ISRAEL MATHEMATICAL CONFERENCE PROCEEDINGS Vol. 00, 0000

COMPLEX METHOD INTERPOLATION DEFINED USING A HALF-PLANE OR FULL DISC

SVANTE JANSON

Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden E-mail address: svante.janson@math.uu.se

The standard definition of the complex interpolation space [X_{0}, X_{1}]_{θ} due to Calderón [2] uses (X_{0}+X_{1})-valued analytic functions in the strip {z : 0 < Re z < 1}, see below for details. Is it possible to use only functions that are analytic in the half-plane {z : 0 < Re z}?

There is a variant of this question which arises since, as shown by Cwikel [3], the space [X_{0}, X_{1}]_{θ} may also be defined using analytic functions in the annulus {z : R_{1} < |z| < R_{0}}. Is it possible to use only functions that are analytic in the entire disc {z : |z| < R_{0}}?

The main purpose of this paper is to show that, in general, these questions have a negative answer, even if we suppose X_{1} ⊆ X_{0}. This is done by an explicit counter example.

Section 1 contains some definitions and an equivalence theorem showing that the two questions above are equivalent.

The counter example is given in Section 2. The reason for considering this particular example is given in Section 3. Section 4 contains some additional results. Some of these concern special cases where the answer to the above question is positive. One such case is when (X_{0}, X_{1}) is a couple of Banach lattices with X_{1} ⊆ X_{0}. Section 5 is an appendix written by Michael Cwikel which presents another case where the answer is positive. Acknowledgment. I thank Michael Cwikel for posing the questions studied here, as well as for encouraging me, many years later, to write up my counter example.

1. Preliminaries We introduce some notation. If X and Y are Banach spaces, then X = Y means

that the spaces contain the same elements and that the norms are equivalent (but not necessarily equal); similarly, X ⊆ Y means that the inclusion is continuous (but not necessarily isometric). C will be used to denote unspecified constants (possibly depending on some parameters that are kept fixed); the meaning may change from one occurrence to the next.

1991 Mathematics Subject Classification. Primary 46B70, Secondary 42A16.

Key words and phrases. Complex interpolation, Minimal interpolation functor, Orbit space.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Suppose that (X_{0}, X_{1}) is a Banach couple. We define F_{S} = F_{S}(X_{0}, X_{1}) as the space of all functions F from the closed strip \overline{S} = \{0 ≤ Re z ≤ 1\} into X_{0} + X_{1} that are bounded and continuous on \overline{S} and analytic on the interior S, and such that t ↦ F (j + it) is a bounded and continuous map of the real line into X_{j} for j = 0, 1. We let

∥F ∥_{F_{S}} = sup_{j=0,1} sup_{−∞<t<∞} ∥F (j + it)∥_{X_{j}}.

The complex interpolation space [X_{0}, X_{1}]_{θ} is defined for 0 < θ < 1 by

[X_{0}, X_{1}]_{θ} = {F (θ) : F ∈ F_{S}},

equipped with the natural quotient norm ∥x∥ = inf{∥F ∥_{F_{S}} : F (θ) = x}, cf. Calderón [2] and Bergh and Löfström [1].

Remark. It is customary to impose also the condition that F (j + it) → 0 in X_{j} as t → ∞, which is convenient for some purposes but not for ours; it is easily seen that this yields the same interpolation space. (On the other hand, the continuity condition on the boundary is essential, see [4].)

For the half-plane version, we let F_{H} = F_{H}(X_{0}, X_{1}) be the space of bounded continuous functions on the closed half-plane \overline{H} = \{z : Re z ≥ 0\} that are analytic on the open half-plane H = \{z : Re z > 0\} and such that the restriction to \overline{S} belongs to F_{S}. We regard F_{H} as a subspace of F_{S}; it is easily seen that F_{H} is a closed subspace of F_{S}, and thus a Banach space. We define, again for 0 < θ < 1,

C_{θ}^{+}(X_{0}, X_{1}) = {F (θ) : F ∈ F_{H}(X_{0}, X_{1})};

this is a Banach space with the natural quotient norm. It should be clear that C_{θ}^{+} is an interpolation method.

Moreover, it follows easily, e.g. by applying linear functionals in (X_{0} +X_{1})^{∗}, that if F ∈ F_{H}(X_{0}, X_{1}) and z ∈ H, then F (z) is given by the integral ∫_{−∞}^{∞} F (it)P_{z}(t)dt, where P_{z}(t) denotes the appropriate Poisson kernel. Since t → F (it) by assumption is a bounded continuous map into X_{0}, this integral converges in X_{0}, and z ↦ F (z) is a bounded continuous map of \overline{H} into X_{0}. Hence

F_{H}(X_{0}, X_{1}) = F_{H}(X_{0}, X_{0} ∩ X_{1})

and thus

C_{θ}^{+}(X_{0}, X_{1}) = C_{θ}^{+}(X_{0}, X_{0} ∩ X_{1}).  (1.1)

It is obvious that

C_{θ}^{+}(X_{0}, X_{1}) ⊆ [X_{0}, X_{1}]_{θ}  (1.2)

and (1.1) thus implies

C_{θ}^{+}(X_{0}, X_{1}) ⊆ [X_{0}, X_{0} ∩ X_{1}]_{θ}.  (1.3)

The first question given in the introduction is whether equality holds in (1.2). We see from (1.1) that it is natural to consider the case X_{1} ⊆ X_{0} only; this is equivalent to asking whether equality always holds in (1.3). A counter example is given in the next section, but first we introduce the annulus and disc versions of the definitions above.

Let R_{0} and R_{1} be two fixed real numbers with 0 < R_{1} < R_{0}, and define R_{θ} = R_{0}^{1−θ}R_{1}^{θ}, 0 < θ < 1. We consider the annulus A = {z : R_{1} < |z| < R_{0}} and the disc D = {z : |z| < R_{0}}.

