Interpolation Between Hilbert Spaces 3

arXiv:1401.6090v5 [math.FA] 29 Jun 2019 When esalso e that see soon shall We intermediate both 1.1. If L T norm saBnc pc,we qipdwt h norm the with (1.1) equipped when space, Banach a is or symbols saHletsaeudrtenorm the under space Hilbert a is The uhta h quantity the that such H ( 0 to k X osdrapi fHletspaces Hilbert of pair a Consider edfiethe define We 1 aahspace Banach A map A T ∩H oepeiey hsi the is this precisely, More T ; sum K neplto norms. Interpolation H : H Y k → H X i 1 ( L i ) maps ,x t, Abstract. eut noeao-hoyadi function-theory. relat in and their operator-theory and in spaces results particular interpolation In of spaces. characterizations Hilbert between interpolation exact ( sdnein dense is odnt h oaiyo one iermaps linear bounded of totality the denote to r meddi oecmo asofftplgclvco spa vector topological Hausdorff common some in embedded are = ftespaces the of H T NEPLTO EWE IBR SPACES HILBERT BETWEEN INTERPOLATION = ) ; Y K Σ( : K ihrsett h pair the to respect with H ) odnt that denote to esml write simply we K ≤ i H onel into boundedly K 1 ,x t, ) hsnt opie ytei fcranrslsi h the the in results certain of synthesis a comprises note This -functional H eseko a of speak we → 1. k X 0 T k ; H swl sin as well as Σ( Σ neplto hoei notions theoretic Interpolation H T uhthat such k k 0 L saHletsae(e em .) The 1.1). Lemma (see space Hilbert a is k x K udai version quadratic and ( L X inf = k ) ( Σ 2 ; H T Σ( = Σ Y scle a called is ∆( = ∆ When x H L = ( ) ; := sacul a.I ses ocekthat check to easy is It map. couple a is 1 K K 1 x ( sup = o h couple the for ) X 0 ) AI AMEUR YACIN i contraction sdfie ob h pc ossigo all of consisting space the be to defined is + ∆ k K =max := for ) H H x x . X 1 H (1 1 ⊂ k { k H H ( = j easm httepi is pair the that assume We . i , {k ∆ =0 2 x , = ) . opemap couple 0 = = ) Y X 1 x := H , x T ftecasclPeetre classical the of ) 0 1 k { r omdsae,w s h symbol the use we spaces, normed are 0 H k , sfiie ednt hssaeb the by space this denote we finite; is ⊂ H k , 1 0 k 2 0 H euetenotations the use We . x Y T 0 from + Σ + 1 k H ∩ ; k ) 0 t 2 H H k L cniuu nlsos scalled is inclusions) (continuous hc is which k + ( x 1 from x H 1 H by . k k 1 j T X k to x ; K 1 2 : k ≤ eeaievarious examine we , j ost ubrof number a to ions H K } 1 X 2 ) > t , . regular 1 . } to } → . K . K -functional. Y ftersrcinof restriction the if ihteoperator the with ntesnethat sense the in intersection compatible 0 T x , r of ory L ∈ M ∈ L ce ( ( x H H M . M ∈ i.e., , ; ; . K K ) ) 2 YACIN AMEUR

Let X, Y be intermediate spaces with respect to couples , , respectively. We say that X, Y are (relative) interpolation spaces if there is aH constantK C such that T : implies that T : X Y and H → K → (1.2) T C T . k k L(X;Y ) ≤ k k L(H;K) In the case when C = 1 we speak about exact interpolation. When = and X = Y we simply say that X is an (exact) interpolation space with respectH K to . H Let H be a suitable function of two positive variables and X, Y spaces intermediate to the couples , , respectively. We say that the spaces X, Y are of type H (relative to , ) ifH forK any positive numbers M , M we have H K 0 1 (1.3) T Mi, i =0, 1 implies T H(M0,M1). k k L(Hi;Ki) ≤ k k L(X;Y ) ≤ The case H(x, y) = max x, y corresponds to exact interpolation, while H(x, y)= x 1−θy θ corresponds to the{ convexity} estimate

(1.4) T T 1−θ T θ . k k L(X;Y ) ≤k k L(H0;K0) k k L(H1;K1) In the situation of (1.4), one says that the interpolation spaces X, Y are of exponent θ with respect to the pairs , . H K 1.2. K-spaces. Given a regular Hilbert couple and a positive Radon measure ̺ on the compactified half-line [0, ] we define anH intermediate quadratic norm by ∞ (1.5) x 2 = x 2 = 1+ t−1 K t, x; d̺(t). k k∗ k k̺ H Z[0,∞] Here the integrand k(t) = 1+ t−1 K(t, x) is defined at the points 0 and by ∞ k(0) = x 2 and k( ) = x 2; we shall write or for the Hilbert space 1 0 ∗ ̺ definedk byk the norm (1.5).∞ k k H H Let T ; and suppose that T Mi; then ∈ L H K k k L(Hi;Ki) ≤ (1.6) K t,Tx ; M 2 K M 2t/M 2, x; , x Σ. K ≤ 0 1 0 H ∈ In particular, Mi 1 for i= 0, 1 implies T x x for all x ̺. It ≤ k k K̺ ≤ k k H̺ ∈ H follows that the spaces , are exact interpolation spaces with respect to , . H̺ K̺ H K Geometric interpolation. When the measure ̺ is given by t−θ π d̺(t)= c dt, c = , 0 <θ< 1, θ 1+ t θ sin θπ we denote the norm (1.5) by ∞ dt (1.7) x 2 := c t−θK (t, x) . k kθ θ t Z0 The corresponding space is easily seen to be of exponent θ with respect to . Hθ H In §3.1, we will recognize θ as the geometric interpolation space which has been studied independently by severalH authors, see [27, 40, 25]. INTERPOLATION BETWEEN HILBERT SPACES 3

2 1.3. Pick functions. Let be a regular Hilbert couple. The squared norm x 1 is a densely defined quadraticH form in , which we represent as k k H0 x 2 = Ax , x = A 1/2x 2 k k1 h i0 k k0 where A is a densely defined, positive, injective (perhaps unbounded) operator in . The domain of the positive square-root A1/2 is ∆. H0 Lemma 1.1. We have in terms of the functional calculus in H0 tA (1.8) K (t, x )= x , x , t> 0. 1+ tA 0 tA In the formula (1.8), we have identified the bounded operator 1+tA with its extension to . H0 Proof. Fix x ∆. By a straightforward convexity argument, there is a unique ∈ decomposition x = x0,t + x1,t which is optimal in the sense that

(1.9) K(t, x)= x 2 + t x 2 . k 0,t k0 k 1,t k1 It follows that x ∆ for i =0, 1. Moreover, for all y ∆ we have i,t ∈ ∈ d x + ǫy 2 + t x ǫy 2 =0, dǫ{k 0,t k0 k 1,t − k1 }|ǫ=0 i.e., A−1/2x tA1/2x , A1/2y =0, y ∆. h 0,t − 1,t i0 ∈ −1/2 1/2 By regularity, we conclude that A x0,t = tA x1,t, whence tA 1 (1.10) x = x and x = x. 0,t 1+ tA 1,t 1+ tA

(Note that the operators in (1.10) extend to bounded operators on 0.) Inserting the relations (1.10) into (1.9), one ﬁnishes the proof of the lemma. H

Now ﬁx a positive Radon measure ̺ on [0, ]. The norm in the space ̺ (see (1.5)) can be written ∞ H

(1.11) x 2 = h(A)x , x , k k̺ h i0 where (1 + t)λ (1.12) h(λ)= d̺(t). 1+ tλ Z[0,∞] The class of functions representable in this form for some positive Radon measure ′ ̺ is the class P of Pick functions, positive and regular on R+. Notice that for the deﬁnition (1.11) to make sense, we just need h to be deﬁned on σ(A) 0 , where σ(A) is the spectrum of A. (The value h(0) is irrelevant since A is injective).\{ } A calculus exercise shows that for the space (see (1.7)) we have Hθ (1.13) x 2 = Aθx , x . k kθ h i0 4 YACIN AMEUR

1.4. Quadratic interpolation norms. Let ∗ be any quadratic intermediate space relative to . We write H H x 2 = Bx, x k k∗ h i0 where B is a positive injective operator in (the domain of B1/2 is ∆). H0 For a map T ( ) we shall often use the simpliﬁed notations ∈ L H T = T , T = T , T = T . k k k k L(H0) k kA k k L(H1) k kB k k L(H∗) The reader can check the identities T = A1/2T A−1/2 and T = B1/2TB−1/2 . k kA k k k kB k k We shall refer to the following lemma as Donoghue’s lemma, cf. [14, Lemma 1].

Lemma 1.2. If ∗ is exact interpolation with respect to , then B commutes with every projectionH which commutes with A and B = h(A)Hwhere h is some positive Borel function on σ(A). Proof. For an orthogonal projection E on , the condition E 1 is equivalent H0 k kA ≤ to that EAE A, i.e., that E commutes with A. The hypothesis that ∗ be exact interpolation thus≤ implies that every spectral projection of A commutesH with B. It now follows from von Neumann’s bicommutator theorem that B = h(A) for some positive Borel function h on σ(A).

In view of the lemma, the characterization of the exact quadratic interpolation norms of a given type H reduces to the characterization of functions h : σ(A) R → + such that for all T ∈ L H (1.14) T M and T M T H(M ,M ), k k≤ 0 k kA ≤ 1 ⇒ k kh(A) ≤ 0 1 or alternatively, (1.15) T ∗T M 2 and T ∗AT M 2 A T ∗h(A)T H(M ,M ) 2 h(A). ≤ 0 ≤ 1 ⇒ ≤ 0 1 The set of functions h : σ(A) R satisfying these equivalent conditions forms a → + convex cone CH,A; its elements are called interpolation functions of type H relative to A. In the case when H(x, y) = max x, y we simply write CA for CH,A and speak of exact interpolation functions relative{ } to A.

1.5. Exact Calderón pairs and the K-property. Given two intermediate normed spaces Y , X relative to , , we say that they are (relatively) exact K-monotonic if the conditions H K x0 X and K t,y0; K t, x0; , t> 0 ∈ H ≤ K imply that y0 Y and y0 x0 . ∈ k kY ≤k kX It is easy to see that exact K-monotonicity implies exact interpolation.

Proof of this. If T 1 then x, t: K t,Tx; K t, x; whence k k L(K;H) ≤ ∀ H ≤ K T x x , by exact K-monotonicity. Hence T 1. k kY ≤k kX k k L(X;Y ) ≤ INTERPOLATION BETWEEN HILBERT SPACES 5

Two pairs , are called exact relative Calderón pairs if any two exact interpolation (Banach-)H spacesK Y , X are exact K-monotonic. Thus, with respect to to exact Calderón pairs, exact interpolation is equivalent to exact K-monotonicity. The term "Calderón pair" was coined after thorough investigation of A. P. Calderón’s study of the pair (L1,L∞), see [10] and [11]. In our present discussion, it is not convenient to work directly with the definition of exact Calderón pairs. Instead, we shall use the following, closely related notion. We say that a pair of couples , has the relative (exact) K-property if for all H K x0 Σ( ) and y0 Σ( ) such that ∈ K ∈ H (1.16) K t,y0; K t, x0; , t> 0, H ≤ K there exists a map T ( ; ) such that T x0 =y0 and T 1. ∈ L K H k k L(K;H) ≤ Lemma 1.3. If , have the relative K-property , then they are exact relative Calderón pairs. H K Proof. Let Y , X be exact interpolation spaces relative to , and take x0 X and y0 Σ( ) such that (1.16) holds. By the K-property thereH K is T : ∈such that T x∈0 = yH0 and T 1. Then T 1, and so y0 =K →T xH0 k k≤ k k L(X;Y ) ≤ k kY k kY ≤ x0 . We have shown that Y , X are exact K-monotonic. k kX In the diagonal case = , we simply say that is an exact Calderón couple if for intermediate spaces HY,X,K the property of beingH exact interpolation is equivalent to being exact K-monotonic. Likewise, we say that has the K-property if the pair of couples , has that property. H H H Remark 1.4. For an operator T : to be a contraction, it is necessary and sufficient that K → H (1.17) K t,Tx; K t, x; , x Σ( ),t> 0. H ≤ K ∈ K Indeed, the necessity is immediate. To prove the sufficiency it suffices to observe that letting t in (1.17) gives T x 0 x 0, and dividing (1.17) by t, and then letting t →0 ∞, gives that T x k kx ≤. k k → k k1 ≤k k1 2. Mapping properties of Hilbert couples 2.1. Main results. We shall elaborate on the following main result from [2]. Theorem I. Any pair of regular Hilbert couples , has the relative K-property . H K Before we come to the proof of Theorem I, we note some consequences of it. We first have the following corollary, which shows that a strong form of the K-property is true. Corollary 2.1. Let be a regular Hilbert couple and x0,y0 Σ elements such that H ∈ (2.1) K t,y0 M 2 K M 2t/M 2 , x0 , t> 0. ≤ 0 1 0 Then 0 0 (i) There exists a map T such that T x = y and T Mi, ∈ L H k k L(Hi) ≤ i =0, 1. (ii) If x0 X where X is an interpolation space of type H, then ∈ y0 H (M ,M ) x0 . k kX ≤ 0 1 k kX 6 YACIN AMEUR

