Complemented Subspaces of Spaces Obtained by Interpolation

D.J.H. Garling

St. John’s College, Cambridge CB2 1TP, England.

S.J. Montgomery-Smith

Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.

ABSTRACT: If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A_{0}, A_{1}) such that A_{0} and A_{1} are isometric to X ⊕ V, and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon–Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces.

There are many Banach space properties that pass to spaces obtained by the complex method of interpolation. For example, it is known that if a couple (A_{0}, A_{1}) is such that A_{0} and A_{1} both have the UMD (unconditional martingale difference sequence) property, and if A_{θ} is the space obtained using the complex interpolation method with parameter θ, then A_{θ} has the UMD property whenever 0 < θ < 1. Another example is type of Banach spaces: if A_{0} has type p_{0} and A_{1} has type p_{1}, then A_{θ} has type p_{θ}, where 1/p_{θ} = (1−θ)/p_{0} +θ/p_{1}.

Similar results are true for the real method of interpolation. If we denote by A_{θ,p} the space obtained using the real interpolation method from a couple (A_{0}, A_{1}) with parameters θ and p, then A_{θ,p} has the UMD property whenever A_{0} and A_{1} have the UMD property, 0 < θ < 1, and 1 < p < ∞. Similarly, if A_{0} has type p_{0} and A_{1} has type p_{1}, then A_{θ,p} has type p_{θ}, where 1/p_{θ} = (1 − θ)/p_{0} + θ/p_{1} and p = p_{θ} (see [5] 2.g.22).

However, there are other properties for which it has been hitherto unknown whether they pass to the intermediate spaces. Examples include the Radon–Nikodym property, the AUMD (analytic unconditional martingale difference sequence) property, and having non-trivial cotype.

The second named author was supported in part by N.S.F. Grant DMS 9001796.

A.M.S. (1980 ) subject classification: 46B99.

SPACES OBTAINED BY INTERPOLATION

This paper deals with these properties, showing that they do not pass to the intermediate spaces. Indeed, we show the surprising fact that there is a couple (A_{0}, A_{1}) such that A_{0} and A_{1} are both isometric to l_{1}, but all the real or complex intermediate spaces contain a complemented subspace isomorphic to c_{0}. This improves a result due to Pisier, who gave an example of a couple (A_{0}, A_{1}) for which A_{0} is isometric to L_{1}, A_{1} is isometric to a dense subspace of c_{0}, and c_{0} is finitely represented in every intermediate space A_{θ} obtained by the complex interpolation method (see [3]).

Notation Here we outline the notation we will use about interpolation couples. The reader is referred to [1] or [2] for details.

A Banach couple is a pair of Banach spaces (A_{0}, A_{1}) such that A_{0} and A_{1} both embed into a common topological vector space, Ω, which we shall call the ambient space. Given such a couple, we define Banach spaces A_{0} + A_{1} (with norm ∥x∥ = inf{ ∥x_{0}∥_{A_{0}} + ∥x_{1}∥_{A_{1}} : x_{0} ∈ A_{0}, x_{1} ∈ A_{1}, x_{0} + x_{1} = x}) and A_{0} ∩ A_{1} (with norm ∥x∥ = max{∥x∥_{A_{0}} , ∥x∥_{A_{1}}}). A map between two couples T : (A_{0}, A_{1}) → (B_{0}, B_{1}) is a linear map T : A_{0} + A_{1} → B_{0} + B_{1} such that T (A_{0}) ⊆ B_{0} and T (A_{1}) ⊆ B_{1}, and such that ∥T ∥_{A_{0}→B_{0}}, ∥T ∥_{A_{1}→B_{1}} < ∞.

An interpolation method, I, is a functor that takes a Banach couple (A_{0}, A_{1}) to a single Banach space A_{I}, such that A_{0} ∩ A_{1} ⊆ A_{I} ⊆ A_{0} + A_{1} with

c^{−1} ∥x∥_{A_{0}+A_{1}} ≤ ∥x∥_{A_{I}} ≤ c ∥x∥_{A_{0}∩A_{1}} ,  (1)

and so that if T : (A_{0}, A_{1}) → (B_{0}, B_{1}) is a map between couples, then T (A_{I}) ⊆ B_{I} with ∥T ∥_{A_{I}→B_{I}} < ∞.

An interpolation method is called exponential with exponent θ if 0 < θ < 1, and whenever T : (A_{0}, A_{1}) → (B_{0}, B_{1}) is a map between couples, then

∥T ∥_{A_{I}→B_{I}} ≤ c ∥T ∥_{A_{0}→B_{0}}^{1−θ} ∥T ∥_{A_{1}→B_{1}}^{θ} .  (2)

An interpolation method is called exact exponential if it is exponential and c = 1 in inequalities (1) and (2).

The most well known interpolation methods are the real interpolation method, and the complex interpolation method. They are both exponential, and the complex interpolation method is exact exponential. Another interpolation method, parameterized by 0 < θ < 1, is the following: if x ∈ A_{0} ∩ A_{1}, then let

|||x||| = inf { ∑_{i=1}^{n} ∥x_{i}∥_{A_{0}}^{1−θ} ∥x_{i}∥_{A_{1}}^{θ} : ∑_{i=1}^{n} x_{i} = x} .