We define F_{A} = F_{A}(X_{0}, X_{1}) as the space of all bounded, continuous functions F from \overline{A} into X_{0} + X_{1} that are analytic on A, and such that t ↦ F (R_{j}e^{it}) is a continuous map of the real line into X_{j} for j = 0, 1; we let

∥F ∥_{F_{A}} = sup_{j=0,1} sup_{|z|=R_{j}} ∥F (z)∥_{X_{j}}.

We also define F_{D} = F_{D}(X_{0}, X_{1}) as the space of all bounded continuous functions from \overline{D} into X_{0} + X_{1} that are analytic on D and such that the restriction to \overline{A} belongs to F_{A}(X_{0}, X_{1}). We regard F_{D} as a subspace of F_{A}, and we use the subspace norm; F_{D} is a closed subspace and thus a Banach space. It is easily seen, as for F_{H} above, that F_{D}(X_{0}, X_{1}) = F_{D}(X_{0}, X_{0} ∩ X_{1}).

Cwikel [3] showed that the complex method may be defined using analytic functions in an annulus; more precisely,

[X_{0}, X_{1}]_{θ} = {F (R_{θ}) : F ∈ F_{A}(X_{0}, X_{1})}.

Cwikel’s method also shows the corresponding result for the half-plane and disc.

Proposition 1. For every Banach couple (X_{0}, X_{1}),

C_{θ}^{+}(X_{0}, X_{1}) = {F (R_{θ}) : F ∈ F_{D}(X_{0}, X_{1})}.

Proof. Let γ = ln(R_{0}/R_{1}) > 0. First, if F ∈ F_{D}, then G(z) = F (R_{0}e^{−γz}) ∈ F_{H}, and thus F (R_{θ}) = G(θ) ∈ C_{θ}^{+}(X_{0}, X_{1}).

Conversely, if F ∈ F_{H}, let F_{1}(z) = (\frac{e^{−γ(z−θ)}−1}{γ(z−θ)})^{2}F (z). Then F_{1} ∈ F_{H}, with F_{1}(θ) = F (θ), and ∥F_{1}(z)∥_{X_{0}+X_{1}} ≤ C/(1 + |z|^{2}). Hence the sum

F_{2}(z) = ∑_{k=−∞}^{∞} F_{1}(z + 2πik/γ)

converges for all z ∈ \overline{H}; moreover, F_{2} ∈ F_{H} and F_{2}(θ) = F_{1}(θ) = F (θ), because F_{1}(θ + 2πik/γ) = 0 when k ≠ 0, and ∥F_{2}(z)∥_{X_{0}+X_{1}} ≤ C/(1 + Re z). Since F_{2} is periodic with period 2πi/γ, we may define

G(z) = F_{2}(\frac{ln(R_{0}/z)}{γ}), 0 < |z| ≤ R_{0},

regardless of the branch of the logarithm. Then G is an analytic (X_{0} + X_{1})-valued function in the punctured disc D \ {0}, and since ∥G(z)∥_{X_{0}+X_{1}} → 0 as z → 0, the origin is a removable singularity and if we define G(0) = 0, G becomes analytic in D. It is easily seen that G ∈ F_{D} and G(R_{θ}) = F_{2}(θ) = F (θ), which completes the proof. □

Consequently, the two questions in the introduction are equivalent.

2. A counter example Let R_{0}, R_{1} and θ be given with 0 < R_{1} < R_{0} and 0 < θ < 1, and let α_{0} = ln R_{0},

α = ln R_{1}. Thus R_{θ} = R_{0}^{1−θ}R_{1}^{θ} = e^{(1−θ)α_{0}+θα_{1}}. We may without loss of generality assume R_{θ} = 1, and thus α_{0} > 0, α_{1} < 0 and

(1 − θ)α_{0} + θα_{1} = 0.  (2.1)

Let F L denote the space of Fourier coefficients of functions in L^{1}(T):

F L = {(f̂(n))_{n=−∞}^{∞} : f ∈ L^{1}(T)};

this is a Banach space with the norm ∥(f̂(n))_{−∞}^{∞}∥_{F L} = ∥f∥_{L^{1}}.

Define also corresponding weighted spaces by

F L_{α} = {(x_{n})_{−∞}^{∞} : (e^{−nα}x_{n})_{−∞}^{∞} ∈ F L},

where α is a real number, with ∥(x_{n})_{−∞}^{∞}∥_{F L_{α}} = ∥(e^{−nα}x_{n})_{−∞}^{∞}∥_{F L}. Note that if (x_{n})_{−∞}^{∞} ∈ F L_{α}, then

|x_{n}| ≤ e^{nα}∥(x_{k})_{−∞}^{∞}∥_{F L_{α}}.  (2.2)

It is known that [F L_{α_{0}}, F L_{α_{1}}]_{θ} = F L_{(1−θ)α_{0}+θα_{1}} = F L, see e.g. [7].

Let X_{0} = F L_{α_{0}} + F L_{α_{1}}, and X_{1} = F L_{α_{1}}; we claim that, cf. (1.2),

C_{θ}^{+}(X_{0}, X_{1}) ⊊ [X_{0}, X_{1}]_{θ}.  (2.3)

In order to see this, we first observe that

[X_{0}, X_{1}]_{θ} ⊇ [F L_{α_{0}}, F L_{α_{1}}]_{θ} = F L.

On the other hand, we claim that if (x_{n})_{−∞}^{∞} ∈ C_{θ}^{+}(X_{0}, X_{1}) and q > 1/θ, then

∑_{1}^{∞} |x_{n}|^{q}/n < ∞.  (2.4)

Consequently, in order to verify (2.3) it suffices to show that there is a sequence (x_{n})_{−∞}^{∞} ∈ F L such that (2.4) fails. This can be done by explicit examples, such as 1/ln ln(3 + |n|), cf. Zygmund [14], Theorem V.1.5, or by observing that otherwise the closed graph theorem would imply ∑_{1}^{∞} |f̂(n)|^{q}/n ≤ C for some C < ∞ and all f ∈ L^{1}(T) with ∥f∥_{L^{1}} ≤ 1, but that is false as the sequence of Fejér kernels shows.