Proof. (i) Introduce a new couple by letting x = Mi x H . The relation K k kKi k k i (2.1) then says that K t,y0; K t, x0; , t> 0. H ≤ K By Theorem I there is a contraction T : such that T x0 = y0. It now suffices K → H to note that T = Mi T ; (ii) then follows from Lemma 1.3. k k L(Hi) k k L(Ki;Hi) We next mention some equivalent versions of Theorem I, which uses the families of functionals K and E defined (for p 1 and t,s > 0) via p p ≥ p p Kp(t)= Kp(t, x)= Kp t, x; = inf x0 0 + t x1 1 H x=x0+x1 {k k k k } (2.2) p Ep(s)= Ep(s, x)= Ep s, x; = infp x x0 1 . H k x0 k0 ≤s {k − k } 1/p p Note that K = K2 and that Ep(s) = E1 s ; the E-functionals are used in approximation theory. One has that E is decreasing and convex on R and that p + Kp(t) = inf s + tEp(s) , s>0 { } which means that Kp is a kind of Legendre transform of Ep. The inverse Legendre transformation takes the form K (t) s E (s) = sup p . p t − t t>0 It is now immediate that, for all x Σ and y Σ , we have ∈ K ∈ H (2.3) K (t,y) K (t, x), t> 0 E (s,y) E (s, x), s> 0. p ≤ p ⇔ p ≤ p 2/p p/2 Since moreover Ep(s)= E2 s , the conditions in (2.3) are equivalent to that K(t,y) K(t, x) for all t> 0. We have shown the following result. ≤ Corollary 2.2. In Theorem I, one can substitute the K-functional for any of the functionals Kp or Ep. Define an exact interpolation norm relative to by k·k̺,p H x p = 1+ t−1 K (t, x) d̺(t) k k̺,p p Z[0,∞] where ̺ is a positive Radon measure on [0, ]. This norm is non-quadratic when p =2, but is of course equivalent to the quadratic∞ norm corresponding to p =2. 6 2.2. Reduction to the diagonal case. It is not hard to reduce the discussion of Theorem I to a diagonal situation. Lemma 2.3. If the K-property holds for regular Hilbert couples in the diagonal case = , then it holds in general. H K Proof. Fix elements y0 Σ( ) and x0 Σ( ) such that the inequality (1.16) ∈ H ∈ K holds. We must construct a map T : such that T x0 = y0 and T 1. K → H k k≤ To do this, we form the direct sum = ( 0 0, 1 1). It is clear that + = ( + ) ( + ), and thatS H ⊕ K H ⊕ K S0 S1 H0 H1 ⊕ K0 K1 K t, x y; = K t, x; + K t,y; . ⊕ S H K Then K t, 0 y0; K t, x0 0; . ⊕ S ≤ ⊕ S INTERPOLATION BETWEEN HILBERT SPACES 7

Hence assuming that the couple has the K-property, we can assert the existence of a map S ( ) such that SS(x0 0) = 0 y0 and S 1. Letting P : ∈ L S ⊕ ⊕ k k ≤ 0 + 1 0 + 1 be the orthogonal projection, the assignment T x = PS(x 0) Snow definesS → K a mapK such that T x0 = y0 and T 1. ⊕ k k L(H;K) ≤ 2.3. The principal case. The core content of Theorem I is contained in the following statement. Theorem 2.4. Suppose that a regular Hilbert couple is finite dimensional and that all eigenvalues of the corresponding operator A areH of unit multiplicity. Then has the K-property . H We shall settle for proving Lemma 2.4 in this section, postponing to Section 5 the general case of Theorem I. To prepare for the proof, we write the eigenvalues λi of A in increasing order, σ(A)= λ n where 0 < λ < < λ . { i}1 1 ··· n Let e be corresponding eigenvectors of unit length for the norm of . Then for a i H0 vector x = xiei we have n n P x 2 = x 2 , x 2 = λ x 2. k k0 | i| k k1 i| i| 1 1 X X Working in the coordinate system (e ), the couple becomes identified with the i H n-dimensional weighted ℓ2 couple n n n ℓ2 (λ) := (ℓ2 ,ℓ2 (λ)) , n where we write λ for the sequence (λi)1 . n C n We will henceforth identify a vector x = xiei with the point x = (xi)1 in ; n ∗ C accordingly, the space (ℓ2 ) is identified with the C -algebra Mn( ) of complex n n matrices. L P × n It will be convenient to reparametrize the K-functional for the couple ℓ2 (λ) and write n (2.4) kλ(t, x) := K 1/t,x; ℓ2 (λ) . By Lemma 1.1 we have n λ (2.5) k (t, x)= i x 2, x C n. λ t + λ | i| ∈ i=1 i X n 2.4. Basic reductions. To prove that the couple ℓ2 (λ) has the K-property, we introduce an auxiliary parameter ρ> 1. The exact value of ρ will change meaning during the course of the argument, the main point being that it can be chosen arbitrarily close to 1. Initially, we pick any ρ > 1 such that ρλi < λi+1 for all i; we assume also that we are given two elements x0,y0 C n such that ∈ 1 (2.6) k t,y0 < k t, x0 , t 0. λ ρ λ ≥ We must construct a matrix T M (C)such that ∈ n (2.7) T x0 = y0 and k (t,Tx) k (t, x) , x C n,t> 0. λ ≤ λ ∈ 8 YACIN AMEUR

Deﬁne x˜0 = ( x0 )n and y˜0 = ( y0 )n and suppose that | i | 1 | i | 1 1 k (t, y˜0) < k (t, x˜0), t 0. λ ρ λ ≥ 0 0 Suppose that we can ﬁnd an operator T0 Mn(C) such that T0x˜ =y ˜ and C n ∈ 0 iθk 0 0 iϕk 0 kλ (t,T0x) < kλ(t, x) for all x and t> 0. Writing xk = e x˜k and yk = e y˜k where θ , ϕ R, we then have∈ T x0 = y0 and k (t,Tx) < k (t, x) where k k ∈ λ λ iϕk −iθk T = diag(e )T0 diag(e ). 0 0 0 0 0 0 Replacing x ,y by x˜ , y˜ we can thus assume that the coordinates xi and yi are non-negative; replacing them by small perturbations if necessary, we can assume that they are strictly positive, at the expense of slightly diminishing the number ρ.

Now put βi = λi and αi = ρλi. Our assumption on ρ means that 0 <β < α < <β < α . 1 1 ··· n n Using the explicit expression for the K-functional, it is plain to check that k (t, x) k (t, x) ρk (t, x), x Cn, t 0. β ≤ α ≤ β ∈ ≥ Our assumption (2.6) therefore implies that (2.8) k (t,y0) < k (t, x0), t 0. α β ≥ We shall verify the existence of a matrix T = Tρ = Tρ,x0,y0 such that (2.9) T x0 = y0 and k (t,Tx) k (t, x) , x Cn,t> 0. α ≤ β ∈ It is clear by compactness that, as ρ 1, the corresponding matrices Tρ will cluster at some point T satisfying T x0 = y0↓and T 1. (See Remark 1.4.) k k L( H ) ≤ In conclusion, the proof of Theorem 2.4 will be complete when we can construct a matrix T satisfying (2.9) with ρ arbitrarily close to 1.

2.5. Construction of T . Let k denote the linear space of complex polynomials of degree at most k. We shall useP the polynomials n n Lα(t)= (t + αi) ,Lβ(t)= (t + βi) , 1 1 Y Y and the product L = LαLβ. Notice that (2.10) L′( α ) < 0 ,L′( β ) > 0. − i − i Recalling the formula (2.5), it is clear that we can deﬁne a real polynomial P by ∈ P2n−1 P (t) (2.11) = k t, x0 k t,y0 . L(t) β − α Clearly P (t) > 0 when t 0. Moreover, a consideration of the residues at the poles of the right-hand member≥ shows that P is uniquely deﬁned by the values (2.12) P ( β ) = (x0) 2β L′ ( β ) , P ( α )= (y0) 2α L′ ( α ) . − i i i − i − i − i i − i Combining with (2.10), we conclude that (2.13) P ( α ) > 0 and P ( β ) > 0. − i − i INTERPOLATION BETWEEN HILBERT SPACES 9

Perturbing the problem slightly, it is clear that we can assume that P has exact degree 2n 1, and that all zeros of P have multiplicity 1. (We here diminish the value of ρ>− 1 somewhat, if necessary.) Now, P has 2n 1 simple zeros, which we split according to − P −1 ( 0 )= r 2m−1 c , c¯ n−m , { } {− i}i=1 ∪ {− i − i}i=1 where the ri are positive and the ci are non-real, and chosen to have positive imaginary parts. The following is the key observation. Lemma 2.5. We have that (2.14) L′ ( β ) P ( β ) > 0 ,L′ ( α ) P ( α ) < 0 − i − i − i − i and there is a splitting r 2m−1 = δ m γ m−1 such that { i}i=1 { i}i=1 ∪{ i}i=1 (2.15) L ( δ ) P ′ ( δ ) > 0 ,L ( γ ) P ′ ( γ ) < 0. − j − j − k − k Proof. The inequalities (2.14) follow immediately from (2.13) and (2.10). It remains to prove (2.15). Let h denote the leftmost real zero of the polynomial LP (of degree 4n 1). We claim− that P ( h)=0. If this were not the case, we would have h = α . Since− − n the degree of P is odd, P ( t) is negative for large values of t, and so P ( αn) < 0 contradicting (2.13). We have− shown that P ( h)=0. Since all zeros of−LP have multiplicity 1, we have (LP )′( h) =0, whence− − 6 L( h)P ′( h) = (LP )′( h) > 0. − − − We write δm = h and put P∗(t) = P (t)/(t + δm). Since t + δm > 0 for t α , β n, we have by (2.13) that for all i ∈ {− i − i}1 P ( α ) > 0 and P ( β ) > 0. ∗ − i ∗ − i Denote by r ∗ 2m−2 the real zeros of P . Since the degree of LP is even and {− j }j=1 ∗ ∗ the polynomial (LP )′ has alternating signs in the set α , β n r ∗ 2m−2, ∗ {− i − i}i=1 ∪{− i }i=1 we can split the zeros of P as δ , γ m−1, where ∗ {− i − i}i=1 (2.16) L( δ )P ′ ( δ ) > 0 ,L( γ )P ′ ( γ ) < 0. − i ∗ − i − i ∗ − i ′ ∗ ∗ ′ ∗ ∗ ′ ∗ Since P ( rj ) = (δm rj )P∗( rj ) and δm > rj , the signs of P ( rj ) and P ′( r ∗) −are equal, proving− (2.15).− − ∗ − j n−m Recall that ci 1 denote the zeros of P such that Im ci > 0. We put (with the convention{− that} an empty product equals 1)

m m−1 n−m Lδ(t)= (t + δi) ,Lγ(t)= (t + γi) ,Lc(t)= (t + ci). i=1 i=1 i=1 Y Y Y We define a linear map F : Cn+m Cn+m−1 in the following way. First define a subspace U by → ⊂P2n−1 U = L q ; q . { c ∈Pn+m−1 } Notice that U has dimension n + m 1 and that P U; in fact P = aL L ∗L L − ∈ c c δ γ where a is the leading coefficient and the -operation is defined by L ∗(z)= L(¯z). ∗ 10 YACIN AMEUR

For a polynomial Q U we have ∈ n n Q(t) 2 β δ | | = x 2 i + x′ 2 i L(t)P (t) | i| t + β | i| t + δ i=1 i i=1 i X X (2.17) n m−1 α γ y 2 i y′ 2 i , − | i| t + α − | i| t + γ i=1 i i=1 i X X where, for deﬁniteness,

Q( βi) ′ Q( δj ) (2.18) xi = − ; x = − ′ j ′ β L ( β )P ( β ) δj L ( δj)P ( δj ) i − i − i − − Q( αi) ′ Q( γj) (2.19) yi = p − ; y = p − . ′ j ′ α L ( α )P ( α ) γj L ( γj )P ( γj ) − i − i − i − − − The identitiesp in (2.18) give rise to a linear mapp (2.20) M : C n C m U ;[x; x′] Q. ⊕ → 7→ We can similarly regard (2.19) as a linear map