GARLING – MONTGOMERY-SMITH

Let A_{min,θ} be the completion of (A_{0} ∩ A_{1}, ||| · |||). This interpolation method is exact exponential of exponent θ. Furthermore, if I is another exponential method of exponent θ, then ∥x∥_{A_{I}} ≤ c ∥x∥_{A_{min,θ}}, with c = 1 if I is exact exponential. (In fact, this interpolation method gives a space equivalent to the space A_{θ,1} obtained by the real interpolation method.)

We denote by X ⊕ V the Banach space of ordered pairs (x, v) with norm ∥(x, v)∥ = ∥x∥ + ∥v∥.

All the results given work for Banach spaces over the real or complex scalars. (Obviously the complex interpolation method will require the complex scalars.)

The Main Result Here we present the main result of this paper. Theorem 1. Suppose that Z is a Banach space and that Q_{i} is a quotient mapping onto Z from a closed linear subspace Y_{i} of a separable Banach space X_{i}, for i = 0, 1, such that the following dimensional constraints hold:

a) the dimensions of Y_{0} and Y_{1} are equal;

b) the codimensions of Y_{0} in X_{0} and Y_{1} in X_{1} are equal;

c) the dimensions of the kernels of Q_{0} and Q_{1} are equal. Suppose further that 0 < ϵ < 1, that 0 < θ_{0} ≤ θ_{1} < 1 and that V_{0} and V_{1} are any infinite dimensional separable Banach spaces.

Then there is a Banach couple (A_{0}, A_{1}), a subspace W of A_{0} + A_{1}, and linear maps P : W → Z and E : Z → W with the following properties:

i) A_{0} is isometric to X_{0} ⊕ V_{0}, and A_{1} is isometric to X_{1} ⊕ V_{1}; ii) P ∘ E = Id_{Z}; if I is an exponential interpolation method of exponent θ, then iii) A_{I} ⊆ W and ∥P ∥_{A_{I}→Z} < ∞ (with ∥P ∥_{A_{I}→Z} < 1 + ϵ/3 if I is exact and θ_{0} ≤ θ ≤ θ_{1}); iv) E(Z) ⊆ A_{I} and ∥E∥_{Z→A_{I}} < ∞ (with ∥E∥_{Z→A_{I}} < 1 + ϵ/3 if I is exact and θ_{0} ≤ θ ≤

θ_{1}). Thus Z is isomorphic to a complemented subspace of A_{I}. (If I is exact and θ_{0} ≤ θ ≤ θ_{1}, then Z is (1 + ϵ) isomorphic to a (1 + ϵ)-complemented subspace of A_{I}.)

This has the following corollaries. Corollary. Given any separable Banach space X, there is a Banach couple (A_{0}, A_{1}) such that A_{0} and A_{1} are isometric to l_{1}, and for every exponential interpolation method I, the intermediate space A_{I} contains a complemented subspace isomorphic to X.

Corollary. Given any separable Banach space X, there is a Banach couple (A_{0}, A_{1}) such that A_{0} and A_{1} are isometric to C([0, 1]), and for every exponential interpolation method I, the intermediate space A_{I} contains a complemented subspace isomorphic to X. Corollary. The following properties do not pass to intermediate spaces by any exponential interpolation methods, in the sense that there is a couple (A_{0}, A_{1}) such that A_{0} and A_{1} both have the property, but none of the intermediate spaces do:

i) the Radon–Nikodym property; ii) the analytic Radon-Nikodym property; iii) having non-trivial cotype; iv) the AUMD property;

v) having a dual with non-trivial cotype; vi) having a dual with the AUMD property.

Note that parts (v) and (vi) of the above corollary follow because C([0, 1])^{∗} is finitely represented in l_{1}.

A More Elementary Result Before presenting the proof of Theorem 1, we first prove a more elementary result that is, in fact, a corollary of Theorem 1. The reason for this is that the proof of Theorem 1 involves many technicalities, disguising the essential idea of the proof, which is very simple. Theorem 2. There is a Banach couple (A_{0}, A_{1}) such that A_{0} and A_{1} are isometric to l_{1}, and for every exponential interpolation method I, the intermediate space A_{I} contains a complemented subspace isomorphic to c_{0}.

Proof: Let e_{n} be the unit vectors in l_{1} and c_{0}, and let r_{n} be an enumeration of the ‘corners of the unit cube’ in c_{0}, that is, all vectors of the form (±1, ±1, …, ±1, 0, 0, 0, …). Let ϵ_{n} be the sequence of numbers defined by

ϵ_{n} = (2 + ∥r_{n}∥_{1})^{−n}.

