q(1/f(2m))

](B0, B1) . (6.2)

Another interesting example is Ovchinnikov’s ϕu-method, that we denote here by Hf and that is defined by

Hf (B0, B1) = H[(�1, �1(2−m)); �1(1/f(2m))

](B0, B1) (6.3)

(see [19, 23]). For f(t) = tθ it turns out that [B0, B1]θ ↪→ Hθ(B0, B1), and if (B0, B1) is a couple of complexi-fied Banach lattices with the Fatou property, then [B0, B1]θ = Hθ(B0, B1).

The Aronszajn-Gagliardo maximal method is a dual construction to the minimal method. The precise state-ments can be found in [19, §2]. Norms of interpolated operators can be also estimated by shift operators.



Lemma 6.1 Let (Y0, Y1) and Y as before. Let A = (A0, A1), B = (B0, B1) be Banach couples and letT : A→ B. Then for any n ∈ N:

(i) if ‖T ‖A0,B0 ≤ 2−n and ‖T ‖A1,B1 ≤ 1, then ‖T ‖H(A),H(B) ≤ 2−n ‖τn‖Y,Y ;(ii) if ‖T ‖A0,B0 ≤ 1 and ‖T ‖A1,B1 ≤ 2−n, then ‖T ‖H(A),H(B) ≤ ‖τ−n‖Y,Y .

P r o o f. The proof is similar to [21, Lemma 2.8]. Take any a ∈ H(A)

and any ε > 0. There exists S : B → Ywith ‖S‖B,Y ≤ 1 such that ‖Ta‖H(B) ≤ ‖STa‖Y + ε. Operator S can be written as Sb = {fm(b)}m∈Z with{fm} ⊆ (B0 + B1)∗. Let gm = 2nfm−nT and put Ru = {gm(u)}. If ‖T ‖A0,B0 ≤ 2−n and ‖T ‖A1,B1 ≤ 1,then R : A→ Y with ‖R‖A,Y ≤ 1. Moreover

‖Ta‖H(B) ≤ ‖STa‖Y + ε = ‖{fmTa}‖Y + ε

= ‖τn{fm−nTa}‖Y + ε ≤ 2−n ‖τn‖Y,Y ‖{2nfm−nTa}‖Y + ε

= 2−n ‖τn‖Y,Y ‖Ra‖Y + ε ≤ 2−n ‖τn‖Y,Y ‖a‖H(A) + ε .

As ε > 0 is arbitrary in this reasoning, it follows that ‖T ‖H(A),H(B) ≤ 2−n‖τn‖Y,Y .The case (ii) is analogous.

Remark 6.2 If the norms of the shift operators in Y satisfy

2−n ‖τn‖Y,Y −→ 0 and ‖τ−n‖Y,Y −→ 0 as n → ∞ ,

it follows that if Tn : A → B for n = 1, 2, . . . , with supn∈N‖Tn‖A,B < ∞ and either{‖Tn‖A0,B0

}or{‖Tn‖A1,B1

}converges to 0, then

{‖Tn‖H(A),H(B)

}converges to 0.

In order to characterize compact operators between maximal spaces, given any Banach coupleB = (B0, B1),any w = {wS} ∈ �∞[Y0 + Y1] and any n ∈ N we put

Rnw = {PnwS}S∈V , S+n w =

{Q+

nwS

}S∈V

, S−n w =

{Q−

nwS

}S∈V

.

Here Pn , Q+n , Q

−n are the projections on Y0 +Y1 defined in the previous section. Similarly to (E1), (E2), (E3)

and (E4), these operators satisfy:

E5 : The identity operator I on �∞[Y0 + Y1] can be decomposed as I = Rn + S+n + S−

n for each n ∈ N.

E6 : For each n ∈ N, maxj=0,1

{∥∥Rn

∥∥�∞[Yj],�∞[Yj ]

,∥∥S+

n

∥∥�∞[Yj ],�∞[Yj ]

,∥∥S−

n

∥∥�∞[Yj ],�∞[Yj ]

}≤ 1 .

E7: For each n ∈ N, max{∥∥S+

n

∥∥�∞[Y0],�∞[Y1]

,∥∥S−

n

∥∥�∞[Y1],�∞[Y0]

}≤ 2−(n+1) and∥∥Rn

∥∥�∞[Y0]+�∞[Y1],�∞[Y0]∩�∞[Y1]

≤ 2n .