(2.21) N : U C n C m−1 ; Q [y; y′] . → ⊕ 7→ Our desired map F is deﬁned as the composite F = NM : C n C m C n C m−1 ;[x; x′] [y; y′]. ⊕ → ⊕ 7→ Notice that if Q = M [x; x′] and [y; y′]= F [x; x′] then (2.17) means that

Q(t) 2 k (t, [x; x′]) k (t, F [x; x′]) = | | 0, t 0. β⊕δ − α⊕γ L(t)P (t) ≥ ≥

n+m n+m−1 This implies that F is a contraction from ℓ2 (β δ) to ℓ2 (α γ). We now define T as a "compression" of F . Namely,⊕ let E : C n ⊕C m−1 C n be the projection onto the first n coordinates, and define an operator⊕ T on C→n by T x = EF [x; 0] , x C n. ∈ Taking Q = P in (2.17) we see that T x0 = y0. Moreover,

n n β α k (t, x) k (t,Tx)= x 2 i y 2 i β − α | i| t + β − | i| t + α i=1 i i=1 i X X n n m−1 β α γ x 2 i y 2 i y′ 2 i ≥ | i| t + β − | i| t + α − | i| t + γ i=1 i i=1 i j=1 i X X X Q(t) 2 = k (t, [x; 0]) k (t, F [x; 0]) = | | . β⊕δ − α⊕γ L(t)P (t) Since the right-hand side is non-negative, we have shown that k (t,Tx) k (t, x), t> 0, x Cn, α ≤ β ∈ as desired. The proof of Theorem 2.4 is ﬁnished. q.e.d. INTERPOLATION BETWEEN HILBERT SPACES 11

2.6. Real scalars. Theorem 2.4 holds also in the case of Euclidean spaces over the real scalar field. To see this, assume without loss of generality that the vectors x0,y0 Cn have real entries (still satisfying k t,y0 k t, x0 for all t> 0). ∈ λ ≤ λ By Theorem 2.4 we can find a (complex) contraction T of ℓn(λ) such that T x0 = 2 y0. It is clear that the operator T ∗ defined by T ∗x = T (¯x) satisfies those same 1 ∗ R conditions. Replacing T by 2 (T + T ) we obtain a real matrix T Mn( ), which n 0 0 ∈ is a contraction of ℓ2 (λ) and maps x to y . 2.7. Explicit representations. We here deduce an explicit representation for the operator T constructed above. Let x0 and y0 be two non-negative vectors such that k t,y0 k t, x0 , t> 0. λ ≤ λ For small ρ > 0 we perturb x0, y0 slightly to vectors x˜ 0, y˜0 which satisfy the conditions imposed the previous subsections. We can then construct a matrix T = Tρ such that (2.22) T x˜ 0 =y ˜ 0 and k (t,Tx) k (t, x) , t> 0, x C n, α ≤ β ∈ where β = λ and α = ρλ. As ρ, x˜0, y˜0 approaches 1, x0, resp. y0, it is clear that any cluster point T of the set of contractions Tρ will satisfy T x0 = y0 and k (t,Tx) k (t, x) , t> 0, x C n. λ ≤ λ ∈ n Theorem 2.6. The matrix T = T̺ = (τik)i,k=1 where 1 x˜0 β L ( α )L ( α )L ( β ) (2.23) τ = Re k k δ − i c − i α − k ik α β y˜0 α L ( β )L ( β )L′ ( α ) i − k i i δ − k c − k α − i satisfies (2.22). Proof. The range of the map C n U, x M [x; 0] (see 2.20) is precisely the n-dimensional subspace → 7→ (2.24) V := L L = L L R; R U. δ c ·Pn−1 { δ c ∈Pn−1}⊂ n We introduce a basis (Qk)k=1 for V by L (t)L (t)L (t) β L′( β )P ( β ) Q (t)= δ c β k − k − k . k t + β L ( β )L ( β )L′ ( β ) k δp− k c − k β − k Then Q ( β ) 1 i = k, k − i = ′ βiL ( βi)P ( βi) (0 i = k. − − 6 C n Denoting by (ei) the canonicalp basis in and using (2.18), (2.19) we get

Qk( αi) τik = (Tek)i = − α L′( α )P ( α ) i − i − i ′ 1/2 1 Lpδ( αi)Lc( αi)Lβ( αi) βkL ( βk)P ( βk) = − − − − − . β α L ( β )L ( β )L′ ( β ) α L′( α )P ( α ) k − i δ − k c − k β − k − i − i − i Inserting the expressions (2.12) for P ( αi) and P ( βk) and taking real parts (see the remarks in §2.6), we obtain the formula− (2.23).− 12 YACIN AMEUR

Remark 2.7. It is easy to see that, if we pick all matrix-elements real, some elements 0 0 τik of the matrix T in (2.23) will be negative, even while the numbers xi and yk are positive. It was proved in [2], Theorem 2.3, that this is necessarily so. Indeed, 5 one there constructs an example of a ﬁve-dimensional couple ℓ2 (λ) and two vectors x0,y0 R5 having non-negative entries such that no contraction T = (τ )5 on ∈ ik i,k=1 ℓ 5(λ) having all matrix entries τ 0 can satisfy T x0 = y0. On the other hand, if 2 ik ≥ one settles for using a matrix with T √2, then it is possible to ﬁnd one with only non-negative matrix entries. Indeed,k k≤ such a matrix was used by Sedaev [35], see also [39].

2.8. On sharpness of the norm-bounds. We shall show that if m

Q(1)(t) 2 k (t, x(1)) k (t,y(1))= | | , t> 0. β − α L(t)P (t)

(1) 2 (1) 2 Choosing t =0 we conclude that x n T x n =0, whence T 1, k kℓ2 −k kℓ2 k k L(H0) ≥ proving our claim. Similarly, the condition m

Q(2)(t) 2 k (t, x(2)) k (t,y(2))= | | , t> 0. β − α L(t)P (t)

(2) 2 Multiplying this relation by t and then sending t , we ﬁnd that x n → ∞ k kℓ2 (β) − (2) 2 −1/2 T x n =0, which implies T L H ρ . k kℓ2 (α) k k ( 1) ≥

2.9. A remark on weighted ℓ -couples. As far as we are aware, if 1

n n p tλi Kp t, x; ℓp (λ) = xi 1 . | | p−1 p−1 i=1 (1 + (tλi) ) X 0 n 0 n It was proved by Sedaev [35] (cf. [39]) that if Kp t,y ; ℓp (λ) Kp t, x ; ℓp (λ) ≤ ′ for all t > 0 then there is T : ℓn(λ) ℓn(λ) of norm at most 21/p such that p → p T x0 = y0. (Here p′ is the exponent conjugate to p.) INTERPOLATION BETWEEN HILBERT SPACES 13

Although our present estimates are particular for the case p =2, our construction still shows that, if we re-define P (t) to be the polynomial n n P (t) β α (2.25) = (˜x0) p i (˜y0) p i , L(t) i t + β − i t + α 1 i 1 i X X then the matrix T defined by 1 (˜x0 )p−1 β L ( α )L ( α )L ( β ) (2.26) τ = Re k k δ − i c − i α − k ik α β (˜y0)p−1 α L ( β )L ( β )L′ ( α ) i − k i i δ − k c − k α − i will satisfy T x˜0 =y ˜0, at least, provided that P (t) > 0 when t 0. (Here L and ≥ δ Lc are constructed from the zeros of P as in the case p =2.) The matrix (2.26) differs from those used by Sedaev [35] and Sparr [39]. In- deed the matrices from [35, 39] have non-negative entries, while this is not so for the matrices (2.26). It seems to be an interesting problem to estimate the norm

T n for the matrix (2.26), when p = 2. The motivation for this type of k k L(ℓp (λ)) 6 question is somewhat elaborated in §6.7, but we shall not discuss it further here.

2.10. A comparison with Löwner’s matrix. In this subsection, we briefly ex- plain how our matrix T is related to the matrix used by Löwner [26] in his original work on monotone matrix functions. (2) We shall presently display four kinds of partial isometries; Löwner’s matrix will be recognized as one of them. In all cases, operators with the required properties can alternatively be found using the more general construction in Theorem 2.4. The following discussion was inspired by the earlier work of Sparr [38], who seems to have been the first to note that Löwner’s matrix could be constructed in a similar way. In this subsection, scalars are assumed to be real. In particular, when we write n "ℓ2 " we mean the (real) Euclidean n-dimensional space. Suppose that two vectors x0,y0 Rn satisfy ∈ k t,y0 k t, x0 , t> 0. λ ≤ λ Let n Lλ(t)= (t + λi) , 1 Y and let P be the polynomial fulfilling ∈Pn−1 n P (t) λ = k t, x0 k t,y0 = i (x0) 2 (y0) 2 . L (t) λ − λ t + λ i − i λ i=1 i X By assumption, P (t) 0 for t 0. Moreover, P is uniquely determined by the n conditions ≥ ≥ 0 2 0 2 (xi ) (yi ) P ( λi)= ′ − . − λiLλ( λi) − n Let u1, v1,u2, v2,... denote the canonical basis of ℓ2 and let ℓn = O E 2 ⊕ 2 By "Löwner’s matrix", we mean the unitary matrix denoted "V " in Donoghue’s book [12], on p. 71. A more explicit construction of this matrix is found in [26], where it is called "T ". 14 YACIN AMEUR be the corresponding splitting, i.e., O = span u , E = span v . { i} { i} Notice that dim O = (n 1)/2 +1 , dim E = (n 2)/2 +1, ⌊ − ⌋ ⌊ − ⌋ where x is the integer part of a real number x. We⌊ shall⌋ construct matrices T M (R) such that ∈ n (2.27) T x0 = y0 and k (t,Tx) k (t, x), t> 0, x Rn, λ ≤ λ ∈ in the following special cases: (1) P (t)= q(t)2 where q (R), x0 O, and y0 E, ∈P(n−1)/2 ∈ ∈ (2) P (t)= tq(t)2 where q (R), x0 E, and y0 O. ∈P(n−2)/2 ∈ ∈ Here should be interpreted as . Px P⌊x⌋ Remark 2.8. In this connection, it is interesting to recall the well-known fact that 2 any polynomial P which is non-negative on R+ can be written P (t) = q0(t) + 2 tq1(t) for some real polynomials q0 and q1.

To proceed with the solution, we rename the λi as λi = ξi when i is odd and λi = ηi when i is even. We also write

Lξ(t)= (t + ξi) ,Lη(t)= (t + ηi), odd i even iY Y and write L = L L . Notice that L′ ( ξ ) > 0 and L′ ( η ) < 0. ξ η λ − i λ − i Case 1. Suppose that P (t)= q(t)2, q (R), x0 O, and y0 E. Then ∈P(n−1)/2 ∈ ∈ q(t)2 ξ η = k (x0 ) 2 i (y0) 2, L (t) t + ξ k − t + η i λ odd k i even i kX X where ε q( ξ ) ζ q( η ) (2.28) x0 = k − k , y0 = i − i k ξ L′ ( ξ ) i η L′ ( η ) k λ − k − i λ − i for some choice of signs εpk, ζi 1 . p By (2.28) are deﬁned linear∈ maps {± } O (R): x Q ; (R) E : Q y. →P(n−1)/2 7→ P(n−1)/2 → 7→ The composition is a linear map T : O E : x y. 0 → 7→ We now deﬁne T M (R) by ∈ n T : O E O E :[x; v] [0; T x]. ⊕ → ⊕ 7→ 0 Then clearly T x0 = y0 and k (t, [x; v]) k (t,T [x; v]) λ − λ kξ(t, x) kη (t,T0x) (2.29) ≥ − Q(t)2 = 0, t> 0, x O, v E. Lλ(t) ≥ ∈ ∈ INTERPOLATION BETWEEN HILBERT SPACES 15

We have veriﬁed (2.27) in case 1. A computation similar to the one in the proof of Theorem 2.6 shows that, with respect to the bases uk and vi, ′ 1/2 ε ζ L ( η ) ξkL ( ξk)Lη( ξk) (T ) = k i ξ − i ξ − − . 0 ik ξ η L′ ( ξ ) η L ( η )L′ ( η ) k − i ξ − k − i ξ − i η − i Notice that, multiplying (2.29) by t, then letting t implies that → ∞ x2 ξ (T x)2η =0. k k − 0 i i odd i even kX X This means that T is a partial isometry from O to E with respect to the norm of n ℓ2 (λ).