(Any sequence of numbers tending sufficiently rapidly to zero will do.) We form an ambient space Ω = c_{0} ⊕ l_{1} ⊕ l_{1}. We define subspaces

A_{0} = { (∑_{n} a_{n}e_{n} + b_{n}r_{n}, ∑_{n} ϵ_{n}b_{n}e_{n}, ∑_{n} c_{n}e_{n}) : (a_{n}), (b_{n}), (c_{n}) ∈ l_{1} }
A_{1} = { (∑_{n} a_{n}e_{n} + c_{n}r_{n}, ∑_{n} b_{n}e_{n}, ∑_{n} ϵ_{n}c_{n}e_{n}) : (a_{n}), (b_{n}), (c_{n}) ∈ l_{1} } ,

with norms

∥(∑_{n} a_{n}e_{n} + b_{n}r_{n}, ∑_{n} ϵ_{n}b_{n}e_{n}, ∑_{n} c_{n}e_{n})∥_{A_{0}} = ∑_{n} (|a_{n}| + |b_{n}| + |c_{n}|)
∥(∑_{n} a_{n}e_{n} + c_{n}r_{n}, ∑_{n} b_{n}e_{n}, ∑_{n} ϵ_{n}c_{n}e_{n})∥_{A_{1}} = ∑_{n} (|a_{n}| + |b_{n}| + |c_{n}|) .

The idea is that the unit balls of A_{0} and A_{1} are ‘slightly perturbed’ versions of the unit balls of c_{0}, where the perturbations for A_{0} and A_{1} go in different directions. These perturbations cause the unit balls of A_{0} and A_{1} to be convex hulls of linearly independent vectors, and hence they are affine to the unit ball of l_{1}. The size of these perturbations is controlled by the quantities ϵ_{n}. The vectors that are used to perturb the unit ball of A_{0} are also contained in A_{1} with no control on their size. Similarly, the perturbing vectors of A_{1} are contained in A_{0}. Thus, when we form the intermediate space A_{I}, the perturbations in A_{0} are ‘swamped out’ or ‘washed away’ by the vectors in A_{1}, and similarly for the perturbations in A_{1}, and we are left with a complemented copy of c_{0}. So much for the idea—now we give the proof.

We define a projection P : Ω → c_{0} by

P (x, y, z) = x.

It is easy to see that ∥P ∥_{A_{i}→c_{0}} ≤ 1 for i = 0, 1, and hence ∥P ∥_{A_{I}→c_{0}} < ∞ for any interpolation method I.

We define an embedding E : c_{0} → Ω by

E(x) = (x, 0, 0).

We prove that for every exponential interpolation method that E(x) ∈ A_{I} (x ∈ c_{0}), with ∥E∥_{c_{0}→A_{I}} < ∞. To do this, it is sufficient to show that there is a constant c, depending on the exponential interpolation method I only, such that for every n ≥ 1 we have

∥(r_{n}, 0, 0)∥_{A_{I}} ≤ c.

We first note the following facts.

∥(r_{n}, ϵ_{n}e_{n}, 0)∥_{A_{0}} = ∥(r_{n}, 0, ϵ_{n}e_{n})∥_{A_{1}} = 1
∥(0, 0, ϵ_{n}e_{n})∥_{A_{0}} = ∥(0, ϵ_{n}e_{n}, 0)∥_{A_{1}} = ϵ_{n}

∥(r_{n}, 0, 0)∥_{A_{0}} = ∥(r_{n}, 0, 0)∥_{A_{1}} = ∥r_{n}∥_{1} .

Let us suppose that I is of exponent θ. We shall give the details in the case where I is exact exponential: a similar argument works in the general case, with less control of the constants. We have the following inequalities.

∥(r_{n}, 0, 0)∥_{A_{I}} ≤ ∥(r_{n}, ϵ_{n}e_{n}, ϵ_{n}e_{n})∥_{A_{I}} + ∥(0, ϵ_{n}e_{n}, 0)∥_{A_{I}} + ∥(0, 0, ϵ_{n}e_{n})∥_{A_{I}} ,

and

∥(r_{n}, ϵ_{n}e_{n}, ϵ_{n}e_{n})∥_{A_{I}} ≤ ∥(r_{n}, ϵ_{n}e_{n}, ϵ_{n}e_{n})∥_{A_{0}}^{1−θ} ∥(r_{n}, ϵ_{n}e_{n}, ϵ_{n}e_{n})∥_{A_{1}}^{θ}
≤( ∥(r_{n}, ϵ_{n}e_{n}, 0)∥_{A_{0}} + ∥(0, 0, ϵ_{n}e_{n})∥_{A_{0}} )^{1−θ}
×( ∥(r_{n}, 0, ϵ_{n}e_{n})∥_{A_{1}} + ∥(0, ϵe_{n}, 0)∥_{A_{1}} )^{θ}
≤(1 + ϵ_{n})^{1−θ}(1 + ϵ_{n})^{θ}
≤(1 + ϵ_{n}),

and

∥(0, ϵ_{n}e_{n}, 0)∥_{A_{I}} ≤ ∥(0, ϵ_{n}e_{n}, 0)∥_{A_{0}}^{1−θ} ∥(0, ϵ_{n}e_{n}, 0)∥_{A_{1}}^{θ} .