E8: There is a constant C such that for each n ∈ N∥∥S+n

∥∥�∞[Y0],�∞[Y ]

≤ C ‖τ−n‖Y,Y ,∥∥S−

n

∥∥�∞[Y1],�∞[Y ]

≤ C 2−n ‖τn‖Y,Y .

Now we state the characterization of compact operators. We denote by H(A)◦

the closure of A0 ∩ A1 inH(A).

Theorem 6.3 Let (Y0, Y1) and Y be as before. Assume in addition that

2−n ‖τn‖Y,Y −→ 0 and ‖τ−n‖Y,Y −→ 0 as n → ∞ . (6.4)

Let A = (A0, A1), B = (B0, B1) be Banach couples and let T : A → B. Then T : H(A)◦ → H

(B)

iscompact if and only if the following conditions hold.

(a) T : H(A)◦ → B0 +B1 is compact.

(b)∥∥νT − RnνT

∥∥H(A)◦,�∞[Y ]

→ 0 as n→ ∞.



P r o o f. If (a) and (b) are satisfied, then for each n ∈ N the operator RnνT : H(A)◦ → �∞[Y ] is compact

because it can be factorized in the form

H(A)◦ T−−→ B0 +B1

ν−−→ �∞[Y0] + �∞[Y1]Rn−−→ �∞[Y0] ∩ �∞[Y1] ↪→ �∞[Y ] .

Using (b) it follows that νT : H(A)◦ → �∞[Y ] is compact and so T : H

(A)◦ → H

(B)

compactly becauseν : H

(B)→ �∞[Y ] is a metric injection.

Conversely, suppose that T : H(A)◦ → H

(B)

is compact. Then (a) clearly holds. To establish (b), by the

compactness of T : H(A)◦ → H

(B)

and by Lemma 2.3, it suffices to see that limn→∞∥∥(I−Rn

)νb∥∥

�∞[Y ]= 0

for every b ∈ B0 ∩B1. In fact, B0 ∩B1 ⊇ T (A0 ∩A1) and A0 ∩A1 is dense in H(A)◦

. Let b ∈ B0 ∩B1, then∥∥(I − Rn

)νb∥∥

�∞[Y ]≤ ∥∥S+

n νb∥∥

�∞[Y ]+∥∥S−

n νb∥∥

�∞[Y ]

≤ ∥∥S+n

∥∥�∞[Y0],�∞[Y ]

‖b‖B0 +∥∥S−

n

∥∥�∞[Y1],�∞[Y ]

‖b‖B1

which goes to 0 as n→ ∞ by (E8) and (6.4).

Let now F be another symmetric Banach lattice of sequences, put X0 = F , X1 = F (2−m) and let X be aBanach lattice intermediate with respect to X = (X0, X1). Subsequently we assume that

G[X;X

](A)↪→ H

[Y ;Y

](A)

for any Banach couple A . (6.5)

Condition (6.5) guarantees that for any T : A → B the restriction T : G[X ;X

](A) → H

[Y ;Y

](A)

isbounded. An example of interpolation methods satisfying (6.5) are H

[Y ;Y

](·) = Hf (·) and G

[X;X

](·) =

〈·, ·〉f

or 〈·, ·, f〉 (see [19]).Compact operators from a minimal space into a maximal space can be characterized as follows.

Theorem 6.4 Assume that G(·) = G[X ;X

](·) and H(·) = H

[Y ;Y

](·) satisfy condition (6.5) and that

2−n ‖τn‖X,X −→ 0 and ‖τ−n‖X,X −→ 0 as n → ∞ . (6.6)

Let A = (A0, A1), B = (B0, B1) be Banach couples and let T : A→ B. Then T : G(A)→ H

(B)

is compactif and only if the following conditions hold.

(a) T : G(A)→ B0 +B1 is compact.

(b) T : A0 ∩A1 → H(B)

is compact.

(c)∥∥νTπ − νTπPn − RnνTπ

(Q+

n + Q−n

)∥∥�1[X],�∞[Y ]

→ 0 as n→ ∞.

P r o o f. The factorizations

�1[X ] Pn−−−→ �1[X0] ∩ �1[X1]π−−→ A0 ∩A1

T−−→ H(B) ν−−→ �∞[Y ] ,

�1[X ]Q+

n +Q−n−−−−−→ �1[X ] π−−→ G

(A) T−−→ B0 +B1

ν−−→ �∞[Y0] + �∞[Y1]Rn−−−→ �∞[Y ]

and the conditions (a) and (b) yield that for any n ∈ N the operators νTπPn and RnνTπ(Q+

n + Q−n

)act

compactly from �1[X ] into �∞[Y ]. Using (c), we get that νTπ : �1[X ] → �∞[Y ] is compact, and this impliescompactness of T : G

(A)→ H

(B).