2 0 0 Case 2. Now assume that P (t) = tq(t) , q (n−2)/2(R), x E, and y O. Then ∈ P ∈ ∈ 2 tq(t) 0 2 ξi ηk 0 2 = (yi ) + (xk) , Lλ(t) − odd t + ξi even t + ηk iX kX where ε′ q( ξ ) ζ′ q( η ) (2.30) y0 = i − i , x0 = − k − k i L′ ( ξ ) k L′ ( η ) λ − i − λ − k ′ ′ for some εi, ζk 1 . p p By (2.30) are∈ defined{± } linear maps E (R): x Q ; (R) O : Q y. →P(n−2)/2 7→ P(n−2)/2 → 7→ We denote their composite by T : E O : x y. 1 → 7→ Define T M (R) by ∈ n T : O E O E :[u; x] [T x; 0] . ⊕ → ⊕ 7→ 1 We then have k (t,T [u; x]) + k (t, [u; x]) − λ λ kξ (t,T1x)+ kη(t, x) (2.31) ≥− tQ(t)2 = 0, t> 0, u O, x E, Lλ(t) ≥ ∈ ∈ and (2.27) is verified also in case 2. A computation shows that, with respect to the bases vk and ui, 1/2 ′ ′ ′ ε ζ L ( ξ ) Lξ( ηk)L ( ηk) (T ) = i k η − i − − η − . 1 ik η ξ L′ ( η ) L′ ( ξ )L ( ξ ) k − i η − k ξ − i η − i ! Inserting t =0 in (2.31) we find that

2 2 (T1x)i + (xk) =0, − odd even iX kX n i.e., T is a partial isometry form E to O with respect to the norm of ℓ2 .

In the case of even n, the matrix T1 coincides with Löwner’s matrix. 16 YACIN AMEUR

3. Quadratic interpolation spaces 3.1. A classification of quadratic interpolation spaces. Recall that an intermediate space X with respect to is said to be of type H if T Mi for H k k L(Hi) ≤ i = 0, 1 implies that T H (M ,M ). We shall henceforth make a mild k k L(X) ≤ 0 1 restriction, and assume that H be homogeneous of degree one. This means that we can write (3.1) H(s,t) 2 = s 2 H(t 2/s 2) for some function H of one positive variable. In this situation, we will say that X 2 is of type H. The definition is chosen so that the estimates T Mi for k k L(Hi) ≤ i =0, 1 imply T 2 M H (M /M ). k k L(X) ≤ 0 1 0 In the following we will make the standing assumptions: H is an increasing, continuous, and positive function on R+ with H(1) = 1 and H(t) max 1,t . Notice that our assumptions imply that all spaces of type H are≤ exact interpola-{ } tion. Note also that H(t)= t θ corresponds to geometric interpolation of exponent θ. Suppose now that is a regular Hilbert couple and that ∗ is an exact interpolation space with correspondingH operator B. By Donoghue’sH lemma, we have that B = h(A) for some positive Borel function h on σ(A). The statement that is intermediate relative to is equivalent to that H∗ H A (3.2) c B c (1 + A) 1 1+ A ≤ ≤ 2 for some positive numbers c1 and c2. Let us momentarily assume that 0 be separable. (This restriction is removed in Remark 3.1.) We can then defineH the scalar-valued spectral measure of A,

ν (ω)= 2−k E(ω)e ,e A h k ki0 X where E is the spectral measure of A, ek; k =1, 2,... is an orthonormal basis for , and ω is a Borel set. Then, for Borel{ functions h} ,h on σ(A), one has that H0 0 1 h1 = h2 almost everywhere with respect to νA if and only if h1(A)= h2(A). Note that the regularity of means that ν ( 0 )=0. H A { } Theorem II. If is of type H with respect to , then B = h(A) where the H∗ H function h can be modiﬁed on a null-set with respect to νA so that (3.3) h(λ)/h(µ) H (λ/µ) , λ,µ σ(A) 0 . ≤ ∈ \{ } R ′ ′ Proof. Fix a (large) compact subset K σ(A) + and put 0 = 1 = EK ( 0) where E is the spectral measure of A, and⊂ the norms∩ are deﬁnedH byH restriction,H

′ ′ x H = x H , x H = x H∗ , x EK ( 0) . k k i k k i k k ∗ k k ∈ H It is clear that the operator A′ corresponding to ′ is the compression of A to ′ ′ H ′ 0 and likewise the operator B corresponding to ∗ is the compression of B to H′ ′ H H ′ 0. Moreover, ∗ is of interpolation type with respect to and the operator HB′ = (h ) (A′)H. For this reason, and since the compact set K isH arbitrary, it clearly |K suﬃces to prove the statement with replaced by ′. Then A is bounded above and below. Moreover, by (3.2), also BHis bounded aboveH and below. INTERPOLATION BETWEEN HILBERT SPACES 17

Let c < 1 be a positive number such that σ(A) c,c−1 . For a ﬁxed ε > 0 with ε

mε(λ)= ess inf Eλ h, Mε(λ)= ess sup Eλ h, the essential inf and sup being taken with respect to νA. ′ Now fix a small positive number ε and two unit vectors eλ,eµ supported by Eλ, Eµ respectively, such that e 2 M (λ) ε′, e 2 m (µ)+ ε′. k λ k∗ ≥ ε − k µ k∗ ≤ ε Now fix λ, µ σ(A) and let T x = x, e e . Then ∈ h µi0 λ 2 1 2 T x 2 = x, e e 2 x, e (λ + ε) k k1 h µi0 k λ k1 ≤ (µ ε) 2 h µi1 − (µ + ε)( λ + ε) x 2 . ≤ (µ ε)2 k k1 − Likewise, 2 T x 2 x, e x 2 , k k0 ≤ h µi0 ≤k k0 2 (µ+ε)(λ+ε) so T 1 and T αµ,λ,ε where αµ,λ,ε = 2 . k k≤ k kA ≤ (µ−ε) Since is of type H, we conclude that H∗ T 2 H (α ) , k kB ≤ µ,λ,ε whence ′ 2 2 2 Mε(λ) ε eλ = Teµ H (αµ,λ,ε) eµ (3.4) − ≤k k∗ k k∗ ≤ k k∗ H (α ) (m (µ)+ ε′) . ≤ µ,λ,ε ε In particular, since ε′ was arbitrary, and m (λ) e 2 B , we find that ε ≤k λ k∗ ≤k k M (λ) m (λ) [H (α ) 1] B . ε − ε ≤ µ,λ,ε − k k By assumption, H is continuous and H(1) = 1. Hence, as ε 0, the functions ↓ Mε(λ) diminish monotonically, converging uniformly to a function h∗(λ) which is also the uniform limit of the family mε(λ). It is clear that h∗ is continuous, and since mε h∗ Mε, we have h∗ = h almost everywhere with respect to νA. The ≤ ≤ ′ relation (3.3) now follows for h = h∗ by letting ε and ε tend to zero in (3.4). A partial converse to Theorem II is found below, see Theorem 6.3. Remark 3.1. (The non-separable case.) Now consider the case when is non- H0 separable. (By regularity this means that also 1 and ∗ are non-separable.) H H′ First assume that the operator A is bounded. Let 0 be a separable reducing ′ ′H subspace for A such that the restriction A of A to 0 has the same spectrum as ′ H A. The space 0 reduces B by Donoghue’s lemma; by Theorem II the restriction ′ ′ H ′ ′ ′ ′ B of B to 0 satisfies B = h (A ) for some continuous function h satisfying (3.3) H ′′ on σ(A). Let 0 be any other separable reducing subspace, where (as before) ′′ ′′ ′′ H ′ ′′ B = h (A ). Then 0 0 is a separable reducing subspace on which B = h(A) for some third continuousH ⊕ function H h on σ(A). Then h(A′) h(A′′)= h′(A′) h′′(A′′) and by continuity we must have h = h′ = h′′ on σ(A). The⊕ function h thus⊕ satisfies B = h(A) as well as the estimate (3.3). 18 YACIN AMEUR

If A is unbounded, we replace A by its compression to Pn 0 where Pn is the spectral projection of A corresponding to the spectral set [0,n] Hσ(A), n =1, 2,.... The same reasoning as above shows that B appears as a continuous∩ function of A on σ(A) [0,n]. Since n is arbitrary, we ﬁnd that B = h(A) for a function h satisfying (3.3).∩

3.2. Geometric interpolation. Now consider the particular case when ∗ is of exponent θ, viz. of type H(t) = t θ with respect to . We write B = h(AH) where h is the continuous function provided by Theorem IIH (and Remark 3.1 in the non- separable case). Fix a point λ σ(A) and let C = h(λ )λ −θ. The estimate (3.3) then implies 0 ∈ 0 0 that h(λ) Cλθ and h(µ) Cµθ for all λ, µ σ(A). We have proved the following theorem. ≤ ≥ ∈ Theorem 3.2. ([27, 40]) If is an exact interpolation Hilbert space of exponent H∗ θ relative to , then B = h(A) where h(λ)= Cλ θ for some positive constant C. H Theorem 3.2 says that = up to a constant multiple of the norm, where H∗ Hθ θ is the space deﬁned in (1.7). In the guise of operator inequalities: for any ﬁxed Hpositive operators A and B, the condition T ∗T M ,T ∗AT M A T ∗BT M 1−θM θB ≤ 0 ≤ 1 ⇒ ≤ 0 1 is equivalent to that B = A θ. It was observed in [27] that also equals to the complex interpolation space Hθ Cθ( ). For the sake of completeness, we supply a short proof of this fact in the appendix.H Remark 3.3. An exact quadratic interpolation method, the geometric mean was introduced earlier by Pusz and Woronowicz [33] (it corresponds to the C1/2-method). In [40], Uhlmann generalized that method to a method (the quadratic mean) denoted QIt where 0

Theorem III. An intermediate Hilbert space ∗ relative to is an exact interpolation space if and only if there is a positiveH radon measurHe ̺ on [0, ] such that ∞ x 2 = 1+ t−1 K t, x; d̺(t). k k∗ H Z[0,∞] Equivalently, ∗ is exact interpolation relative to if and only if the corresponding operator B canH be represented as B = h(A) for someH function h P ′. ∈ The statements that all norms of the given form are exact quadratic interpolation norms have already been shown (see §1.2). There remains to prove that there are no others. INTERPOLATION BETWEEN HILBERT SPACES 19

Donoghue’s original formulation of the result, as well as other equivalent forms of the theorem, is found in Section 6 below. Our present approach follows [2] and is based on K-monotonicity. Remark 3.4. The condition that be exact interpolation with respect to means H∗ H that ∗ is of type H where H(t) = max 1,t . In view of Theorem II (and Remark 3.1),H this means that we can represent{ B }= h(A) where h is quasi-concave on σ(A) 0 , \{ } (3.5) h(λ) h(µ)max 1,λ/µ , λ,µ σ(A) 0 . ≤ { } ∈ \{ } In particular, h is locally Lipschitzian on σ(A) R . ∩ + Remark 3.5. A related result concerning non-exact quadratic interpolation was proved by Ovchinnikov [30] using Donoghue’s theorem. Cf. also [4]. 3.4. The proof for simple ﬁnite-dimensional couples. Similar to our approach to Calderón’s problem, our strategy is to reduce Theorem III to a case of "simple couples”. C n n Theorem 3.6. Assume that 0 = 0 = as sets and that all eigenvalues (λi)1 of the corresponding operator AHare ofH unit multiplicity. Consider a third Hermitian 2 C n norm x ∗ = Bx, x 0 on . Then ∗ is exact interpolation with respect to if and onlyk k if B h= h(Ai) where h P ′. H H ∈ Remark 3.7. The lemma says that the class of functions h on σ(A) satisfying (3.6) T ∗T 1 ,T ∗AT A T ∗h(A)T h(A), (T M (C)) ≤ ≤ ⇒ ≤ ∈ n is precisely the set P ′ σ(A) of restrictions of P ′-functions to σ(A). In this way, the condition (3.6) provides| an operator-theoretic solution to the interpolation problem by positive Pick functions on a ﬁnite subset of R+.