But

∥(0, ϵ_{n}e_{n}, 0)∥_{A_{0}} ≤ ∥(r_{n}, ϵ_{n}e_{n}, 0)∥_{A_{0}} + ∥(r_{n}, 0, 0)∥_{A_{0}}
≤1 + ∥r_{n}∥_{1} ,

and so

∥(0, ϵ_{n}e_{n}, 0)∥_{A_{I}} ≤ (1 + ∥r_{n}∥_{1})^{1−θ} ϵ_{n}^{θ}.

Similarly

∥(0, 0, ϵ_{n}e_{n})∥_{A_{I}} ≤ (1 + ∥r_{n}∥_{1})^{θ} ϵ_{n}^{1−θ}.

Therefore

∥(r_{n}, 0, 0)∥_{A_{I}} ≤ 1 + ϵ_{n} + (1 + ∥r_{n}∥_{1})^{1−θ}ϵ_{n}^{θ} + (1 + ∥r_{n}∥_{1})^{θ}ϵ_{n}^{1−θ}.

By our choice of ϵ_{n}, this is bounded by some constant c that depends only upon θ. Thus we have bounded maps

P : A_{I} → c_{0} E : c_{0} → A_{I}

such that P ∘ I = Id_{c_{0}}, and hence c_{0} is isomorphic to a complemented subspace of A_{I}. Finally, it is very clear that A_{0} and A_{1} are both isometric to l_{1}.

The Proof of Theorem 1 In the sequel, we will make much use of biorthogonal systems. A biorthogonal system of a Banach space X is a sequence (x_{n}; ξ_{n}) ∈ X × X^{∗} such that ξ_{n}(x_{m}) = 0 if n ≠ m and 1 if n = m. A biorthogonal system is called fundamental if \overline{span}\{x_{n}\} = X, and it is called total if x = 0 whenever ξ_{n}(x) = 0 for all n. (Of course, if X is finite dimensional, then the sequences will be finite.) A result of Markushevich shows that every separable Banach space has a total, fundamental biorthogonal system (see [6] or [4] 1.f.3).

We shall need the following proposition and its corollary.

Lemma 3. Let Z be a quotient of a separable Banach space Y by the quotient map Q with kernel K. Suppose that (k_{p}; κ′_{p}) is a total fundamental biorthogonal system for K, and that (z_{n}; ζ_{n}) is a total fundamental biorthogonal system for Z. Then there are sequences (y_{n}) in Y, and (ψ_{n}) and (κ_{p}) in Y^{∗}, for which the following hold:

i) (y_{n}, k_{p}; ψ_{n}, κ_{p}) is a total fundamental biorthogonal system for Y; ii) Q(y_{n}) = z_{n} and ζ_{n} ∘ Q = ψ_{n} for all n; iii) y ∈ K if and only if ψ_{n}(y) = 0 for all n.

Proof: Choose ỹ_{n} ∈ Y such that Q(ỹ_{n}) = z_{n}, and let ψ_{n} = ζ_{n}∘Q. Using the Hahn–Banach Theorem, extend κ′_{p} to κ_{p} on Y so that κ_{p}(ỹ_{n}) = 0 for n < p. Set

y_{n} = ỹ_{n} − ∑_{m=1}^{n} κ_{m}(ỹ_{n})k_{m}.

It is now straightforward to verify the conclusions of the lemma.

Lemma 4. Let Y be a closed subspace of a separable Banach space X, and let (y_{n}; ψ′_{n}) be a total fundamental biorthogonal system for Y. Then there are sequences (x_{m}) in X, and (ξ_{m}) and (ψ_{n}) in X^{∗}, such that (x_{m}, y_{n}; ξ_{m}, ψ_{n}) is a total fundamental biorthogonal system for X, and such that an element x in X belongs to Y if and only if ξ_{m}(x) = 0 for all m.

Proof: By Markushevich’s result, Y has a total fundamental system (y_{n}; ψ′_{n}), and X/Y has a total fundamental system (z_{m}; ζ_{m}). Then the result follows by applying Lemma 3 to the quotient mapping Q : X → X/Y.

Proof of Theorem 1: We begin by defining some biorthogonal systems. First, let (z_{n}, ζ_{n}) be a total fundamental biorthogonal system for Z, with ∥ζ_{n}∥ = 1 for all n. Using Lemmas 3 and 4, for i = 0, 1, we can find total fundamental biorthogonal systems

(x_{m}^{i}, y_{n}^{i}, k_{p}^{i}; ξ_{m}^{i}, ψ_{n}^{i}, κ_{p}^{i})

for X_{i} with the following properties:

i) Q_{i}(y_{n}^{i}) = z_{n} and ζ_{n} ∘ Q_{i} = ψ_{n}^{i} for all n; ii) Y_{i} = { x ∈ X_{i} : ξ_{m}^{i}(x) = 0 for all m}; iii) (y_{n}^{i}, k_{p}^{i}; ψ_{n}^{i}|_{Y_{i}}, κ_{p}^{i}|_{Y_{i}}) is a total fundamental biorthogonal system for Y_{i}; iv) (k_{p}^{i}; κ_{p}^{i}|_{K_{i}}) is a total fundamental biorthogonal system for K_{i}, the kernel of Q_{i};

v) ∥ξ_{m}^{i}∥ = ∥ψ_{n}^{i}∥ = ∥κ_{p}^{i}∥ = 1 for all m, n and p. We also choose total fundamental biorthogonal systems (v_{q}^{i}; ϕ_{q}^{i}) for V_{i}, with ∥ϕ_{q}^{i}∥ = 1 for all q. If any of these systems is finite, then we extend it to an infinite system by including zero terms.