Conversely, if T : G(A)→ H

(B)

is compact then (a) and (b) are obviously satisfied. To check (c), note that∥∥νTπ − νTπPn − RnνTπ(Q+

n + Q−n

)∥∥�1[X],�∞[Y ]

=∥∥(S+

n + S−n

)νTπ

(Q+

n + Q−n

)∥∥�1[X],�∞[Y ]

≤ ∥∥Tπ(Q+n + Q−

n

)∥∥�1[X]),H(B)

.

The last expression goes to 0 as n → ∞ by (b) of Theorem 5.4 (which applies with(B = H

(B))

. Thisimplies (c).



We end the paper with an application of Theorem 6.4, which is an improvement of [9, Thm. 2.1].We assume that

‖ξ − Pnξ‖F −→ 0 as n → ∞ for any ξ ∈ F . (6.7)

Then X0 ∩X1 is dense in X0 and in X1 and so, by [19, Lemma 2/(i)], for any Banach couple A = (A0, A1) itholds

G[(X0, X1);X ](A0, A1) = G[(X0, X1);X ](A◦

0, A◦1

). (6.8)

Here A◦j is the closure of A0 ∩ A1 in Aj . Examples of interpolation methods that satisfy (6.7) are 〈·, ·〉

fand

〈·, ·, f〉.Theorem 6.5 Let G(·) = G[(X0, X1);X ](·) and H(·) = H [(Y0, Y1);Y ](·) be as in the previous theorem.

Assume further that (6.7) holds. Let A = (A0, A1), B = (B0, B1) be Banach couples and let T : A → B withT : A0 → B0 compact. Then T : G

(A)→ H

(B)

is compact.

P r o o f. Since T : A0 → B0 compactly, the operators T : A0 → B0 + B1 and T : A0 ∩ A1 → B0 arecompact. Moreover, Lemma 5.3 and (6.6) imply that

limt→∞

ψ(t, G(A), A)

t= 0 = lim

t→0

t

ρ(t, G(B), B) .

Whence condition (a) of Theorem 6.4 follows from [5, Thm. 3.1]. Moreover, by [5, Thm. 3.2], the operatorT : A0 ∩A1 → G

(B)

is compact. So, (6.5) implies that T : A0 ∩A1 → H(B)

is compact. That is, condition(b) of Theorem 6.4 also holds. Let now check (c). We put A◦ =

(A◦

0, A◦1

).

The factorization

�1[X1]Q−

n−−→ �1[X0]νTπ−−−→ �∞[Y0]

S+n−−→ �∞[Y1]

yields ∥∥S+n νTπQ

−n

∥∥�1[X1],�∞[Y1]

≤ 2−n ‖T ‖A◦0,B02

−n −→ 0 as n → ∞ .

On the other hand, applying Lemma 2.1 with Y = B0 +B1 and using (E3), we derive that∥∥S+n νTπQ

+n

∥∥�1[X0],�∞[Y0]

≤ ∥∥TπQ+n

∥∥�1[X0],B0

−→ 0 as n → ∞ ,

and by Lemma 2.3 and (E7), we have∥∥S−n νTπ

(Q+

n + Q−n

)∥∥�1[X0],�∞[Y0]

≤ ∥∥S−n νT

∥∥A◦

0 ,�∞[Y0]−→ 0 as n → ∞ .

Then, according to Lemma 5.1, assumptions (6.6) and (6.5) and [10, Lemmas 2.1 and 3.1], we obtain that the

sequences{∥∥S+

n νTπQ−n

∥∥�1[X],�∞[Y ]

},{∥∥S+

n νTπQ+n

∥∥�1[X],�∞[Y ]

}and{∥∥S−

n νTπ(Q+

n + Q−n

)∥∥�1[X],�∞[Y ]

}converge to 0. Since

∥∥νTπ−νTπPn−RnνTπ(Q+

n +Q−n

)∥∥�1[X],�∞[Y ]

≤ ∥∥(S+n + S−

n

)νTπ

(Q+

n +Q−n

)∥∥�1[X],�∞[Y ]

condition (c) follows.

Acknowledgements The authors would like to thank the referee for his careful and thorough report, including severalsuggestions on additions and changes on the first version of this article.

Authors have been supported in part by Ministerio de Ciencia y Tecnologıa (BFM2001-1424).



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Complex interpolation, minimal methods and compact operators