Proof of Theorem 3.6. We already know that the spaces ∗ of the asserted form are exact interpolation relative to (see subsections 1.2 andH 1.3). H Now let ∗ be any exact quadratic interpolation space. By Donoghue’s lemma H n and the argument in §2.3, we can for an appropriate positive sequence λ = (λi)1 n identify = ℓ2 (λ), A = diag(λi), and B = h(A) where h is some positive function H n deﬁned on σ(A)= λi 1 . { } n n Our assumption is that ℓ2 (h(λ)) is exact interpolation relative to ℓ2 (λ). We must prove that h P ′ σ(A). To this end, write ∈ | (1 + t)λi kλi (t)= , 1+ tλi and recall that (see Lemma 1.1) n −1 K t, x; ℓn(λ) = 1+ t−1 x 2 k (t). 2 | i| λi 1 X Let us denote by C the algebra of continuous complex functions on [0, ] with the ∞ supremum norm u ∞ = supt>0 u(t) . Let V C be the linear span of the kλi for i =1,...,n. Wek k deﬁne a positive| functional| φ⊂on V by n n

φ( aikλi )= ai h(λi). 1 1 X X 20 YACIN AMEUR

We claim that φ is a positive functional, i.e., if u V and u(t) 0 for all t > 0, then φ(u) 0. ∈ ≥ ≥ n R 2 2 To prove this let u = 1 aikλi be non-negative on + and write ai = xi yi for some x, y Cn. The condition that u 0 means that | | −| | ∈ P ≥ n 1+ t−1 K t, x; ℓn(λ) = x 2 k (t) 2 | i| λi i=1 X (3.7) n y 2 k (t) ≥ | i| λi i=1 X −1 n = 1+ t K t,y; ℓ2 (λ) , t> 0. n n Since ℓ2 (λ) is an exact Calderón couple (by Theorem 2.4), the space ℓ2 (h(λ)) is exact K-monotonic. In other words, (3.7) implies that

x n y n , k kℓ2 (h(λ)) ≥k kℓ2 (h(λ)) i.e., n φ(u)= x 2 y 2 h(λ ) 0. | i| − | i| i ≥ 1 X The asserted positivity of φ is thereby proved. Replacing λi by cλi for a suitable positive constant c we can without losing generality assume that 1 σ(A), i.e., that the unit 1(x) 1 of the C∗-algebra C belongs to V . The positivity∈ of φ then ensures that ≡ φ = sup φ(u) = φ(1). k k u∈V ; kuk∞≤1 | | Let Φ be a Hahn-Banach extension of φ to C and note that Φ = φ = φ(1)=Φ(1). k k k k This means that Φ is a positive functional on C (cf. [29], §3.3). By the Riesz representation theorem there is thus a positive Radon measure ̺ on [0, ] such that ∞ Φ(u)= u(t) d̺(t), u C. ∈ Z[0,∞] In particular (1 + t)λ h(λ )= φ (k )=Φ(k )= i d̺(t), i =1, . . . , n. i λi λi 1+ tλ Z[0,∞] i We have shown that h is the restriction to σ(A) of a function of class P ′. 3.5. The proof of Donoghue’s theorem. We here prove Theorem III in full generality. ′ We remind the reader that if S R+ is a subset, we write P S for the convex cone of restrictions of P ′-functions⊂ to S. We ﬁrst collect some simple| facts about this cone. Lemma 3.8. (i) The class P ′ S is closed under pointwise convergence. | n (ii) If S is ﬁnite and if λ = (λi)i=1 is an enumeration of the points of S then ′ n h belongs to P S if and only if ℓ2 (h(λ)) is exact interpolation with respect to the n | pair ℓ2 (λ). INTERPOLATION BETWEEN HILBERT SPACES 21

(iii) If S is infinite, then a continuous function h on S belongs to P ′ S if and only if h P ′ Λ for every finite subset Λ S. | ∈ | ⊂ Proof. (i) Let h be a sequence in P ′ converging pointwise on S and fix λ S. It n ∈ is clear that the boundedness of the numbers hn(λ) is equivalent to boundedness of the total masses of the corresponding measures ̺n on the compact set [0, ]. It now suffices to apply Helly’s selection theorem. ∞ (ii) This is Theorem 3.6. (iii) Let Λn be an increasing sequence of finite subsets of S whose union is dense. ′ Let hn = h Λn where h is continuous on S. If hn P Λn for all n then the sequence h converges| pointwise on Λ to h. By part (i)∈ we| then have h P ′ σ(A). n ∪ n ∈ | We can now finish the proof of Donoghue’s theorem (Theorem III). Let ∗ be exact interpolation with respect to and represent the corresponding operatorH as B = h(A) where h satisfies (3.3). ByH the remarks after Theorem III, the function h is locally Lipschitzian. n In view of Lemma 3.8 we shall be done when we have proved that ℓ2 (h(λ)) is exact n n interpolation with respect to ℓ2 (λ) for all sequences λ = (λi)1 σ(A) of distinct points. Let us arrange the sequences in increasing order: 0 < λ ⊂< < λ . 1 ··· n Fix ε> 0, ε< min c, λ1, 1/λn and let Ei = [λi ε, λi + ε] σ(A); we assume { } − n ∩ that ε is sufficiently small that the Ei be disjoint. Let M = 1 Ei. We can assume that h has Lipschitz constant at most 1 on M. ∪ Let be the reducing subspace of 0 corresponding to the spectral set M, and let A˜ beM the compression of A to .H We define a function g on M by g(λ) = λ M i on E . Then g(λ) λ <ε on σ(A˜), so i | − | (3.8) A˜ g(A˜) ε , h(A˜) h(g(A˜)) ε. k − k≤ k − k≤ Lemma 3.9. Suppose that A′, A′′ ( ) satisfy A′, A′′ δ > 0 and A′ A′′ ∈ L M ≥ k − k≤ ε. Then T ′′ 1+2ε/δ max T , T ′ for all T ( ). k kA ≤ {k k k kA } ∈ L M ∗ ′ Proof. By definition,p T ′ is the smallest number C 0 such that T A T k kA ≥ ≤ C 2A′. Thus T ∗A′′T = T ∗(A′′ A′)T + T ∗A′T − 2 2 ′′ ′ ′′ T ε + T ′ (A + (A A )) ≤k k k kA − 2 2 2 ′′ 2ε max T , T ′ + T ′ A ≤ {k k k kA } k kA 2 2 ′′ max T , T ′ (1+2ε/δ) A . ≤ {k k k kA }

We can find δ > 0 such that the operators A˜, g(A˜), h(A˜), and h(g(A˜)) are δ. Then by repeated use of Lemma 3.9, ≥ T 1+2ε/δ max T , T k kh(g(A˜)) ≤ {k k k kh(A˜)} p1+2ε/δ max T , T ˜ ≤ {k k k kA} (1+2ε/δ)max T , T ,T ( ). ≤ p {k k k kg(A˜)} ∈ L M Let e be a unit vector supported by the spectral set E and define a space i i V⊂M to be the n-dimensional space spanned by the ei. Let A0 be the compression of 22 YACIN AMEUR g(A˜) to ; then V (3.9) T (1+2ε/δ)max T , T ,T ( ) . k kh(A0) ≤ k k k kA0 ∈ L V n Identifying with ℓ2 and A0 with the matrix diag(λi ), we see that (3.9) is inde- V n pendent of ε. Letting ε diminish to 0 in (3.9) now gives that ℓ2 (h(λ)) is exact n interpolation with respect to ℓ2 (λ). In view of Lemma 3.8, this finishes the proof of Theorem III. q.e.d.

4. Classes of matrix functions In this section, we discuss the basic properties of interpolation functions: in particular, the relation to the well known classes of monotone matrix functions. We refer to the books [12] and [34] for further reading on the latter classes.

4.1. Interpolation and matrix monotone functions. Let A1 and A2 be pos- n itive operators in ℓ2 (n = is admitted). Suppose that A1 A2 and form the following operators on ℓn ∞ℓn, ≤ 2 ⊕ 2 0 0 A 0 T = , A = 2 . 0 1 0 0 A 1 A 0 It is then easy to see that T ∗T 1 and that T ∗AT = 1 A. 0 0 ≤ 0 0 0 0 ≤ Now assume that a function h on σ(A) belongs to the class CA defined in §1.4, i.e., that h satisfies (4.1) T ∗T 1 ,T ∗AT A T ∗h(A)T h(A), ≤ ≤ ⇒ ≤ 2n where T denotes an operator on ℓ2 . We then have T ∗h(A)T h(A), or 0 0 ≤ h(A ) 0 h(A ) 0 1 2 . 0 0 ≤ 0 h(A1) In particular, we find that h(A1) h(A2). We have shown that (under the assumptions above) ≤ (4.2) A A h (A ) h (A ) . 1 ≤ 2 ⇒ 1 ≤ 2 We now change our point of view slightly. Given a positive integer n, we let Cn denote the convex of positive functions h on R+ such that (4.1) holds for all n n positive operators A on ℓ2 and all T (ℓ2 ). ′ ∈ L R Similarly, we let Pn denote the class of all positive functions h on + such that n h(A1) h(A2) whenever A1, A2 are positive operators on ℓ2 such that A1 A2. ≤ ′ R ≤ We refer to Pn as the cone of positive functions monotone of order n on +. ′ We have shown above that C2n Pn. ⊂ ′ In the other direction, assume that h P2n. Let A, T be bounded operators on n ∗ ∗ ∈ ℓ2 with A> 0, T T 1 and T AT A. Assume also that h be continuous. We will use the following lemma≤ due to Hansen≤ [19]. We recall the proof for completeness. Lemma 4.1. ([19]) T ∗h(A)T h(T ∗AT ). ≤ Proof. Put S = (1 TT ∗)1/2 and R = (1 T ∗T )1/2 and consider the 2n 2n matrix − − × T S A 0 U = , X = . R T ∗ 0 0 − INTERPOLATION BETWEEN HILBERT SPACES 23

It is well-known, and easy to check, that U is unitary and that T ∗AT T ∗AS U ∗XU = . SAT SAS Next fix a number ε> 0, a constant λ> 0 (to be fixed), and form the matrix T ∗AT + ε 0 Y = 0 2λ which, provided that we choose λ SAS , satisfies ≥k k ε T ∗AS ε D Y U ∗XU = − , − SAT 2λ SAS ≥ D∗ λ − − where we have written D = T ∗AS. If we now also choose λ so− that λ D 2/ε, then we obtain for all ξ, η Cn that ≥ k k ∈ ε D ξ ξ , = ε ξ 2 + Dη,ξ + D∗ξ, η + λ η 2 D∗ λ η η k k h i h i k k ε ξ 2 2 D ξ η + λ η 2 0. ≥ k k − k kk kk k k k ≥ Hence U ∗XU Y and as a consequence U ∗h(X)U = h(U ∗XU) h(Y ), since h is matrix monotone≤ of order 2n. The last inequality means that ≤ T ∗h(A)TT ∗h(A)S h(T ∗AT + ε) 0 , Sh(A)T Sh(A)S ≤ 0 h(2λ) so in particular T ∗h(A)T h(T ∗AT + ε). Since ε> 0 was arbitrary, and since h is assumed to be continuous,≤ we conclude the lemma. We now continue our discussion. Assuming that T ∗T 1 and T ∗AT A, ′ ∗ ≤ ′≤ and that h P2n is continuous, we have h(T AT ) h(A) [since h Pn], so ∗ ∈ ≤ ∈ T h(A)T h(A) by Lemma 4.1. We conclude that h Cn. ≤ ′ ∈ To prove that P2n Cn, we need to remove the continuity assumption on h made above. This is⊂ completely standard: let ϕ be a smooth positive func- R ∞ tion on + such that 0 ϕ(t) dt/t = 1, and define a sequence hk by hk(λ) = k−1 ∞ ϕ λk/tk h(t) dt/t. The class P ′ is a convex cone, closed under pointwise 0 R 2n convergence [12], so the functions h ,h ,... are of class P ′ . They are furthermore R 1 2 2n continuous, so by the argument above, they are of class Cn. By Lemma 3.8, the cone Cn is also closed under pointwise convergence, so h = lim hn Cn. ′ ′ ∈ To summarize, we have the inclusions C2n Pn, P2n Cn, and also Cn+1 Cn, ′ ′ ⊂ ⊂ ∞ ′ ⊂ Pn+1 Pn. In view of Theorem III, we have the identity 1 Cn = P . The inclusions⊂ above now imply the following result, sometimes known∩ as "Löwner’s theorem on matrix monotone functions". Theorem 4.2. We have ∞P ′ = ∞C = P ′. ∩1 n ∩1 n ∞ ′ ′ The identity 1 Pn = P says that a positive function h is monotone of all finite ∩ ′ ′ ′ orders if and only it is of class P . The somewhat less precise fact that P∞ = P is interpreted as that the class of operator monotone functions coincides with P ′. ′ The identity C∞ = P is, except for notation, contained in the work of Foiaş and Lions, from [17]. See §6.4. ′ Note that the inclusion P2n Cn shows that a matrix monotone functions of order 2n can be interpolated by⊂ a positive Pick function at n points. Results of a 24 YACIN AMEUR similar nature, where it is shown, in addition, that an interpolating Pick function can be taken rational of a certain degree, are discussed, for example, in Donoghue’s book [13, Chapter XIII] or (more relevant in the present connection) in the paper [14]. ∞ ′ ′ It seems somewhat inaccurate to refer to the identity 1 Pn = P as "Löwner’s theorem", since Löwner discusses more subtle results conce∩ rning matrix monotone functions of a given finite order n. In spite of this, it is common nowadays to let "Löwner’s theorem" refer to this identity.