Next we define some sequences of rapidly decreasing numbers. Let ν_{n} and μ_{n} be sequences of positive numbers for which

∑ L_{n}ν_{n} ≤ ϵ/12,
max{μ_{n}^{θ}L_{n}, μ_{n}^{1−θ}L_{n}} ≤ ν_{n}  for θ_{0} ≤ θ ≤ θ_{1},

and

∑ μ_{n}^{θ}L_{n} < ∞  for 0 < θ < 1,

where

L_{n} = max{ ∥x_{n}^{i}∥ , ∥y_{n}^{i}∥ , ∥k_{p}^{i}∥ , ∥v_{2n−1}^{i}∥ , ∥v_{2n}^{i}∥ , 1 : i = 0, 1}.

Now we define the Banach couple. We take as the ambient space Ω = l_{∞} ⊕ l_{∞} ⊕ l_{∞} ⊕ l_{∞} ⊕ l_{∞}. We define linear maps M_{i} : X_{i} ⊕ V_{i} → Ω in the following way:

M_{0}(x, v) = ((ψ_{n}^{0}(x)), (ξ_{m}^{0}(x)), (μ_{p}κ_{p}^{0}(x)), (μ_{q}ϕ_{2q−1}^{0}(v)), (ϕ_{2q}^{0}(v))),
M_{1}(x, v) = ((ψ_{n}^{1}(x)), (μ_{q}ϕ_{2q−1}^{1}(v)), (ϕ_{2q}^{1}(v)), (ξ_{m}^{1}(x)), (μ_{p}κ_{p}^{1}(x)), ).

The important feature here is the interchange in the order of the terms. We note that M_{0} and M_{1} are one-one maps into Ω. For i = 0, 1, we set A_{i} = M_{i}(X_{i} ⊕ V_{i}) with their norms inherited from the domains. We take W to be the linear span of all the exponential interpolation spaces A_{I}.

We shall now go through the details for the case when I is exact and θ_{0} ≤ θ ≤ θ_{1}. Otherwise the arguments are similar, with less control of the constants. Let e_{n} denote the nth unit vector in l_{∞}. Then

(0, μ_{n}e_{n}, 0, 0, 0) = μ_{n}M_{0}(x_{n}^{0}) = M_{1}(v_{2n−1}^{1}),

so that

∥(0, μ_{n}e_{n}, 0, 0, 0)∥_{A_{0}} ≤ μ_{n}L_{n},

and

∥(0, μ_{n}e_{n}, 0, 0, 0)∥_{A_{1}} ≤ L_{n}.

Hence

∥(0, μ_{n}e_{n}, 0, 0, 0)∥_{A_{I}} ≤ μ_{n}^{1−θ}L_{n} ≤ ν_{n}.

Similarly,

∥(0, 0, μ_{n}e_{n}, 0, 0)∥_{A_{I}} ≤ ν_{n},
∥(0, 0, 0, μ_{n}e_{n}, 0)∥_{A_{I}} ≤ ν_{n},

and

∥(0, 0, 0, 0, μ_{n}e_{n})∥_{A_{I}} ≤ ν_{n}.

We define linear functionals on Ω as follows: if t = (h, a, b, c, d) ∈ Ω, then let η_{n}(t) = h_{n}, α_{n}(t) = a_{n}, β_{n}(t) = b_{n}, γ_{n}(t) = c_{n} and δ_{n}(t) = d_{n}. Then we have that

∥α_{n}∥_{A_{0}^{∗}} = 1

and

∥α_{n}∥_{A_{1}^{∗}} = μ_{n},

and so

∥α_{n}∥_{A_{I}^{∗}} ≤ μ_{n}^{θ} ≤ ν_{n}.

Similarly, ∥β_{n}∥_{A_{I}^{∗}} ≤ ν_{n}, ∥γ_{n}∥_{A_{I}^{∗}} ≤ ν_{n} and ∥δ_{n}∥_{A_{I}^{∗}} ≤ ν_{n}.

Now we define E : Z → Ω by E(z) = ((ζ_{n}(z)), 0, 0, 0, 0). We assert that E(z) ∈ A_{I}, with ∥E(z)∥_{A_{I}} ≤ (1 + ϵ/3) ∥z∥. To show this, we can suppose that ∥z∥ < 1. For i = 0, 1, choose y_{i} ∈ Y_{i} such that Q_{i}(y_{i}) = z and ∥y_{i}∥ ≤ 1. Let

v_{0} = ∑_{p} μ_{p}κ_{p}^{1}(y_{1})v_{2p}^{0} and v_{1} = ∑_{p} μ_{p}κ_{p}^{0}(y_{0})v_{2p}^{1}.