We turn to the proof of (2.4). We use the characterization with F_{D} in Proposition 1.

Suppose that F ∈ F_{D}(F L_{α_{0}} + F L_{α_{1}}, F L_{α_{1}}) with ∥F ∥ ≤ 1. We write F (z) = (f_{n}(z))_{−∞}^{∞}, thus each f_{n} is analytic in D and continuous on \overline{D}; we further expand each f_{n} as a Taylor series

f_{n}(z) = ∑_{k=0}^{∞} a_{nk}z^{k},

and set a_{nk} = 0 for k < 0.

If |z| = R_{0}, then F (z) ∈ F L_{α_{0}} + F L_{α_{1}}, with norm ≤ 1 and thus, cf. (2.2),

|f_{n}(z)| ≤ max(e^{nα_{0}}, e^{nα_{1}}) = \{ \begin{matrix}e^{nα_{0}}, & n ≥ 0, \\ e^{nα_{1}}, & n < 0. \end{matrix}

Consequently, for any k,

|a_{nk}R_{0}^{k}| ≤ sup_{|z|=R_{0}} |f_{n}(z)| ≤ e^{nα_{0}}, n ≥ 0

or, recalling that R_{0} = e^{α_{0}},

|a_{n,n+m}| ≤ R_{0}^{−n−m}e^{nα_{0}} = e^{−mα_{0}},  n ≥ 0, −∞ < m < ∞.  (2.5)

We now turn to z with |z| = R_{1}. By assumption, F (R_{1}e^{it}) = (f_{n}(R_{1}e^{it}))_{−∞}^{∞} belongs to the unit ball of F L_{α_{1}} for every real t. Hence there exist g_{t} ∈ L^{1}(T), with ∥g_{t}∥_{L^{1}} ≤ 1, such that

ĝ_{t}(n) = e^{−α_{1}n}f_{n}(R_{1}e^{it}) = ∑_{k=0}^{∞} e^{−α_{1}n}a_{nk}R_{1}^{k}e^{ikt} = ∑_{k=0}^{∞} a_{nk}e^{−α_{1}(n−k)}e^{ikt}. (2.6)

Moreover, the mapping t ↦ g_{t} is a continuous map R → L^{1}(T).

Let τ_{s} be the translation operator on L^{1}(T) given by τ_{s}g(t) = g(t − s) and thus

(τ_{s}g)^{∧}(n) = e^{−ins}ĝ(n)  (2.7)

Since, for any t and ε,

∥τ_{t+ε}g_{t+ε} − τ_{t}g_{t}∥_{L^{1}} ≤ ∥τ_{t+ε}(g_{t+ε} − g_{t})∥_{L^{1}} + ∥τ_{t+ε}g_{t} − τ_{t}g_{t}∥_{L^{1}}
= ∥g_{t+ε} − g_{t}∥_{L^{1}} + ∥τ_{ε}g_{t} − g_{t}∥_{L^{1}},

it follows that t ↦ τ_{t}g_{t} is a continuous map into L^{1}(T). Thus we may define, for m ∈ Z, the Bochner integral

h_{m} = \frac{1}{2π} ∫_{0}^{2π} e^{−imt}τ_{t}g_{t} dt ∈ L^{1}(T),

with ∥h_{m}∥_{L^{1}} ≤ 1. Then, by (2.7) and (2.6),

ĥ_{m}(n) = \frac{1}{2π} ∫_{0}^{2π} e^{−imt}e^{−int}ĝ_{t}(n) dt = \frac{1}{2π} ∫_{0}^{2π} ∑_{k=0}^{∞} a_{nk}e^{−α_{1}(n−k)}e^{i(k−m−n)t} dt
= a_{n,n+m}e^{α_{1}m}, n, m ∈ Z.

In particular, ĥ_{m}(n) = 0 if n + m < 0. We also see that

|a_{n,n+m}| = e^{−α_{1}m}|ĥ_{m}(n)| ≤ e^{−α_{1}m}∥h_{m}∥_{L^{1}} ≤ e^{−α_{1}m}.  (2.8)

Let v_{m} be the sequence (a_{n,n+m})_{n=1}^{∞}, and let ℓ^{q}(1/n) be the sequence space {(x_{n})_{1}^{∞} : ∑_{1}^{∞} |x_{n}|^{q}/n < ∞}.

Consider first the case m ≤ 0. Then ĥ_{m}(n) = 0 when n < 0, so h_{m} belongs to the analytic Hardy space H^{1} ⊆ L^{1}. By Hardy’s inequality, see Zygmund [14], Theorem VII.8.7,

∑_{1}^{∞} \frac{1}{n}|a_{n,n+m}| = e^{−α_{1}m} ∑_{1}^{∞} \frac{1}{n}|ĥ_{m}(n)| ≤ Ce^{−α_{1}m}∥h_{m}∥_{L^{1}} ≤ Ce^{−α_{1}m}

and thus, using (2.8),

∑_{1}^{∞} \frac{1}{n}|a_{n,n+m}|^{q} ≤ e^{−(q−1)α_{1}m} ∑_{1}^{∞} \frac{1}{n}|a_{n,n+m}| ≤ Ce^{−qα_{1}m},

or in the notation just introduced,

∥v_{m}∥_{ℓ^{q}(1/n)} ≤ Ce^{−α_{1}m},  m ≤ 0.  (2.9)

Consider next the case m > 0. Then ĥ_{m}(n) = 0 when n < −m, and the shifted sequence (a_{k−m,k})_{k=−∞}^{∞} is the Fourier transform of the function e^{−α_{1}m}e^{imt}h_{m}(t) in H^{1}. Hence Hardy’s inequality yields

∑_{1}^{∞} \frac{1}{n}|a_{n,n+m}| ≤ (m + 1) ∑_{1}^{∞} \frac{1}{n + m}|a_{n,n+m}| ≤ (m + 1) ∑_{1}^{∞} \frac{1}{k} |a_{k−m,k}|
≤ (m + 1)C∥e^{−α_{1}m}e^{imt}h_{m}(t)∥_{L^{1}} ≤ C(m + 1)e^{−α_{1}m}.

