A new proof of Donoghue ’ s interpolation theorem

We give a new proof and new interpretation of Donoghue’s interpolation theorem; for an intermediate Hilbert space H∗ to be exact interpolation with respect to a regular Hilbert couple H it is necessary and sufficient that the norm in H∗ be representable in the form ‖f‖∗ = (∫ [0,∞] (1 + t)K2(t, f ;H)dρ(t) )1/2 with some positive Radon measure ρ on the compactified half-line [0,∞]. The result was re-proved in [1] in the finitedimensional case. The purpose of this note is to extend the proof given in [1] to cover the infinite-dimensional case. Moreover, the presentation of the aforementioned proof in [1] was slightly flawed, because we forgot to include a reference to ’Donoghue’s Lemma’, which is implicitly used in the proof. Hence we take this opportunity to correct that flaw. 1. Donoghue’s Theorem 1 Let H = (H0,H1) be a Hilbert couple over C ∗. With a rather trivial restriction, we will, in the following, make the standing assumption that the couple H is separable and regular, i.e., H0, H1 are separable and H0 ∩H1 is ∗With minor modifications, one can include the case of real scalars. 254 A new proof of Donoghue’s interpolation theorem dense in H0, H1 †. In the following, all involutions and inner products will be taken with respect to the norm of H0. One may refer to [3] for a general background to interpolation theory. The squared norm ‖ · ‖1 is an unbounded, densely defined quadratic form on H0 and can hence be expressed as ‖f‖1 = (Af, f)0, where A is a densely defined, positive injective operator in H0 (the domain of A 1 2 is H0 ∩ H1). A useful tool in the study of interpolation of Hilbert spaces is the functional K2, given by K2(t, f) = K2(t, f ;H) = inf f=f0+f1 (‖f0‖0 + t‖f1‖1). In [1] (for instance) was noted the basic formula for K2, K2(t, f) = ( tA 1 + tA f, f )

dense in H 0 , H 1 † .In the following, all involutions and inner products will be taken with respect to the norm of H 0 .One may refer to [3] for a general background to interpolation theory.
The squared norm • 2  1 is an unbounded, densely defined quadratic form on H 0 and can hence be expressed as where A is a densely defined, positive injective operator in H 0 (the domain of A 1 2 is H 0 ∩ H 1 ).A useful tool in the study of interpolation of Hilbert spaces is the functional K 2 , given by In [1] (for instance) was noted the basic formula for K 2 , Given an intermediate Hilbert space H * , we can, in an analogous fashion, express its norm as for some positive injective operator B in H 0 (domain(B 2 ) = H 0 ∩ H 1 ).The property that H * be intermediate is then equivalent to the existence of a number r > 0 with the property 2) It will be convenient to use the following alternative notations for the operator norms with respect to H 0 , H 1 and H * , (1.3) † The passage to the non-separable and the non-regular case is done as in Donoghue's paper [5], and there is no point in repeating those arguments here.We shall however comment on those cases later on.
Recall that H * is exact interpolation with respect to H if where L(H) denotes the set of linear operators T on H 0 + H 1 such that the restriction of T to H i belongs to L(H i ), i = 0, 1.
Our main result is the following (Donoghue's Theorem 1), which is equivalent to Donoghue ([5], Theorem 1).We remark that the finitedimensional version of this theorem was obtained in ([1], Theorem 2.4) as a consequence of the fact that Hilbert couples constitute a case of 'Calderón couples'.This paper is intended as a follow-up to that paper, where we treat the infinite-dimensional (separable) case.
Remark 1.1.In order to pass to the infinite-dimensional case, it becomes necessary to recast portion of Donoghue's arguments in a form which is suitable for our setting.We take this opportunity to remark that Donoghue's setting is advantageous over ours in the following two ways: (i) it treats spaces in a symmetric way, and (ii) it is well-suited for generalization to the non-regular case.

Donoghue's Theorem 1.
For a Hilbert space H * to satisfy (ExInt), it is necessary and sufficient that there exitst a positive Radon measure ρ on the compactified half line [0, ∞] such that (Here the function k : t → (1 + t −1 )K 2 (t, f ) 2 is defined by continuity at the points t = 0 and t = ∞, i.e. k(0) = f 2  1 and k(∞) = f 2 0 , where we have used the convention: Remark 1.2 (Importance of Donoghue's Lemma).In ([1], Theorem 2.4), we obtained the finite-dimensional version of Donoghue's Theorem 1.However, in our proof in ([1], Section 7), we mistakenly left out a reference to the beautiful 'Donoghue's Lemma' ( [5], Lemma 1, or part (1) of Lemma 1.1 below).Indeed, in order to obtain ([1], Theorem 2.4) from ([1], Theorem 7.1), it is necessary to refer to that lemma at the end of Section 7.However, in this article, we shall give a proof the general case of ([1], Theorem 2.4) depending only on ([1], Theorem 7.1), which is correctly proved in [1], i.e., the following proof is not affected by our previously neglected reference.
Proof of Donoghue's Theorem 1.The sufficiency is straightforward and left to the reader.We, therefore, turn to the necessity.
Following Donoghue, we start with the following lemma.(2) Moreover, h can be modified on a null-set for a scalar-valued spectral measure ‡ of A to a quasi-concave function:

In particular, this version of h is locally Lipschitzian on
Proof.(1): We first note that, by (1.3), (ExInt) can be rephrased in the following way we infer that (1.6) yields that B belongs to the bicommutator algebra of the von-Neumann algebra generated by the spectral projections of A. The result now follows from the bicommutant theorem.
(2): We shall show that the function h from part (1) can be modified on a null set to satisfy (1.5).To simplify the problem, we note that it suffices for every compact subset K ⊂ R + ∩ σ(A) to show that the restriction h|K is quasi-concave on K. Now fix K and let , where E is the spectral measure of A, and the norms are defined by restriction, It is plain that the operator A K corresponding to K is the compression of A to E K (H 0 ), and likewise, the operator B K corresponding to K * is the compression of B to E K (H 0 ).Moreover, K * is exact interpolation with respect to K, and it is easy to see that B K = (h|K)(A K ).By these considerations, if necessary by replacing H by K, we can w.l.o.g.assume that the operator A is bounded above and below.By (1.2), then also B is bounded above and below.
the essential inf and sup being taken with respect to a scalar-valued spectral measure of A. Take ε > 0 and a Borel set E 0 ⊂ σ(A) ∩ (λ 0 − ε, λ 0 + ε) supporting a unit vector e 0 (for the norm of H 0 ) such that h(λ) > M ε (λ)−ε on E 0 .Likewise, there exists another Borel set Let M be the reducing subspace for A corresponding to the spectral set E 0 ∪ E 1 and consider the map P defined on M by P f = (f, e 1 ) 0 e 0 .Then Extending P to H 0 by P = 0 on the orthogonal complement of M, and , we obtain the bound P A ≤ 1 + 4ε/c.
Since ε > 0 was arbitrary, it follows that As ε approaches 0, the functions M ε (λ) diminish monotonically, converging uniformly to a function h * (λ) which is also the uniform limit of the increasing family m ε (λ).Clearly, h * is continuous on σ(A), and by (1.7), it is increasing.Moreover, since m ε (λ) ≤ h * (λ) ≤ M ε (λ) holds almost everywhere with respect to the spectral measure of A, h * is equivalent to h for that measure.To see that h * is quasi-concave, we make use of the fact that the space H * is exact interpolation with respect to the reversed couple (H 1 , H 0 ).The latter couple has corresponding operator A −1 , and we have the relation Thus the first part of the proof can be applied to show that the function h r * is increasing on σ(A) −1 .Hence h * is quasi-concave on σ(A).
To obtain an operator-theoretic interpretation of our problem we will need another lemma: Lemma 1.2.For a space H * to be of the form (1.4), it is necessary and sufficient that B = h(A) where h is of the form with ρ is some positive Radon measure on [0, ∞] ("a.e." denotes a.e. with respect to a scalar-valued spectral measure of A).
Proof.Let E denote the spectral measure of A. Using a change of the order of integration and the formula (1.1) for K 2 , we infer that H * is of the form (1.4) if and only if By Lemma 1.2, our problem reduces to showing that a function h corresponding to an exact interpolation space H * is necessarily of the form (1.8).We will need some facts about the convex cone functions of the form (1.8), which is henceforth denoted by the letter P (cf.[2]).Proof.Let h n be a sequence of P -functions converging pointwise on S. By (1.8) the boundedness of the numbers h n (λ)(1 + λ −1 ) imply the boundedness of the total masses of the corresponding measures ρ n , whence a subsequence of ρ n converges weak* to a positive Radon measure ρ on [0, ∞], by Helly's theorem.Putting infer that the corresponding subsequence of the h n converges pointwise on R + to h, proving (1).To prove (2), let Λ n be an increasing sequence of finite subsets of σ(A), whose union is dense.Put The following lemma gives a useful characterization of the set P | Λ.
Lemma 1.4.For a function h to belong to P | Λ it is necessary and sufficient that the space n 2 (h(λ)) is exact interpolation with respect to ( n 2 , n 2 (λ)).Proof.This is Theorem 7.1 of [1].
Recall that H * is exact interpolaton with respect to H and B = h(A) where h is quasi-concave on σ(A).Multiplying h by a positive number, we can assume that its Lipshitz constant is 1.By the above lemmata, Donoghue's Theorem 1 reduces to showing that n 2 (h(λ)) is exact interpolation with respect to ( n 2 , n 2 (λ)) for arbitrary sequences λ = (λ i ) n i=1 ⊂ σ(A).This is done below.Take ε > 0 and let , where ε < 2λ 1 is sufficiently small that the E i 's be disjoint.Let M be the reducing subspace of H 0 corresponding to the spectral set ∪ n i=1 E i , and let A be the compression of Proof.By definition, T A is the smallest number C ≥ 0 such that We can find c > 0 such that A, g( A), h( A) and h(g( A)) are ≥ c.Hence, repeated use of Lemma 1.5 and (1.9) yield Let e i be a unit vector in M supported by the spectral set E i and define a subspace V ⊂ M as the space spanned by the e i 's, i = 1, . . ., n.Let A • be the compression of g( A) to V; then Identifying V with n 2 and A • with the matrix diag(λ i ), we infer that (1.10) is independent of ε.Letting ε diminish to 0 now gives that n 2 (h(λ)) is exact interpolation with respect to ( n 2 , n 2 (λ)), which finishes the proof of Donoghue's Theorem 1.
Remark 1.3.For a proof of the non-separable case, from the separable case as obtained above, we refer to the remarks of Donoghue, ( [5] on bottom of p. 256 and top of p. 257, c.f. Section 3) to translate between the articles.