4.2. More on the cone CA. We can now give an short proof of the following result due to Donoghue [14]. Theorem 4.3. For a positive function h on σ(A) we define two positive functions h˜ and h∗ on σ A−1 by h˜(λ)= λh (1/λ) and h∗(λ)=1/h (1/λ). Then the following conditions are equivalent, (i) h CA, ˜ ∈ (ii) h CA−1 , ∗∈ (iii) h C −1 . ∈ A Proof. Let ∗ be a quadratic intermediate space relative to a regular Hilbert couple ; let B =H h(A) be the corresponding operator. It is clear that is exact H H∗ interpolation relative to if and only if ∗ is exact interpolation relative to the (r) H H reverse couple = ( 1, 0). The latter couple has corresponding operator A−1 and it is clearH thatH theH identity x 2 = h(A)x, x is equivalent to that k k∗ h i0 2 −1˜ −1 x ∗ = A h A x, x . We have shown the equivalence of (i) and (ii). k k 1 ∗ ∗ ∗ ∗ Next letD = ( 0, 1E) be the dual couple, where we identify 0 = 0. With H H ∗H 2H H−1 this identification, becomes associated with the norm x ∗ = A x, x , 1 H1 0 ∗ H 2 −1 k k and ∗ is associated with x ∗ = B x, x . It remains to note that ∗ is H∗ 0 H k k ∗ H exact interpolation relative to if and only if ∗ is exact interpolation relative to ∗, proving the equivalenceH of (i) and (iii). H H Combining with Theorem III, one obtains alternative proofs of the interpolation theorems for P ′-functions discussed by Donoghue in the paper [14]. Remark 4.4. The exact quadratic interpolation spaces which are fixed by the dual- ∗ ′ ity, i.e., which satisfy ∗ = ∗, correspond precisely to the class of P -functions which are self-dual: h∗H= h. ThisH class was characterized by Hansen in the paper [20].

4.3. Matrix concavity. A function h on R+ is called matrix concave of order n if we have Jensen’s inequality λh (A )+(1 λ)h (A ) h (λA + (1 λ)A ) 1 − 2 ≤ 1 − 2 for all positive n n matrices A , A , and all numbers λ [0, 1]. Let us denote by × 1 2 ∈ Γn the convex cone of positive concave functions of order n on R+. The fact that ′ nΓn = P follows from the theorem of Kraus [23]. Following [2] we now give an ∩alternative proof of this fact.

′ Proposition 4.5. For all n we have the inclusion C3n Γn Pn. In particular ∞Γ = P ′. ⊂ ⊂ ∩1 n INTERPOLATION BETWEEN HILBERT SPACES 25

Proof. Assume first that h C and pick two positive matrices A and A . Define ∈ 3n 1 2 A3 = (1 λ)A1 + λA2 where λ [0, 1] is given, and define matrices A and T of order 3n −by ∈

A3 0 0 0 00 A = 0 A 0 ,T = √1 λ 0 0 .  1   −  0 0 A2 √λ 0 0 It is clear that T ∗T 1 and    ≤ A3 0 0 T ∗AT = 0 00 A,  0 00 ≤ so, since h C , we have T ∗h(A)T  h(A), or  ∈ 3n ≤ (1 λ)h(A1)+ λh(A2) 0 0 h(A3)0 0 − 0 00 0 h(A ) 0 .   ≤  1  0 00 0 0 h(A2)     Comparing the matrices in the upper left corners, we find that h Γn. Assume now that h Γ , and take positive definite matrices∈A , A of order ∈ n 1 2 n with A1 A2. Also pick λ (0, 1). Then λA2 = λA1 + (1 λ)A3 where A = λ(1 ≤λ)−1(A A ). By matrix∈ concavity, we then have − 3 − 2 − 1 h(λA ) λh(A )+(1 λ)h(A ) λh(A ), 2 ≥ 1 − 3 ≥ 1 where we used non-negativity to deduce the last inequality. Being concave, h is certainly continuous. Letting λ 1 one thus finds that h(A1) h(A2). We have shown that h P ′ . ↑ ≤ ∈ n For a further discussion of classes of convex matrix functions and their relations to monotonicity, we refer to the paper [21]. 4.4. Interpolation functions of two variables. In this section, we briefly discuss a class of interpolation functions of two matrix variables. We shall not completely characterize the class of such generalized interpolation functions here, but we hope that the following discussion will be of some use for a future investigation.

Let H1 and H2 be Hilbert spaces. We turn H1 H2 into a Hilbert space ⊗ ′ ′ by defining the inner product on elementary tensors via x1 x2, x1 x2 := x , x ′ x , x ′ (then extend via sesqui-linearity). Similarly,h ⊗ if T⊗arei op- h 1 1 i1 · h 2 2 i2 i erators on Hi, the tensor product T1 T2 is defined on elementary tensors via (T T ) (x x ) = T x T x . It⊗ is then easy to see that if A are positive 1 ⊗ 2 1 ⊗ 2 1 1 ⊗ 2 2 i operators on Hi for i =1, 2, then A1 A2 0 as an operator on the tensor product. Furthermore, we have A A A′⊗ A′≥if A A′ for i =1, 2. 1 ⊗ 2 ≤ 1 ⊗ 2 i ≤ i Given two positive definite matrices A of orders n and a function h on σ(A ) i i 1 × σ(A2), we define a matrix h(A1, A2) by h(A , A )= h (λ , λ ) E1 E2 1 2 1 2 λ1 ⊗ λ2 (λ1,λ2)∈σX(A1)×σ(A2) j where E is the spectral resolution of the matrix Aj . We shall say that h gives rise to exact interpolation relative to (A1, A2), and write h C , if the condition ∈ A1,A2 (4.3) T ∗T 1 ,T ∗A T A , j =1, 2 j j ≤ j j j ≤ j 26 YACIN AMEUR implies ∗ h(A1, A2) + (T1 T2) h(A1, A2)(T1 T2) (4.4) ⊗ ⊗ (T 1)∗h(A , A )(T 1) (1 T )∗h(A , A )(1 T ) 0. − 1 ⊗ 1 2 1 ⊗ − ⊗ 2 1 2 ⊗ 2 ≥

Taking T1 = T2 =0 we see that h 0 for all h CA1,A2 . It is also clear that CA1,A2 is a convex cone closed under pointwise≥ convergence∈ on the ﬁnite set σ(A ) σ(A ). 1 × 2 If h = h1 h2 is an elementary tensor where hj CAj is a function of one vari- ⊗ ∗ ∈ ∗ able, then (4.3) implies Tj hj (Aj )Tj hj (Aj ), whence (h1(A1) T1 h1(A1)T1) (h (A ) T ∗h (A )T ) 0, which implies≤ (4.4). We have shown− that C C ⊗ 2 2 − 2 2 2 2 ≥ A1 ⊗ A2 ⊂ CA1,A2 . Since for each t 0 the P ′-function λ (1+t)λ is of class C , we infer that ≥ 7→ 1+tλ Aj every function representable in the form

(1 + t1)λ1 (1 + t2)λ2 (4.5) h(λ1, λ2)= x d̺(t1,t2) 1+ t1λ1 1+ t2λ2 [0,∞] 2

2 with some positive Radon measure ̺ on [0, ] is in the class CA1,A2 . We shall say that a function h on σ(A ) ∞σ(A ) has the separate interpolation- 1 × 2 property if for each ﬁxed b σ(A2) the function λ1 h(λ1,b) is of class CA1 , and a similar statement holds for∈ all functions λ h(a,7→ λ ). 2 7→ 2

Lemma 4.6. Each function of class CA1,A2 has the separate interpolation-property. ∗ ∗ Proof. Let T2 =0 and take an arbitrary T1 with T1 T1 1 and T1 A1T1 A1. By hypothesis, ≤ ≤ (T 1)∗h(A , A )(T 1) h(A , A ). 1 ⊗ 1 2 1 ⊗ ≤ 1 2 Fix an eigenvalue b of A2 and let y be a corresponding normalized eigenvec- tor. Then for all x H1 we have h(A1, A2)x y, x y = h(A1,b)x, x H1 and ∗ ∈ h ∗ ⊗ ⊗ i h i (T1 1) h(A1, A2)(T1 1)x y, x y = T1 h(A1,b)T1x, x so h ⊗ ⊗ ⊗ ⊗ i h iH1 ∗ T1 h(A1,b)T1x, x h(A1,b)x, x . h iH1 ≤ h iH1 The functions h(a, λ2) can be treated similarly.

1/2 Example. The function h(λ1, λ2) = (λ1+λ2) clearly has the separate interpolation- property for all A1, A2. However, it is not representable in the form (4.5). Indeed, Re h(i,i) h( i,i) =1 while it is easy to check that Re h(λ , λ ) h(λ¯ , λ ) 0 { − − } { 1 2 − 1 2 }≤ whenever Im λ1, Im λ2 > 0 and h is of the form (4.5).

Let us say that a function h(λ1, λ2) deﬁned on R+ R+ is an interpolation function (of two variables) if h C for all A , A . Lemma× 4.6 implies that in- ∈ A1,A2 1 2 terpolation functions are separately real-analytic in R+ R+ and that the functions h(a, ) and h( ,b) are of class P ′ (cf. Theorem III). × The· above· notion of interpolation function is close to Korányi’s deﬁnition of monotone matrix function of two variables: f(λ1, λ2) is matrix monotone in a R ′ rectangle I = I1 I2 (I1, I2 intervals in ) if A1 A1 (with spectra in I1) and A A′ (with spectra× in I ) implies ≤ 2 ≤ 2 2 f(A′ , A′ ) f(A′ , A ) f(A , A′ )+ f(A , A ) 0. 1 2 − 1 2 − 1 2 1 2 ≥ Lemma 4.7. Each interpolation function is matrix monotone in R R . + × + INTERPOLATION BETWEEN HILBERT SPACES 27

′ ′ ˜ Ai 0 0 0 ∗ ˜ Proof. Let 0 < Ai Ai and put Ai = ,Ti = . Since Ti AiTi ≤ 0 Ai 1 0 ≤ A˜i, an interpolation function h will satisfy the interpolation inequality (4.4) with x x A replaced by A˜ . Applying this inequality to vectors of the form 1 2 i i 0 ⊗ 0 we readily obtain h(A′ , A′ )x x , x x h(A , A′ )x x , x x h 1 2 1 ⊗ 2 1 ⊗ 2i − h 1 2 1 ⊗ 2 1 ⊗ 2i h(A′ , A )x x , x x + h(A , A )x x , x x 0. − h 1 2 1 ⊗ 2 1 ⊗ 2i h 1 2 1 ⊗ 2 1 ⊗ 2i≥ ′ ′ The same result obtains with x1 x2 replaced by a sum x1 x2 + x1 x2 + ..., i.e., h is matrix monotone. ⊗ ⊗ ⊗

Remark 4.8. Assume that f is of the form f(λ1, λ2) = g1(λ1)+ g2(λ2). Then f is matrix monotone for all g1, g2 and f is an interpolation function if and only if g ,g P ′. In order to disregard "trivial" monotone functions of the above type, 1 2 ∈ Korányi [22] imposed the normalizing assumption (a) f(λ1, 0) = f(0, λ2)=0 for all λ1, λ2. It follows from Lemma 4.7 and the proof of [22, Theorem 4] that, if h is a C2-smooth interpolation function, then the function h(x , x ) h(x ,y ) h(y , x )+ h(y ,y ) k(x , x ; y ,y )= 1 2 − 1 2 − 1 2 1 2 1 2 1 2 (x y )(x y ) 1 − 1 2 − 2 is positive definite in the sense that k(x ,y ; x ,y )α α¯ 0 for all finite m n m m n n m n ≥ sequences of positive numbers xj ,yk and all complex numbers αl. (The proof uses Löwner’s matrix.) Korányi uses essentiallyP P this positive definiteness condition (and condition (a) in the remark above) to deduce an integral representation formula for h as an integral of products of Pick functions. See Theorem 3 in [22]. However, in contrast to our situation, Korányi considers functions monotone on the rectangle ( 1, 1) ( 1, 1), so this last result cannot be immediately applied. (It easily implies local− representation× − formulas, valid in finite rectangles, but these representations do not appear to be very natural from our point of view.) This is not the right place to attempt to extend Korányi’s methods to functions on R R ; it would seem more appropriate to give a more direct characterization + × + of the classes CA1,A2 or of the class of interpolation functions. At present, we do not know if there is an interpolation function which is not representable in the form (4.5).