It follows from the choice of μ_{p} that these sums converge in V_{0} and V_{1}, and that ∥v_{0}∥ ≤ ϵ/12 and ∥v_{1}∥ ≤ ϵ/12. Now

M_{0}(y_{0}, v_{0}) = M_{1}(y_{1}, v_{1})
= ((ζ_{n}(z)), 0, (μ_{p}κ_{p}^{0}(y_{0})), 0, (μ_{p}κ_{p}^{1}(y_{1})))
= w,

say, so that w ∈ A_{I} with ∥w∥_{A_{I}} ≤ 1 + ϵ/12. But

∥(0, 0, (μ_{p}κ_{p}^{0}(y_{0})), 0, 0)∥_{A_{I}} ≤ ∑ μ_{p}L_{p} ≤ ϵ/12,

and similarly,

∥(0, 0, 0, 0, (μ_{p}κ_{p}^{0}(y_{0})))∥_{A_{I}} ≤ ϵ/12.

Hence E(z) ∈ A_{I} with ∥E(z)∥ ≤ 1 + ϵ/3, as desired.

Now we turn to the problem of defining P. To do this, we will first show how we may consider Z as a subspace of what we shall call X_{0} ⊕_{Q} X_{1}. We let

D = { (y_{0}, y_{1}) : y_{i} ∈ Y_{i}, Q_{0}(y_{0}) + Q_{1}(y_{1}) = 0},

and let X_{0} ⊕_{Q} X_{1} = X_{0} ⊕ X_{1}/D. We denote by Q_{D} the quotient map from X_{0} ⊕ X_{1} to X_{0} ⊕_{Q} X_{1}. We define J : Z → X_{0} ⊕_{Q} X_{1} by setting J(Q_{0}(y)) = Q_{D}(y, 0) for y ∈ Y_{0}. It is easy to see that J is well defined, and that J(Q_{1}(y)) = Q_{D}(0, y) for y ∈ Y_{1}.

Suppose that w = Q_{D}(x_{0}, x_{1}) ∈ X_{0} ⊕_{Q} X_{1}. Then it is easy to verify that π_{m}^{0}(w) = ξ_{m}^{0}(x_{0}) and π_{m}^{1}(w) = ξ_{m}^{1}(x_{1}) are well defined maps. It is also easy to check that Z is isometric to the space

J(Z) = { w : π_{m}^{0}(w) = π_{m}^{1}(w) = 0}.

Now, we will define two bounded operators S and T from A_{I} to X_{0} ⊕_{Q} X_{1}. First we will concentrate on S, which will, in fact, be defined on A_{0} + A_{1}.

If t = M_{0}(x, v) ∈ A_{0}, we let

R_{0}(t) = ∑ μ_{m}ϕ_{2m−1}^{0}(v)x_{m}^{1}.

By the choice of μ_{m}, the sum converges in X_{1} with ∥R_{0}(t)∥ ≤ ∥v∥. Thus if we set S_{0}(t) = Q_{D}(x, R_{0}(t)), then S_{0} is a norm decreasing map from A_{0} to X_{0} ⊕_{Q} X_{1}. We define S_{1} in a similar way.

Now, if t = M_{0}(x_{0}, v_{0}) = M_{1}(x_{1}, v_{1}) ∈ A_{0} ∩ A_{1}, then

ξ_{m}^{0}(x_{0}) = μ_{m}ϕ_{2m−1}^{1}(v_{1}) = ξ_{m}^{0}(R_{1}(t))

and

ξ_{m}^{1}(x_{1}) = μ_{m}ϕ_{2m−1}^{0}(v_{0}) = ξ_{m}^{1}(R_{0}(t)),

so that x_{0} − R_{1}(t) ∈ Y_{0} and x_{1} − R_{0}(t) ∈ Y_{1}. Further,

ζ_{n}(Q_{0}(x_{0} − R_{1}(t))) = ψ_{n}^{0}(x_{0} − R_{1}(t)) = ψ_{n}^{0}(x_{0}),

and similarly

ζ_{n}(Q_{1}(x_{1} − R_{0}(t))) = ψ_{n}^{1}(x_{1}).

Then, since ψ_{n}^{0}(x_{0}) = ψ_{n}^{1}(x_{1}) = η_{n}(t), and since (ζ_{n}) is total, it follows that Q_{0}(x_{0} − R_{1}(t)) = Q_{1}(x_{1} − R_{0}(t)). Thus we see that

S_{0}(t) = Q_{D}(x_{0} − R_{1}(t)) = Q_{D}(x_{1} − R_{0}(t)) = S_{1}(t),

and so we may define S : A_{0} + A_{1} → X_{0} ⊕_{Q} X_{1} by setting S|_{X_{0}} = S_{0} and S|_{X_{1}} = S_{1}. Clearly, S maps A_{I} into X_{0} ⊕_{Q} X_{1} with ∥S∥_{A_{I}→X_{0}⊕_{Q}X_{1}} ≤ 1.