We combine this with (2.5) and obtain

∑_{1}^{∞} \frac{1}{n}|a_{n,n+m}|^{q} ≤ e^{−(q−1)α_{0}m} ∑_{1}^{∞} \frac{1}{n}|a_{n,n+m}| ≤ C(m + 1)e^{−(q−1)α_{0}m−α_{1}m}. (2.10)

From (1 − θ)α_{0} + θα_{1} = 0 we obtain qα_{0} = qθ(α_{0} − α_{1}) and (q − 1)α_{0} + α_{1} = (qθ − 1)(α_{0} − α_{1}). Hence (2.10) yields

∥v_{m}∥_{ℓ^{q}(1/n)} ≤ Cm^{1/q}e^{−(θ−1/q)(α_{0}−α_{1})},  m > 0.  (2.11)

Since α_{1} < 0 and (θ − 1/q)(α_{0} − α_{1}) > 0, (2.9) and (2.11) together imply

∑_{m=−∞}^{∞} ∥v_{m}∥_{ℓ^{q}(1/n)} < ∞,

and thus ∑_{−∞}^{∞} v_{m} ∈ ℓ^{q}(1/n). But

∑_{−∞}^{∞} v_{m} = ( ∑_{m=−∞}^{∞} a_{n,n+m})_{n=1}^{∞} = (∑_{k=0}^{∞} a_{n,k})_{n=1}^{∞} = (f_{n}(1))_{n=1}^{∞}.

This proves (2.4) for the sequence (x_{n})_{−∞}^{∞} = (f_{n}(1))_{−∞}^{∞} = F (1) = F (R_{θ}), and the proof is complete.

3. Background The example in Section 2 solves our problem, but how was it found? The couple

(F L_{α_{0}} +F L_{α_{1}}, F L_{α_{1}}) may look rather unnatural at the first sight, but the following argument shows that this is in fact the canonical (counter) example.

If Ã, X̃ and A are Banach spaces such that A ⊆ Ã, and J ⊆ B(Ã, X̃) is a Banach space of bounded linear operators Ã → X̃, then the orbit of A under J is the subspace of X̃ given by {∑_{1}^{∞} T_{i}a_{i} : ∑_{i}^{∞} ∥T_{i}∥_{J}∥a_{i}∥_{A} < ∞} (with the sum ∑ T_{i}a_{i} converging in X̃). This is a Banach space with ∥x∥ = inf ∑_{i} ∥T_{i}∥_{J}∥a_{i}∥_{A} over all such representations of x.

A particularly important example is when (A_{0}, A_{1}) and (X_{0}, X_{1}) are two Banach couples, Ã = A_{0} + A_{1}, X̃ = X_{0} + X_{1} and J = {T ∈ B(Ã, X̃) | T: (A_{0}, A_{1}) → (X_{0}, X_{1})} is the space of all linear operators from A_{0} + A_{1} to X_{0} + X_{1} that map A_{0} into X_{0} and A_{1} into X_{1}. We then denote the orbit of a space A ⊆ A_{0} + A_{1} by G(A_{0}, A_{1}, A; X_{0}, X_{1}). For fixed A_{0}, A_{1} and A ≠ {0}, this is an interpolation method, and it is easily seen that this is the minimal interpolation method that satisfies F (A_{0}, A_{1}) ⊇ A.

Now consider the action of this interpolation method on the couple (X_{0}, X_{0}∩X_{1}). Since

T: (A_{0}, A_{1}) → (X_{0}, X_{0} ∩ X_{1}) ⟺ T: A_{0} → X_{0}, T: A_{1} → X_{0}, T: A_{1} → X_{1},
⟺ T: (A_{0} + A_{1}, A_{1}) → (X_{0}, X_{1}),

we obtain the following identity.

Proposition 2. Let (A_{0}, A_{1}) and (X_{0}, X_{1}) be Banach couples, and A ⊆ A_{0} + A_{1}. Then

G(A_{0}, A_{1}, A; X_{0}, X_{0} ∩ X_{1}) = G(A_{0} + A_{1}, A_{1}, A; X_{0}, X_{1}).

Proposition 3. Let (A_{0}, A_{1}) be a Banach couple and A ⊆ A_{0} + A_{1}. If F is an interpolation method, then the following are equivalent.

(i) F (X_{0}, X_{1}) ⊇ G(A_{0}, A_{1}, A; X_{0}, X_{1}) for all Banach couples (X_{0}, X_{1}) such

that X_{0} ⊇ X_{1}.

(ii) F (X_{0}, X_{0}∩X_{1}) ⊇ G(A_{0}, A_{1}, A; X_{0}, X_{0}∩X_{1}) for all Banach couples (X_{0}, X_{1}). (iii) F (A_{0} + A_{1}, A_{1}) ⊇ A.

Proof. The equivalence of (i) and (ii) is clear.

By the comments before Proposition 2, (iii) holds if and only if

G(A_{0} + A_{1}, A_{1}, A; X_{0}, X_{1}) ⊆ F (X_{0}, X_{1})

for every couple (X_{0}, X_{1}), which by Proposition 2 easily is seen to be equivalent to (ii). □

The complex method of interpolation [X_{0}, X_{1}]_{θ} can be characterized as the orbit G(F L_{α_{0}}, F L_{α_{1}}, F L_{(1−θ)α_{0}+θα_{1}}; X_{0}, X_{1}) for any real α_{0} and α_{1} with α_{0} ≠ α_{1}, see [7]. We thus have the following corollaries.

Corollary 1. Let α_{0} ≠ α_{1} and 0 < θ < 1. Then, for any Banach couple (X_{0}, X_{1}),

[X_{0}, X_{0} ∩ X_{1}]_{θ} = G(F L_{α_{0}} + F L_{α_{1}}, F L_{α_{1}}, F L_{(1−θ)α_{0}+θα_{1}}; X_{0}, X_{1}).