5. Proof of the K-property In this section we extend the result of Theorem 2.4 to obtain the full proof of Theorem I. The discussion is in principle not hard, but it does require some care to keep track of both norms when reducing to a finite-dimensional case. Recall first that, by Lemma 2.3, it suffices to consider the diagonal case = . To prove Theorem I we fix a regular Hilbert couple ; we must prove thatH it hasK the K-property (see §1.5). By Theorem 2.4, we know thatH this is true if is finite dimensional and the associated operator only has eigenvalues of unit multiplicity.H We shall use a weak* type compactness result ([2]). To formulate it, let ( ) L1 H be the unit ball in the space ( ). Moreover, let Σ be the sum + normed L H t H0 H1 28 YACIN AMEUR

2 by x := K(t, x). Note that Σ is an equivalent norm on Σ and that Σ 1 =Σ k kΣ t k·k t isometrically. We denote by 1 (Σ t ) the unit ball in the space (Σ t ). In view of Remark 1.4, oneL has the identity L

(5.1) 1( )= 1 (Σ t ) . L H R L t∈\+ We shall use this to deﬁne a compact topology on ( ). L1 H Lemma 5.1. The subset 1 1 ( Σ ) is compact relative to the weak operator topology inherited from L( Σ )H. ⊂ L L Recall that the weak operator topology on ( H ) is the weakest topology such L that a net Ti converges to the limit T if the inner product Tix, y H converges to Tx,y for all x, y H. h i h iH ∈ Proof of Lemma 5.1. The weak operator topology coincides on the unit ball 1 ( Σ ) with the weak*-topology, which is compact, due to Alaoglu’s theorem (seeL [29], Chap. 4 for details). It is clear that for a ﬁxed t> 0, the subset 1 ( Σ ) 1 (Σ t ) is weak operator closed in ( Σ ); hence it is also compact. In viewL of (5.1),∩ L the set L1 1 is an intersection of compact sets. Hence the set 1 is itself compact, LprovidedH that we endow it with the subspace topology inheritL Hed from ( Σ ). L1

Denote by Pn the projections Pn = Eσ(A)∩[n−1,n] on 0 where E is the spectral resolution of A and n =1, 2, 3,.... Consider the coupleH (n) = (P ( ) , P ( )), H n H0 n H1 the associated operator of which is the compression A of A to the subspace P ( ). n n H0 Note that the norms in the couple (n) are equivalent, i.e., the associated operator H An is bounded above and below. We shall need two lemmas. Lemma 5.2. If (n) has the K-property for all n, then so does . H H Proof. Note that P = 1 for all n, and that P 1 as n relative to k n k L(H) n → → ∞ the strong operator topology on (Σ). Suppose that x0,y0 Σ are elements such that, for some ρ> 1, L ∈ 1 (5.2) K t,y0 < K t, x0 , t> 0. ρ Then K t, P y0 K t,y0 <ρ−1K t, x0 . Moreover, the identity K t, P y0 = n ≤ n tAn 0 0 0 1+tA Pny , Pny shows that we have an estimate of the form K(t, Pny ) n 0 ≤ CD n min 1,t for t>E 0 and large enough Cn (this follows since An is bounded above and below).{ } 0 The functions K t, Pmx increase monotonically, converging uniformly on compact subsets of R to K t, x0 when m . By concavity of the function + → ∞ t K t, P x0 we will then have 7→ m 1 (5.3) K t, P y0 < K t, P x0 , t R , n ρ˜ m ∈ + provided that m is suﬃciently large, where ρ˜ is any number in the interval 1 < ρ<ρ˜ . INTERPOLATION BETWEEN HILBERT SPACES 29

0 0 Indeed, let A = limt→∞ K t, Pny and B = limt→0 K t, Pny /t. Take points 0 ′ 0 ′ t0 < t1 such that K(t, Pny ) A/ρ when t t1 and K(t, Pny )/t Bρ when ′ ≥ ≥ ′ ≤ t t0. Here ρ is some number in the interval 1 <ρ <ρ. ≤ 0 0 Next use (5.2) to choose m large enough that K(t, Pmx ) > ρK(t, Pny ) for all 0 ′ 0 t [t0,t1]. Then K(t, Pmx ) > (ρ/ρ )K(t, Pny ) for t = t1, hence for all t t1, ∈ 0 ′ 0 ≥ and K(t, Pmx )/t > (ρ/ρ )K(t, Pny )/t for t = t0 and hence also when t t0. Choosing ρ′ = ρ/ρ˜ now establishes (5.3). ≤ (N) Put N = max m,n . If has the K-property, we can find a map Tnm { } 0 H 0 ∈ 1( ) such that TnmPmx = Pny . (Define Tmn = 0 on the orthogonal comple- LmentH of P ( ) in Σ.) In view of Lemma 5.1, the maps T must cluster at some N H0 nm point T ( ). It is clear that T x0 = y0. Since ρ > 1 was arbitrary, we have ∈ L1 H shown that has the K-property . H Lemma 5.3. Given x0,y0 (n) and a number ǫ> 0 there exists a positive integer ∈ H0 n and a finite-dimensional couple (n) such that x0,y0 + and V ⊂ H ∈V0 V1 (5.4) (1 ǫ)K t, x; K t, x; (1 + ǫ)K t, x; , t> 0, x + . − H ≤ V ≤ H ∈V0 V1 Moreover, can be chosen so that all eigenvalues of the associated operator AV are of unit multiplicity.V

(n) Proof. Let An be the operator associated with the couple ; thus 1/n An n. N H ≤ ≤N Take η > 0 and let λi 1 be a ﬁnite subset of σ (An) such that σ (An) 1 Ei where E = (λ η/2,{ λ +} η/2). We deﬁne a Borel function w : σ (A ) ⊂σ ∪(A ) i i − i n → n by w(λ)= λi on Ei σ (An); then w (An) An L(H0) η. tλ ∩ k − k ≤ Let kt(λ)= 1+tλ . It is easy to check that the Lipschitz constant of the restriction kt σ (An) is bounded above by C1 min 1,t where C1 = C1(n) is independent of t. Hence { }

k (w (A )) k (A ) C η min 1,t . k t n − t n k L(H0) ≤ 1 { } It follows readily that (k (w (A )) k (A )) x, x C η min 1,t x 2 , x P ( ) . |h t n − t n i0|≤ 1 { }k k0 ∈ n H0 Now let c> 0 be such that A c. The elementary inequality kt(c) (1/2) min 1,ct shows that ≥ ≥ { } k (A ) x, x C min 1,t x 2 , x P ( ) , h t n i0 ≥ 2 { }k k0 ∈ n H0 where C = (1/2) min 1,c . Combining these estimates, we deduce that 2 { } (5.5) k (w (A )) x, x k (A ) x, x C η k (A ) x, x , x P ( ) |h t n i0 − h t n i0|≤ 3 h t n i0 ∈ n H0 for some suitable constant C3 = C3(n). Now pick unit vectors ei,fi supported by the spectral sets Ei σ(An) such that 0 0 N ∩ x and y belong to the space spanned by ei,fi 1 . Put 0 = 1 = and deﬁne norms on those spaces byW { } W W W x = x , x 2 = w (A) x, x . k kW0 k kH0 k kW1 h iH0 The operator associated with is then the compression of w(A ) to , i.e., W n W0 2 x = A x, x = w(An)x, x , x . k kW1 h W iW0 h iH0 ∈W Let ǫ =2C3η and observe that, by (5.5) (5.6) K t, x; K t, x; (ǫ/2)K t, x; , f . W − H ≤ H ∈W

30 YACIN AMEUR

The eigenvalues of AW typically have multiplicity 2. To obtain unit multiplicity, we perturb A slightly to a positive matrix A such that A A < ǫ/2C3. W V k W − V k L(H0) Let be the couple associated with A , i.e., put = for i =0, 1 and V V Vi W x = x and x 2 = A x, x . k kV0 k kW0 k kW1 h V iV0 It is then straightforward to check that K t,f; K t,f; (ǫ/2)K t,f; , f . W − V ≤ H ∈W Combining this with the estimate (5.6), one ﬁnishes the proo f of the lemma.

Proof of Theorem I. Given two elements x0,y0 Σ as in (5.2) we write xn = P x0 and yn = P y0 . By the proof of Lemma∈ 5.2 we then have K (t,yn) n n ≤ ρ˜−1K (t, xn) for large enough n, where ρ˜ is any given number in the interval (1,ρ). We then use Lemma 5.3 to choose a finite-dimensional sub-couple (n) such that V ⊂ H K t,yn; (1 + ǫ)K t,yn; V ≤ H −1 n n n < ρ˜ K t, x ; +ǫ K t, x ; + K t,y ; . V H H Here ǫ> 0 is at our disposal. Choosing ǫ sufficiently small, we can arrange that (5.7) K t,yn; K(t, xn; ), t> 0. V ≤ V By Theorem 2.4, the condition (5.7) implies the existence of an operator T ′ ∈ such that T ′xn = yn. Considering the canonical inclusion and projection L1 V I : Σ ( ) Σ ( ) and Π:Σ( ) Σ ( ) , V → H H → V we have, by virtue of Lemma 5.3, I 2 (1 ǫ)−1 and Π 2 1+ ǫ. k k L(V;H) ≤ − k k L(H;V) ≤ Now let T = T := IT ′Π ( (n) ). Then T 2 1+ǫ and T xn = yn. As ǫ 0 ε ∈ L H k k ≤ 1−ǫ ↓ (n) n n the operators Tǫ will cluster at some point T 1( ) such that T x = y (cf. Lemma 5.1). ∈ L H We have shown that (n) has the K-property . In view of Lemma 5.2, this implies that has the same property.H The proof of Theorem I is therefore complete. H 6. Representations of interpolation functions 6.1. Quadratic interpolation methods. Let us say that an interpolation method defined on regular Hilbert couples taking values in Hilbert spaces is a quadratic interpolation method. (Donoghue [13] used the same phrase in a somewhat wider sense, allowing the methods to be defined on non-regular Hilbert couples as well.) If F is an exact quadratic interpolation method, and a Hilbert couple, then by Donoghue’s theorem III there exists a positive RadonH measure ̺ on [0, ] such ∞ that F = , where the latter space is defined by the familiar norm x 2 = H H̺ k k̺ 1+ t−1 K(t, x) d̺(t). [0,∞] A priori, the measure ̺ could depend not only on F but also on the particular R . That ̺ is independent of can be realized in the following way. Let ′ be a Hregular Hilbert couple such thatH every positive rational number is an eigenvalueH of INTERPOLATION BETWEEN HILBERT SPACES 31 the associated operator. Let B′ be the operator associated with the exact quadratic interpolation space F ′ . There is then clearly a unique P ′-function h on σ (A′) such that B′ = h (A′),H viz. there is a unique positive Radon measure ̺ on [0, ] ∞ such that F ′ = ′ (see §1.2 for the notation). H Hρ If is any regular Hilbert couple, we can form the direct sum = ′ . Denote H S H ⊕H by A˜ the corresponding operator and let B˜ = h˜(A˜) be the operator corresponding to the exact quadratic interpolation space F . Then h˜(A˜) = h˜(A′) h˜ (A) = S ⊕ h (A′) h˜(A). This means that h˜ (A′) = h (A′), i.e. h˜ = h. In particular, the ⊕ operator B corresponding to the exact interpolation space F ( ) is equal to h (A). H We have shown that F ( )= ̺. We emphasize our conclusion with the following theorem. H H Theorem 6.1. There is a one-to-one correspondence ̺ F between positive Radon measures and exact quadratic interpolation methods.7→ We will shortly see that Theorem 6.1 is equivalent to the theorem of Foiaş and Lions [17]. As we remarked above, a more general version of the theorem, admitting for non-regular Hilbert couples, is found in Donoghue’s paper [13]. 6.2. Interpolation type and reiteration. In this subsection, we prove some general facts concerning quadratic interpolation methods; we shall mostly follow Fan [15]. Fix a function h P ′ of the form ∈ (1 + t)λ h(λ)= d̺(t). 1+ tλ Z[0,∞] It will be convenient to write h for the corresponding exact interpolation space . Thus, we shall denote H H̺ x 2 = h(A)x, x = 1+ t−1 K (t, x) d̺(t). k kh h i0 Z[0,∞] More generally, we shall use the same notation when h is any quasi-concave function on R ; then is a quadratic interpolation space, but not necessarily exact. + Hh Recall that, given a function H of one variable, we say that ∗ is of type H with 2 2 H respect to if T Mi implies T M0 H (M1/M0). H k k L(Hi) ≤ k k L(H∗) ≤ We shall say that a quasi-concave function h on R is of type H if is of + Hh type H relative to any regular Hilbert couple . The following result somewhat generalizes Theorem 3.2. The class of functionsH of type H clearly forms a convex cone. Theorem 6.2. Let h be of type H, where (i) H(1) = 1 and H(t) max 1,t , and ′ ≤ { } (ii) H has left and right derivatives θ± = H (1 ) at the point 1, where θ− θ+. Then for any positive constant c, ± ≤ h(cλ) (6.1) min λθ− , λθ+ max λθ− , λθ+ , λ R . ≤ h(c) ≤ ∈ + In particular, if H (t) is differentiable at t = 1 and H ′(1) = θ, then h(λ) = λ θ, λ R . ∈ + Proof. Replacing A by cA, it is easy to see that if h is of type H, then so is hc(t)= h(ct)/h(c). Fix µ> 0 and consider the function h0(t)= hc(µt)/hc(µ). By 32 YACIN AMEUR

Theorem II, we have h0(t) H(t) for all t. Furthermore h0(1) = H(1) = 1 by (i). Since h is diﬀerentiable, the≤ assumption (ii) now gives θ h′ (1) θ , or 0 − ≤ 0 ≤ + ′ µhc(µ) θ− θ+. ≤ hc(µ) ≤ Dividing through by µ and integrating over the interval [1, λ], one now veriﬁes the inequalities in (6.1).