Donoghue's Theorem 2
We shall now draw attention to the second theorem of Donoghue, ([5], Theorem 2), in which the quadratic interpolation methods are characterized.These methods were defined by Donoghue to be those exact interpolation methods F which have the property that for each regular Hilbert couple H, the space F (H) is a Hilbert space.A direct consequence of Donoghue's Theorem 1, is that to a quadratic method F and a regular Hilbert couple H there corresponds a positive Radon measure ρ such that A priori ρ may depend not only on F but also on the particular H.That this is not the case is the content of Donoghue's second theorem.(For a given ρ we define K 2 (ρ) be the quadratic interpolation method F defined by (2.1).)Donoghue's Theorem 2. The assignment ρ → K 2 (ρ) yields a one-one correspondence from the set of positive Radon measures on [0, ∞] onto the set of quadratic interpolation methods.

Connection to Donoghue's versions
In this section, we explain how the results of the previous section are related to Donoghue's original versions of the theorems in [5].Consider a regular Hilbert couple H.In Donoghue's setting, the principal object is the space Δ = H 0 ∩ H 1 with norm defined by In the following, all involutions are understood as being taken with respect to the norm of Δ, and we express the inner products of the spaces H i in the following way: Here The relations between H, K and the operators A, B of the previous section can then be expressed in the following way Since H * fulfills (ExInt) iff B = h(A) where h ∈ P , we obtain the following: and thus The norm in the space J 2 (ν)(H) is defined by We have the following version of the Foiaş-Lions theorem.
Theorem 4.1.Let h be defined by Moreover, every quadratic interpolation method is representable in this form ( for some positive Radon measure ν on [0, ∞]).Remark 4.6.A nice feature of the Foiaş-Lions proof is that it yields an explicit minimizer u of (4.2): We have the following proposition.
Proposition 4.1.For an intermediate Hilbert space H * to be exact interpolation with respect to a regular Hilbert couple H, it is necessary and sufficient that there exist a positive Radon measure ν such that H * = J 2 (ν)(H) (isometrically).
Proof.We use the fact that the set of functions of the form (4.3) coincides with the cone P (use Löwner's theorem (cf.[1]), stating that P equals the set of matrix monotone functions on R + and the simple fact that a function h is matrix monotone iff the function h(λ −1 ) −1 is matrix monotone).The proposition is now immediate from Donoghue's Theorem 1, Lemma 1.2 and Theorem 4.1.
Thus every exact interpolation Hilbert space H * can be described isometrically both as H * = K 2 (ρ)(H) and H * = J 2 (ν)(H) for some positive Radon measures ρ and ν.In general, there does not seem to exist any simple relation between these measures.However, we shall now consider two particular cases where such a relation exists.
Example 4.2.Given a Hilbert couple H, we have the following definition (c.f.[8]) of the complex interpolation spaces H [θ] , 0 < θ < 1, It can be shown that the definition (4.5) is consistent (with equality of norms) with the standard definition of the complex interpolation spaces as stated in, e.g., [3].Now (4.4) and (4.5) yield i.e., the complex θ-method is (on the category of Hilbert couples) proportional to the K θ -method.

Lemma 1 . 3 . ( 1 )( 2 )
Let S be a nonvoid subset of R + .Then the convex cone P | S of restrictions of P -functions to S is closed under pointwise convergence.In particular, a continuous function h belongs to the set P | σ(A) ∩ R + iff the restriction h | Λ ∈ P | Λ for every finite subset Λ ⊂ σ(A) ∩ R + .
and the regularity of H yields that H has no eigenvalues in the set {0, 1}.Likewise, if H * is an intermediate Hilbert space, we have f 2 * = (Kf, f ) Δ for some positive operator K ∈ B(Δ).It is easy to see that H * fulfills (ExInt) iff t, u(t); H) 2 dt t (over u such that f =