Corollary 2. Let α_{0} ≠ α_{1} and 0 < θ < 1 and let F be an interpolation method. Then F (X_{0}, X_{1}) ⊇ [X_{0}, X_{1}]_{θ} for all Banach couples (X_{0}, X_{1}) with X_{0} ⊇ X_{1}, if and only if

F (F L_{α_{0}} + F L_{α_{1}}, F L_{α_{1}}) ⊇ F L_{(1−θ)α_{0}+θα_{1}}.

Corollary 2 thus shows that if C_{θ}^{+}(X_{0}, X_{1}) ⊇ [X_{0}, X_{1}]_{θ} fails for any couple (X_{0}, X_{1}), then it fails for (F L_{α_{0}} + F L_{α_{1}}, F L_{α_{1}}).

4. Further comments The interpolation method C_{θ}^{+} is probably not of much practical use, but let

us nevertheless give a couple of results for it. First, as another example of the propositions in Section 3, consider the ±-method defined by Peetre [12]. It was shown in [7] that this method, there and here denoted by G_{1}, can be characterized as G(c_{0,α_{0}}, c_{0,α_{1}}, c_{0,(1−θ)α_{0}+α_{1}}; X_{0}, X_{1}), where c_{0,α} is a weighted version of c_{0} = {(x_{n})_{−∞}^{∞} : x_{n} → 0 as |n| → ∞}.

It is easily verified (we omit this) that C_{θ}^{+}(c_{0,α_{0}} + c_{0,α_{1}}, c_{0,α_{1}}) ⊇ c_{0,(1−θ)α_{0}+θα_{1}}. Proposition 3 thus implies, together with (1.2), the following.

Proposition 4. For any Banach couple (X_{0}, X_{1}) such that X_{0} ⊇ X_{1},

G_{1}(X_{0}, X_{1}) ⊆ C_{θ}^{+}(X_{0}, X_{1}) ⊆ [X_{0}, X_{1}]_{θ}.

In particular, this shows that if (X_{0}, X_{1}) is a Banach couple such that X_{0} ⊇ X_{1} and

G_{1}(X_{0}, X_{1}) = [X_{0}, X_{1}]_{θ},  (4.1)

then [X_{0}, X_{1}]_{θ} may be defined using X_{0}-valued functions F that are analytic in a half-plane or disc, as defined in detail in Section 1.

Remark. One important case when (4.1) holds, pointed out to me by Michael Cwikel, is for a couple of lattices on the same measure space (Ω, Σ, μ) i.e. when X_{j} = Y_{j}(C) is the complexification of a Banach lattice Y_{j} of real valued measurable functions on Ω for j = 0, 1. This fact has been observed by a number of authors. (Sometimes they impose additional conditions.) For the reader’s convenience let us list some results which can be combined to immediately prove it:

(i) the continous inclusion [X_{0}, X_{1}]_{θ} ⊂ Y_{0}^{1−θ}Y_{1}^{θ}(C) (see [2] section 13.6 (i) p. 125), (ii) the continuous inclusion Y_{0}^{1−θ}Y_{1}^{θ}(C) ⊂ G_{2}(X_{0}, X_{1}) (see [11] Lemma 8.2.1 p. 453.)

The rest of these “ingredients” also hold for arbitrary Banach couples (X_{0}, X_{1}). (iii) the density of X_{0} ∩ X_{1} in [X_{0}, X_{1}]_{θ} ( [2] p. 116) and the fact that the closure of X_{0} ∩ X_{1} in G_{2}(X_{0}, X_{1}) is G_{1}(X_{0}, X_{1}). ([7] Theorem 8 p. 60.) (iv) the continuous inclusion G_{1}(X_{0}, X_{1}) ⊂ [X_{0}, X_{1}]_{θ}. (See [12] p. 176 or [7] p. 67).

The interpolation method C_{θ}^{+} can also be represented as an orbit method. Define P_{+}(x_{n})_{−∞}^{∞} = (x_{n})_{0}^{∞}, the restriction of a doubly infinite sequence to non-negative indices, and let F L_{α}^{+} = P_{+}(F L_{α}), equipped with the quotient norm.

Proposition 5. Let α_{0} > α_{1}. Then for any Banach couple (X_{0}, X_{1}),

C_{θ}^{+}(X_{0}, X_{1}) = G(F L_{α_{0}}^{+}, F L_{α_{1}}^{+}, F L_{(1−θ)α_{0}+θα_{1}}^{+}; X_{0}, X_{1}).

Proof. We may assume (1 − θ)α_{0} + θα_{1} = 0. We use Proposition 1, choosing R_{0} = e^{α_{0}} and R_{1} = e^{α_{1}}, and thus R_{θ} = 1. First, suppose that x = (x_{n})_{−∞}^{∞} is a finite sequence of complex numbers, i.e. a sequence with all but finitely many elements 0. Then F (z) = (x_{n}z^{n})_{0}^{∞} defines an entire analytic function into F L_{α_{0}}^{+} ∩ F L_{α_{1}}^{+} and if z = R_{j}e^{it}, j = 0 or 1, then

∥F (z)∥_{F L_{α_{j}}^{+}} = ∥(x_{n}e^{int})_{0}^{∞}∥_{F L^{+}} ≤ ∥(x_{n}e^{int})_{−∞}^{∞}∥_{F L} = ∥(x_{n})_{−∞}^{∞}∥_{F L}.

Hence F (z) ∈ F_{D}(F L_{α_{0}}^{+}, F L_{α_{1}}^{+}) with norm ≤ ∥x∥_{F L}, and thus P_{+}x = F (1) ∈ C_{θ}^{+}(F L_{α_{0}}^{+}, F L_{α_{1}}^{+}) with norm ≤ ∥x∥_{F L}. The set of all finite sequences is dense in F L, so by continuity

F L^{+} = P_{+}(F L) ⊆ C_{θ}^{+}(F L_{α_{0}}^{+}, F L_{α_{1}}^{+}),

which implies, by the minimality of the interpolation functor G,

G(F L_{α_{0}}^{+}, F L_{α_{1}}^{+}, F L^{+}; X_{0}, X_{1}) ⊆ C_{θ}^{+}(X_{0}, X_{1})

for every Banach couple (X_{0}, X_{1}).