Now we establish the effect of π_{m}^{0} and π_{m}^{1} on S. If t = (h, a, b, c, d) = M_{0}(x, v) ∈ A_{0}, then

π_{m}^{0}(S_{0}(t)) = π_{m}^{0}(Q_{D}(x − R_{0}(t))) = ξ_{m}^{0}(x) = a_{m},

and

π_{m}^{1}(S_{0}(t)) = π_{m}^{0}(Q_{D}(x − R_{0}(t))) = ξ_{m}^{0}(R_{0}(t)) = μ_{m}ϕ_{2m−1}(v) = c_{m}.

Similarly, if t = (h, a, b, c, d) = M_{1}(x, v) ∈ A_{1}, then π_{m}^{0}(S_{1}(t)) = a_{m} and π_{m}^{1}(S_{1}(t)) = c_{m}. Thus, if t = (h, a, b, c, d) ∈ A_{I}, then

π_{m}^{0}(S(t)) = a_{m} and π_{m}^{1}(S(t)) = c_{m}.
Next, we define T : A_{I} → X_{0} ⊕_{Q} X_{1} by
T (t) = Q_{D} (∑ α_{m}(t)x_{m}^{0}, ∑ γ_{m}(t)x_{m}^{1}) .

Note that the sums converge, and that

∥T ∥ ≤ ∑ ν_{m}(∥x_{m}^{0}∥ + ∥x_{m}^{1}∥) ≤ ϵ/6.

We see that, if t = (h, a, b, c, d), then π_{m}^{0}(T (t)) = a_{m} and π_{m}^{1}(T (t)) = c_{m}.

Hence (S −T)(t) ∈ J(Z). So now we set P = J^{−1} ∘(S −T). We have that ∥P ∥_{A_{I}→Z} ≤ 1 + ϵ/3, and

P ∘ E(z_{n}) = P (u_{n}, 0, 0, 0, 0)
= J^{−1} ∘ (S − T)(u_{n}, 0, 0, 0, 0)
= J^{−1} ∘ S(u_{n}, 0, 0, 0, 0)
= J^{−1} ∘ S_{0} ∘ M_{0}(y_{n}^{0}, 0)
= J^{−1} ∘ Q_{D}(y_{n}^{0}, 0) = z_{n},

so that P ∘ E = Id_{Z}.

Further Results and Conjectures An obvious extension to Theorem 1 would be the following.

Conjecture. Suppose that (Z_{0}, Z_{1}) is a Banach couple where Z_{i} is a quotient of a subspace of X_{i}, for i = 0, 1, suppose that X_{0}, X_{1}, V_{0} and V_{1} are separable Banach spaces, and suppose that suitable dimension constraints are satisfied. Then there is a Banach couple (A_{0}, A_{1}) such that

i) A_{0} is isometric to X_{0} ⊕ V_{0}, and A_{1} is isometric to X_{1} ⊕ V_{1}; ii) for every exponential interpolation method I, the space A_{I} contains a complemented

subspace isomorphic to Z_{I}.

The second named author has a tentative proof of a local version of this result for the real interpolation method, which he hopes to publish later.

However, we can show the following.

Theorem 5. Suppose (Y_{0}, Y_{1}) is a Banach couple, where Y_{1} ⊂ Y_{0} with ∥y∥_{Y_{0}} ≤ c ∥y∥_{Y_{1}}. Suppose further that Y_{0} is a quotient space of a separable Banach space X_{0}, and Y_{1} is a complemented subspace of a separable Banach space X_{1}, such that the codimensions of Y_{0} in X_{0} and Y_{1} in X_{1} are equal. Then there is a Banach couple (A_{0}, A_{1}) such that

i) A_{0} is isometric to X_{0}, and A_{1} is isometric to X_{1}; ii) given any exponential interpolation method of exponent θ, there are maps E : Y_{min,θ} →

A_{I} and P : A_{I} → Y_{I}, both of bounded norm, such that P ∘ E = Id_{Y_{min,θ}}

Corollary. There is a Banach couple (A_{0}, A_{1}) such that A_{0} is isometric to l_{1}, and A_{1} is isometric to l_{p}, and for any exponential interpolation method I of exponent θ, the intermediate space A_{I} contains a subspace V such that l_{p/θ,1} ⊆ V ⊆ l_{p/θ,∞} with c^{−1} ∥x∥_{p/θ,∞} ≤ ∥x∥_{V} ≤ c ∥x∥_{p/θ,1}.

Corollary. There is a Banach couple (A_{0}, A_{1}) such that A_{0} has cotype 2, and A_{1} has cotype p and is K-convex, and such that for all 0 < θ < 1, the real interpolation space A_{θ,1} does not have cotype r for any r < p/θ.

This shows that a result of Xu [7] cannot be improved. He showed that, if (A_{0}, A_{1}) is a Banach couple such that A_{1} is K-convex with cotype p, then for all 0 < θ < 1, the real interpolation space A_{θ,1} has cotype p/θ.