In order to prove the converse, suppose that F ∈ F_{D}(X_{0}, X_{1}), and expand F as a Taylor series F (z) = ∑_{0}^{∞} x_{k}z^{k}, with x_{k} ∈ X_{0} ∩ X_{1}. If (a_{n})_{−∞}^{∞} ∈ F L_{α_{0}}, then a_{n} = e^{nα_{0}} f̂(n) for some f ∈ L^{1}(T) and (with X_{0}-valued integrals)

\frac{1}{2π} ∫_{0}^{2π} f(e^{−it})F (R_{0}e^{it}) dt = lim_{r↗1} \frac{1}{2π} ∫_{0}^{2π} f(e^{−it})F (rR_{0}e^{it}) dt = lim_{r↗1} ∑_{n=0}^{∞} f̂(n)r^{n}R_{0}^{n}x_{n}

= lim_{r↗1} ∑_{n=0}^{∞} r^{n}a_{n}x_{n}.  (4.2)

Hence the mapping T : (a_{n})_{−∞}^{∞} → lim_{r↗1} ∑_{n=0}^{∞} r^{n}a_{n}x_{n} is a well-defined linear map F L_{α_{0}} → X_{0}, and

∥T ((a_{n})_{−∞}^{∞})∥_{X_{0}} ≤ ∥f∥_{L^{1}} sup_{t} ∥F (R_{0}e^{it})∥_{X_{0}} ≤ ∥(a_{n})_{−∞}^{∞}∥_{F L_{α_{0}}} ∥F ∥_{F_{D}}.

Since T a obviously depends on P_{+}a only, we can also regard T as a bounded linear map F L_{α_{0}}^{+} → X_{0}.

Similarly if (a_{n})_{0}^{∞} ∈ F L_{α_{1}}^{+}, then a_{n} = e^{nα_{1}} f̂_{1}(n), n ≥ 0, for some f_{1} ∈ L^{1}(T) and, by the same argument as in (4.2),

T ((a_{n})_{0}^{∞}) = \frac{1}{2π} ∫_{0}^{2π} f_{1}(e^{−it})F (R_{1}e^{it}) dt,

with the integral convergent in X_{1}. It follows that T: F L_{α_{1}}^{+} → X_{1}. Thus

T: (F L_{α_{0}}^{+}, F L_{α_{1}}^{+}) → (X_{0}, X_{1}),

and we see also that ∥T ∥ ≤ ∥F ∥_{F_{D}}. Furthermore, if f ∈ L^{1}(T), then by the same argument again,

\frac{1}{2π} ∫_{0}^{2π} f(e^{−it})F (e^{it}) dt = T ((f̂(n))_{−∞}^{∞}).

Thus, if we let f ∗ F denote the X_{0}-valued function

f ∗ F (z) = \frac{1}{2π} ∫_{0}^{2π} f(e^{−it})F (ze^{it}) dt,

and Y = G(F L_{α_{0}}^{+}, F L_{α_{1}}^{+}, F L^{+}; X_{0}, X_{1}), then by the definition of the latter space,

f ∗ F (1) = T (f̂) ∈ Y with ∥f ∗ F (1)∥_{Y} ≤ ∥f∥_{L^{1}}∥F ∥_{F_{D}}.  (4.3)

It is easily seen that f ∗ F ∈ F_{D}(X_{0}, X_{1}) for all f ∈ L^{1}(T), and that, if K_{n} denotes the n:th Fejér kernel, K_{n} ∗ F → F in F_{D}(X_{0}, X_{1}) as n → ∞; moreover, (f ∗ g) ∗ F = f ∗ (g ∗ F) for all f, g ∈ L^{1}(T). Define y_{n} = K_{n} ∗ K_{n} ∗ F (1). By (4.3), y_{n} ∈ Y. Furthermore,

y_{n} − y_{m} = (K_{n} + K_{m}) ∗ (K_{n} − K_{m}) ∗ F (1)

and thus, by (4.3) for (K_{n} − K_{m}) ∗ F,

∥y_{n} − y_{m}∥_{Y} ≤ ∥K_{n} + K_{m}∥_{L^{1}}∥(K_{n} − K_{m}) ∗ F ∥_{F_{D}} ≤ 2∥K_{n} ∗ F − K_{m} ∗ F ∥_{F_{D}} → 0, as n, m → ∞. Hence {y_{n}} is a Cauchy sequence in Y, so y_{n} → y ∈ Y. But K_{n}∗K_{n}∗F → F in F_{D}(X_{0}, X_{1}), and thus y_{n} → F (1) in C_{θ}^{+}(X_{0}, X_{1}). Consequently F (1) = y ∈ Y, which completes the proof. □

Remark. It is also possible to define interpolation methods using harmonic functions in a half-plane or disc, see Janson and Peetre [9]. The resulting interpolation spaces contain the ones given by analytic functions, and are in general larger.

For harmonic interpolation, we do not know if the half-plane and and disc versions always yield the same interpolation spaces, or even if the disc interpolation spaces are independent of the ratio R_{0}/R_{1} as in the analytic case, cf. Proposition 1.

For the disc version, with fixed R_{0} = e^{α_{0}} and R_{1} = e^{α_{1}}, the following analogues of the results above hold. (We omit the proofs.)

First, a somewhat more complicated version of the argument in Section 2 shows that, at least for the disc version, the harmonic interpolation space for the couple (F L_{α_{0}} + F L_{α_{1}}, F L_{α_{1}}) does not contain the standard complex method space.

Furthermore, the method has an orbit description as G(A_{α_{0}}, A_{α_{1}}, A_{α_{θ}}; X_{0}, X_{1}), where A_{α} = {(a_{n})_{−∞}^{∞} : (e^{−α|n|}a_{n})_{−∞}^{∞} ∈ F L}.

An argument similar to the one in Section 2 shows that for the couple (A_{α_{0}}, A_{α_{1}}), the standard complex method space does not contain the harmonic interpolation space. Consequently, even assuming X_{0} ⊃ X_{1}, the harmonic method and the standard complex method are not comparable.

Finally, the harmonic method space is included in the one given by Ovchinnikov’s method ϕ_{u} [10], denoted by H_{1} in [7]. It follows that for ‘tame’ couples, the harmonic and analytic disc methods coincide with the standard complex method.

5. Appendix:

ANOTHER CASE WHERE C_{θ}^{+}(X_{0}, X_{1}) = [X_{0}, X_{1}]_{θ}.

by Michael Cwikel

Here we consider couples (X_{0}, X_{1}) which can be obtained by “one sided reiteration”.

Theorem 5.1. Let (X_{0}, X_{1}) be a Banach couple satisfying X_{1} ⊆ X_{0}. Suppose that there exists another Banach space B such that (X_{0}, B) forms a Banach couple and such that

X_{1} = [X_{0}, B]_{β} for some β ∈ (0, 1).

Then

C_{θ}^{+}(X_{0}, X_{1}) = [X_{0}, X_{1}]_{θ} for all θ ∈ (0, 1).

Proof. Let (Y_{0}, Y_{1}) be the Banach couple obtained by setting Y_{0} = X_{1} and Y_{1} = X_{0}, and let α = 1 − β. I.e. we have Y_{0} ⊆ Y_{1} and

Y_{0} = [B, Y_{1}]_{α}  (5.2)

(We introduce Y_{0} and Y_{1} and α because this is the easiest way of adapting our original version of this proof, which was written without access to a definitive version of the previous sections of this paper, to the format of the notation used in those sections.)

The proof of this theorem amounts to showing that [X_{0}, X_{1}]_{θ} ⊆ C_{θ}^{+}(X_{0}, X_{1}) for all θ ∈ (0, 1). This is of course equivalent to showing that

[Y_{0}, Y_{1}]_{θ} ⊆ C_{1−θ}^{+}(Y_{1}, Y_{0}) for all θ ∈ (0, 1).  (5.3)

We may suppose without loss of generality that B ∩ Y_{1} is dense in B, since if not we can simply replace B by B^{∘} = [B, Y_{1}]_{0}, the closure of B ∩Y_{1} in B, which satisfies [B^{∘}, Y_{1}]_{α} = [B, Y_{1}]_{α}. (Cf. [2] Paragraph 9.3 p. 116.) It follows immediately from (5.2) that

∥a∥_{Y_{0}} ≤ ∥a∥_{B}^{1−α} ∥a∥_{Y_{1}}^{α}

for every a ∈ B ∩ Y_{1}. But since also ∥a∥_{Y_{1}} ≤ C ∥a∥_{Y_{0}} for some constant C, we deduce that ∥a∥_{Y_{0}} ≤ C^{α/(1−α)} ∥a∥_{B} for all such a. Using the density of B ∩ Y_{1} in B we deduce that the previous estimate holds for all a ∈ B and so obtain the continuous embedding

B ⊂ Y_{0}.  (5.4)

By [2] Paragraph 9.3 p. 116, the spaces [B, Y_{1}]_{1} and [Y_{0}, Y_{1}]_{1} are the completions of B and of Y_{0} respectively in Y_{1}. Since B = B ∩ Y_{1} is dense in Y_{0}, it follows that [B, Y_{1}]_{1} = [Y_{0}, Y_{1}]_{1}. Applying [2] Paragraph 9.3 p. 116 once more, we have that [Y_{0}, [Y_{0}, Y_{1}]_{1}]_{θ} = [Y_{0}, Y_{1}]_{θ} and so [Y_{0}, Y_{1}]_{θ} = [[B, Y_{1}]_{α}, [B, Y_{1}]_{1}]_{θ}. Applying the reiteration formula (see [3] p. 1005) to this latter space we deduce that

[Y_{0}, Y_{1}]_{θ} = [B, Y_{1}]_{(1−θ)α+θ}.  (5.5)

Let us now consider an arbitrary fixed element a ∈ [Y_{0}, Y_{1}]_{θ}. By (5.5) it can be represented in the form a = g_{1} ((1 − θ)α + θ) for some g_{1} ∈ F_{S}(B, Y_{1}). Now let g_{2}(z) = g_{1} ((1 − z)α + z). Then g_{2} is a continuous and bounded B + Y_{1} = Y_{1} valued function on the strip Re z ∈ [\frac{−α}{1−α}, 1] and it is analytic in the interior of this strip. Furthermore g_{2}(θ) = a and the restrictions of g_{2} to the lines Re z = \frac{−α}{1−α} and Re z = 0 are continuous bounded B valued and [B, Y_{1}]_{α} = Y_{0} valued functions respectively.

The next step will be to use the construction defined in [3] p. 1008. (Cf. also the proof of Proposition 1 above.) This enables us to construct from g_{2} a new function g_{3} which has all the properties listed above for g_{2} and also the additional property that g_{3}(z + 2πi) = g_{3}(z) for all z in the strip Re z ∈ [\frac{−α}{1−α}, 1] . (The functions w(z) and e^{δ(z−θ)^{2}} which were used in that construction on the strip Re z ∈ [0, 1] of course also satisfy estimates of the required form on the wider strip which we need here.)

Now we define yet another continuous and bounded Y_{1} valued function g_{4}, this time on the annulus |z| ∈ [e^{−α/(1−α)}, e] by setting g_{4}(z) = g_{3}(log z). Here, because of the 2πi periodicity of g_{3}, the choice of branch of log z is irrelevant. Clearly g_{4} is analytic in the interior of the annulus, and furthermore its restrictions to each of the three circles of radius γ := e^{−α/(1−α)}, 1 and e respectively are continuous maps into the Banach spaces B, Y_{0} and Y_{1} respectively.