Proof of Theorem 5: Suppose that Y_{0} is a quotient of X_{0} by a subspace Z_{0}, and that X_{1} = Y_{1} ⊕ Z_{1}. Denote the quotient map from X_{0} to Y_{0} by Q. Let (z_{n}^{0}; ζ̃_{n}^{0}) be a total fundamental biorthogonal sequence for Z_{0}, and let (z_{n}^{1}; ζ̃_{n}^{1}) be a total fundamental biorthogonal sequence for Z_{1}. Extend ζ̃_{n}^{0} to ζ_{n}^{0} on X_{0} using the Hahn–Banach Theorem, and define ζ_{n}^{1} on X_{1} by setting ζ_{n}^{1}(y, z) = ζ̃_{n}^{1}(z). Without loss of generality, ∥ζ_{n}^{0}∥ = ∥ζ_{n}^{1}∥ = 1.

To save chasing constants, let us assume that I is exact. Let ϵ_{n} be a sequence of sufficiently small positive numbers, so that

δ = ∑_{n=1}^{∞} max{ϵ_{n}, ϵ_{n}^{θ}} max{∥z_{n}^{0}∥ , ∥z_{n}^{1}∥} < ∞,

for all 0 < θ < 1. We will let Y_{0} ⊕ l_{∞} be the ambient space. Let M_{i} : X_{i} → Y_{0} ⊕ l_{∞} be the mappings:

M_{0}(x) = (Q(x), (ϵ_{n}ζ_{n}^{0}(x)))
M_{1}(x) = (Q(x), (ζ_{n}^{1}(x))).

Since (z_{n}^{0}; ζ̃_{n}^{0}) and (z_{n}^{1}; ζ̃_{n}^{1}) are both total biorthogonal systems, M_{0} and M_{1} are one to one. We let A_{i} = M_{i}(X_{i}) with norms inherited from the domains of the functions.

Define E : Y_{0} → Y_{0} ⊕ l_{∞} by

E(y) = (y, 0).

We desire to show that E(Y_{min,θ}) ⊆ A_{I}, with ∥E∥_{Y_{min,θ}→A_{I}} < ∞. First, note that

∥(0, ϵ_{n}e_{n})∥_{A_{0}} = ∥z_{n}^{0}∥ ,

and

∥(0, ϵ_{n}e_{n})∥_{A_{1}} = ϵ_{n} ∥z_{n}^{1}∥ .

SPACES OBTAINED BY INTERPOLATION

Hence,

∥(0, ϵ_{n})∥_{A_{I}} ≤ ϵ_{n}^{θ} max{∥z_{n}^{0}∥ , ∥z_{n}^{1}∥},

and so

∥(0, (ϵ_{n}ζ_{n}^{0}(x)))∥_{A_{I}} ≤ δ ∥x∥_{X_{0}} .

Let ν > 0. For y ∈ Y_{1}, let x ∈ X_{0} be such that Q(x) = y with ∥x∥_{X_{0}} ≤ (1 + ν) ∥y∥_{Y_{0}}. Then

∥(y, (ϵ_{n}ζ_{n}^{0}(x)))∥_{A_{0}} = ∥x∥_{X_{0}} ≤ (1 + ν) ∥y∥_{Y_{0}} ,
∥(y, 0)∥_{A_{1}} = ∥y∥_{Y_{1}} ,

and

∥(0, (ϵ_{n}ζ_{n}^{0}(x)))∥_{A_{1}} ≤ ∑_{n} ϵ_{n} ∥z_{n}^{1}∥ ∥x∥_{X_{0}}
≤ (1 + ν)δ ∥y∥_{Y_{0}} ≤ (1 + ν)δ ∥y∥_{Y_{1}} .

So,

∥(y, (ϵ_{n}ζ_{n}^{0}(x)))∥_{A_{1}} ≤ (1 + ν)(1 + δ) ∥y∥_{Y_{1}} .

Therefore,

∥(y, (ϵ_{n}ζ_{n}^{0}(x)))∥_{A_{I}} ≤ (1 + ν)(1 + δ) ∥y∥_{Y_{0}}^{1−θ} ∥y∥_{Y_{1}}^{θ} .

Finally,

∥(y, 0)∥_{A_{I}} ≤ δ ∥x∥_{X_{0}} + (1 + ν)(1 + δ) ∥y∥_{Y_{0}}^{1−θ} ∥y∥_{Y_{1}}^{θ}
≤ (1 + ν)(1 + 2δ) ∥y∥_{Y_{0}}^{1−θ} ∥y∥_{Y_{1}}^{θ} .

This is sufficient to show that ∥E(y)∥_{A_{I}} ≤ (1 + ν)(1 + 2δ) ∥y∥_{Y_{min,θ}} as desired.

Now we let P : Y_{0} ⊕ l_{∞} → Y_{0} be the map

P (y, a) = y.

Then ∥P ∥_{A_{0}→Y_{0}} = ∥P ∥_{A_{1}→Y_{1}} = 1, and hence ∥P ∥_{A_{I}→Y_{I}} ≤ 1. Clearly P ∘ E = Id_{Y_{min,θ}}.

Acknowledgements We would like to express our warm thanks to G. Pisier for suggesting this line of research to us.

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