It will be convenient, for each r > 0, to let C_{r} denote the circle |z| = r oriented in the anticlockwise direction. We can now introduce the function

g_{5}(z) = g_{4}(z) + \frac{1}{2πi} (∫_{Cγ} g_{4}(ζ) (\frac{1}{ζ − z} − \frac{1}{ζ − e^{θ}} ) dζ) .  (5.6)

Obviously g_{5}(e^{θ}) = a. Furthermore, since the integral in (5.6) is a continuous B valued function of z in the region |z| > γ and is also analytic in that region, we deduce that g_{5} is a continuous Y_{1} valued function on the annulus |z| ∈ [1, e] and is analytic in its interior. Furthermore the restrictions of the g_{5} to the circles C_{1} and C_{e} are continuous maps into Y_{0} and Y_{1} respectively. By Cauchy’s integral formula we also have, whenever |z| ∈ (γ, e), that

g_{5}(z) = \frac{1}{2πi} (∫_{C_{e}} \frac{g_{4}(ζ)}{ζ − z} dζ − ∫_{C_{γ}} \frac{g_{4}(ζ)}{ζ − e^{θ}} dζ) .

But the formula on the right defines an analytic Y_{1} valued function in the disk |z| < e.

Let us now define F_{D}(Y_{1}, Y_{0}) = F_{D}(X_{0}, X_{1}) as in Section 1, for the particular choice of radii R_{1} = 1 and R_{0} = e. The preceding discussion shows that g_{5} extends to a function in F_{D}(Y_{1}, Y_{0}). Consequently, by Proposition 1, a = g_{5}(e^{θ}) = g_{5}(R_{0}^{θ}R_{1}^{1−θ}) is an element of C_{1−θ}^{+}(Y_{1}, Y_{0}). This establishes (5.3) and so completes the proof of the theorem. □

Remark. The norms ∥a∥_{C_{θ}^{+}(X_{0},X_{1})} and ∥a∥_{[X_{0},X_{1}]_{θ}} must of course be equivalent. By making obvious appropriate norm estimates at each step of the above proof it is possible to show that the constants of equivalence of these norms depend only on θ, β and the norms of the continuous embeddings of B into X_{0} and X_{1}. Remark. The preceding theorem shows that the scale of spaces used in the counter example of Section 2 provides an apparently new example of a scale of complex interpolation spaces which cannot be continued beyond a certain value of the parameter. Other examples of related phenomena have been considered by Kalton [??], by N. and V. Zobin [13] and also by Zafran (see the Appendix on pp. 297-298 of [8]).

For a discussion of a related phenomenon, where the continuation of the scale exists but is not unique, see [8] pp. 295-297.

In view of Theorem 11 of [5] pp. 273–274 (cf. also Theorem 10 on p. 272) this “non-continuability” of the complex interpolation scale for the Banach couple (F L_{α_{0}} + F L_{α_{1}}, F L_{α_{1}}) suggests the possibility that an appropriate modification of this couple might provide the setting for a counterexample to settle a long standing open question about compact operators and the complex interpolation method. See also [6] for various simplifications and reductions of this problem. The reader who wishes to consider this possibility should probably keep in mind that if there is a counterexample for the above question then there is necessarily a counterexample in the context of domain couples of the form (ℓ^{1}(F L_{α_{0}}^{1}), ℓ^{1}(F L_{α_{1}}^{1})) and/or range couples (ℓ^{∞}(F L_{α_{0}}^{∞}), ℓ^{∞}(F L_{α_{1}}^{∞})) (See [6] Proposition 3, p. 356) and furthermore it should be possible to show via arguments of equicontinuity etc. or from a special case treated in [2] p. 118 that compactness of operators is preserved by the complex method in the context of domain couples of the form (F L_{α_{0}}^{1}, F L_{α_{1}}^{1}) and/or range couples (F L_{α_{0}}^{∞}, F L_{α_{1}}^{∞}).

References

1. J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin, 1976.

2. A. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190.

3. M. Cwikel, Complex interpolation spaces, a discrete definition and reiteration, Indiana J. Math. 27 (1978), 1005–1009.

4. M. Cwikel and S. Janson, Interpolation of analytic families of operators, Studia Math. 79 (1984), 61–71.

5. M. Cwikel and N. J. Kalton, Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre, Proc. Edinburgh Math. Soc. 38 (1995), 261–276.

6. M. Cwikel, N. Krugljak and M. Mastyło, On complex interpolation of compact operators, Illinois J. Math. 40 (1996), 353–364.

7. S. Janson, Minimal and maximal methods of interpolation, J. Funct. Anal. 44 (1981), 50–72.

8. S. Janson, P. Nilsson and J. Peetre, Notes on Wolff’s note on interpolation spaces. With an appendix by Misha Zafran, Proc. London Math. Soc. (3) 48 (1984), 283–299.

9. S. Janson and J. Peetre, Harmonic interpolation, Interpolation spaces and allied topics in analysis (Proceedings, Lund 1983), Lecture Notes Math. 1070, Springer-Verlag, Berlin, 1984, pp. 92–124.

10. V.I. Ovchinnikov, Interpolation theorems resulting from an inequality of Grothendieck, Funk. Anal. i Priložen 10 (1976), 45–54. (Russian)

11. V.I. Ovchinnikov, The method of orbits in interpolation theory, Mathematical Reports 1, part 2 (1984), Harwood Academic Publishers, Chur London Paris Utrecht New York, 349–516.

12. J. Peetre, Sur l’utilisation des suites inconditionellement sommable dans la théorie des espaces d’interpolation, Rend. Sem. Mat. Univ. Padova 46 (1971), 173–190.

13. N. Zobin and V. Zobin, l_{∞} and interpolation between Banach lattices, Proc. Amer. Math. Soc. 125 (1997), 827–833.

14. A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, Cambridge, 1959.