The following result provides a partial converse to Theorem II. ′ H Theorem 6.3. ([15]) Let h P and set (t) = sups>0 h(st)/h(s). Then h is of type H. ∈ 2 Proof. Let T ( ) be a non-zero operator; put Mj = T and M = ∈ L H k k L(Hj ) M1/M0. We then have (by Lemma 1.1)

T x 2 = 1+ t−1 K (t,Tx) d̺(t) k kh Z[0,∞] M 1+ t−1 K (tM,x) d̺(t) ≤ 0 Z[0,∞] (1 + t)MA = M x, x d̺(t) 0 1+ tMA Z[0,∞] 0 = M h (MA) x, x . 0 h i0 Letting E be the spectral resolution of A, we have ∞ h (MA) x, x = h (Mλ) d E x, x . h i0 h λ i0 Z0 Since h (Mλ) /h(λ) H (M), we conclude that ≤ ∞ T x 2 M H (M) h (λ) d E x, x = M H (M) x 2 , k kh ≤ 0 h λ i0 0 k kh Z0 which ﬁnishes the proof.

Given a function h of a positive variable, we deﬁne a new function h˜ by h˜(s,t)= sh (t/s) . The following reiteration theorem is due to Fan. Theorem 6.4. ([15]) Let h,h ,h P ′, and ϕ(λ) = h˜ (h (λ),h (λ)). Then 0 1 ∈ 0 1 = ( , ) with equal norms. Moreover, is an exact interpolation space Hϕ Hh0 Hh1 h Hϕ relative to . H ′ ′ Proof. Let denote the couple ( h0 , h1 ). The corresponding operator A then obeys H H H

′ 1/2 ′ 1/2 ′ 1/2 ′ x H = (A ) x H0 = ϕ0(A) (A ) x 0, x ∆( ). k k h1 k k k k ∈ H 1/2 On the other hand, x H = ϕ1(A) x , so k k h1 0 ′ 1/2 −1/2 1/2 ′ (A ) x = ϕ0( A) ϕ1(A ) x, x ∆ . ∈ H INTERPOLATION BETWEEN HILBERT SPACES 33

′ −1 We have shown that A = ϕ0(A) ϕ1(A), whence (by Lemma 1.1) −1 ′ tϕ0(A) ϕ1(A) K t, x; = −1 x, x H 1+ tϕ0(A) ϕ1(A) ′ H0 (6.2) tϕ1(A) = −1 x, x . 1+ tϕ0(A) ϕ1(A) ′ H0 Now let the function h P ′ be given by ∈ (1 + t)λ h(λ)= d̺(t), 1+ tλ Z[0,∞] and note that the function ϕ = h˜ (h0,h1) is given by (1 + t)h (λ) ϕ(λ)= 1 d̺(t). 1+ th (λ)/h (λ) Z[0,∞] 1 0 Combining with (6.2), we ﬁnd that

2 −1 ′ x ′ = 1+ t K t, x; d̺(t) k kH h H Z[0,∞] ∞ (1 + t)h1(λ) 2 = d̺(t) d Eλx, x = x . 1+ th (λ)/h (λ) h i0 k kHϕ Z0 "Z[0,∞] 1 0 # This ﬁnishes the proof of the theorem. Combining with Donoghue’s theorem III, one obtains the following, purely function- theoretic corollary. Curiously, we are not aware of a proof which does not use interpolation theory. Corollary 6.5. ([15]) Suppose that h P ′ and that h ,h P ′ F , where F is ∈ 0 1 ∈ | some closed subset of R . Then the function ϕ = h˜(h ,h ) is also of class P ′ F . + 0 1 | 6.3. Donoghue’s representation. Let be a regular Hilbert couple. In Donoghue’s H 2 setting, the principal object is the space ∆ = 0 1 normed by x ∆ = 2 2 H ∩ H k k x 0 + x 1 . In the following, all involutions are understood to be taken with krespectk tok thek norm of ∆. We express the norms in the spaces as Hi x 2 = Hx, x and x 2 = (1 H)x, x , k k0 h i∆ k k1 h − i∆ where H is a bounded positive operator on ∆, 0 H 1. The regularity of means that neither 0, nor 1 is an eigenvalue of H. ≤ ≤ H To an arbitrary quadratic intermediate space ∗ there corresponds a bounded positive injective operator K on ∆ such that H x 2 = Kx, x . k k∗ h i∆ It is then easy to see that ∗ is exact interpolation if and only if, for bounded operators T on ∆, the conditionsH T ∗HT H and T ∗(1 H)T 1 H imply T ∗KT K. It is straightforward to check≤ that the relations− between≤ −H, K and the operators≤ A, B used in the previous sections are: 1 1 H B K (6.3) H = , A = − ,K = , B = . 1+ A H 1+ A H (It follows from the proof of Lemma 1.2 that H and K commute.) 34 YACIN AMEUR

By Theorem III we know that ∗ is an exact interpolation space if and only if B = h(A) for some h P ′. By (6.3),H this is equivalent to that K = k(H) where ∈ h(A) 1 H k(H)= = Hh − . 1+ A H In its turn, this means that (1 + t)(1 λ)/λ k(λ)= λ − d̺(t) 1+ t(1 λ)/λ Z[0,∞] − (1 + t)λ(1 λ) = − d̺(t), λ σ(H), λ + t(1 λ) ∈ Z[0,∞] − where ̺ is a suitable Radon measure. Applying the change of variables s =1/(1+t) and deﬁning a positive Radon measure µ on [0, 1] by dµ(s) = d̺(t), we arrive at the expression 1 λ(1 λ) (6.4) k(λ)= − dµ(s), λ σ(H), (1 s)(1 λ)+ sλ ∈ Z0 − − which gives the representation exact quadratic interpolation spaces originally used by Donoghue in [13].

6.4. J-methods and the Foiaş-Lions theorem. We define the (quadratic) J- functional relative to a regular Hilbert couple by H J(t, x)= J t, x; = x 2 + t x 2 , t> 0, x ∆( ). H k k0 k k1 ∈ H Note that J(t, x)1/2 is an equivalent norm on ∆ and that J(1, x)= x 2 . k k∆ Given a positive Radon measure ν on [0, ], we define a Hilbert space J ( ) ∞ ν H as the set of all elements x Σ( ) such that there exists a measurable function u : [0, ] ∆ such that ∈ H ∞ → (6.5) x = u(t) dν(t) (convergence in Σ) Z[0,∞] and J(t,u(t)) (6.6) dν(t) < . 1+ t ∞ Z[0,∞] The norm in the space J ( ) is defined by ν H J(t,u(t)) (6.7) x 2 = inf dν(t) k kJν u 1+ t Z[0,∞] over all u satisfying (6.5) and (6.6). The space (6.7) was (with different notation) introduced by Foiaş and Lions in the paper [17], where it was shown that there is a unique minimizer u(t) of the problem (6.7), namely −1 1+ t 1+ s (6.8) u(t)= ϕ (A)x where ϕ (λ)= dν(s) . t t 1+ tλ 1+ sλ Z[0,∞] ! Inserting this expression for u into (6.7), one finds that x 2 = h(A)x, x k kJν h i0 INTERPOLATION BETWEEN HILBERT SPACES 35 where 1+ t (6.9) h(λ)−1 = dν(t). 1+ tλ Z[0,∞] It is easy to verify that the class of functions representable in the form (6.9) for some positive Radon measure ν coincides with the class P ′. We have thus arrived at the following result.

Theorem 6.6. Every exact quadratic interpolation space can be represented H∗ isometrically in the form ∗ = Jν ( ) for some positive Radon measure ν on [0, ]. Conversely, any space ofH this formH is an exact quadratic interpolation space. ∞

In the original paper [17], Foiaş and Lions proved the less precise statement that each exact quadratic interpolation method F can be represented as F = Jν for some positive Radon measure ν.

6.5. The relation between the K- and J-representations. The assignment K̺ = Jν gives rise to a non-trivial bijection ̺ ν of the set of positive Radon measures on [0, ]. In this bijection, ̺ and ν are7→ in correspondence if and only if ∞ −1 (1 + t)λ 1+ t d̺(t)= dν(t) . 1+ tλ 1+ tλ Z[0,∞] Z[0,∞] !

As an example, let us consider the geometric interpolation space (where cθ = π/ sin(πθ)) ∞ dt x 2 = A θx, x = c t−θK(t, x) . k kθ 0 θ t Z0 −θ cθt The measure ̺ corresponding to this method is d̺θ(t) = 1+t dt. On the other hand, it is easy to check that

∞ 1+ t −1 c tθ dt λθ = dν (t) where dν (t)= θ . 1+ tλ θ θ 1+ t t Z0 We leave it to the reader to check that the norm in θ is the inﬁmum of the expression H ∞ dt c tθJ(t,u(t)) θ t Z0 over all representations ∞ dt x = u(t) . t Z0 We have arrived at the Hilbert space version of Peetre’s J-method of exponent θ.

The identity Jνθ = K̺θ can now be recognized as a sharp (isometric) Hilbert space version of the equivalence theorem of Peetre, which says that the standard Kθ and Jθ-methods give rise to equivalent norms on the category of Banach couples (see [7]). The problem of determining the pairs ̺,ν having the property that the K̺ and Jν methods give equivalent norms was studied by Fan in [15, Section 3]. 36 YACIN AMEUR

6.6. Other representations. As we have seen in the preceding subsections, using the space 0 to express all involutions and inner products leads to a description of the exact quadraticH interpolation spaces in terms of the class P ′. If we instead use the space ∆ as the basic object, we get Donoghue’s representation for interpolation functions. Similarly, one can proceed from any ﬁxed interpolation space ∗ to obtain a diﬀerent representation of interpolation functions. H

6.7. On interpolation methods of power p. Fix a number p, 1

f p = f(x) p w(x) dµ(x). k kLp(w) | | ZX We shall write Lp(w) = (Lp,Lp(w)) for the corresponding weighted Lp couple. Note that the conditions imposed mean precisely that Lp(w) be separable and regular. Let us say that an exact interpolation functor F defined on the totality of separable, regular weighted Lp-couples and taking values in the class of weighted Lp-spaces is of power p. Define, for a positive Radon measure ̺ on [0, ], an exact interpolation functor ∞ F = K̺(p) by the definition

1 p − − p−1 f := (1 + t p 1 ) Kp t,f; Lp(w) d̺(t). k kF (Lp(w)) Z[0,∞] We contend that F is of power p. Indeed, it is easy to verify that

p tw(x) Kp t,f; Lp(w) = f(x) 1 dµ(x), | | p−1 p−1 ZX (1 + (tw(x)) ) so Fubini’s theorem gives that

f p = f(x) p h(w(x)) dµ(x), k kF (Lp(w)) | | ZX where 1 (1 + t p−1 ) p−1λ (6.10) h(λ)= 1 d̺(t), λ w(X). p−1 p−1 ∈ Z[0,∞] (1 + (tλ) )

We have shown that F (Lp(w)) = Lp(h(w)), so F is indeed of power p. Let us denote by (p) the totality of positive functions h on R+ representable in the form (6.10) forK some positive Radon measure ̺ on [0, ]. Further, let (p) denote the class of all (exact) interpolation∞ functions of power I p, i.e., those positive functions h on R+ having the property that for each weighted Lp couple Lp(w) and each bounded operator T on Lp(w), it holds that T is bounded on Lp(h(w)) and T T . k k L(Lp(h(w))) ≤k k L(Lp(w)) The class (p) is in a sense the natural candidate for the class of "operator monotone I functions on Lp-spaces". The class (p) clearly forms a convex cone; it was shown by Peetre [31] that this cone is containedI in the class of concave positive functions on R+ (with equality if p =1). INTERPOLATION BETWEEN HILBERT SPACES 37

We have shown that (p) (p). By Theorem 6.1, we know that equality holds when p = 2. For otherK ⊂ values I of p it does not seem to be known whether the class (p) exhausts the class (p), but one can show that we would have K I (p) = (p) provided that each finite-dimensional L -couple ℓn(λ) has the K - K I p p p property (or equivalently, the K-property , see (2.2)). Naturally, the latter problem (about the Kp-property) also seems to be open, but some comments on it are found in Remark 2.9. Let ν be a positive Radon measure on [0, ]. In [17], Foiaş and Lions introduced ∞ a method, which we will denote by F = Jν (p) in the following way. Define the Jp- functional by J t,f; L (λ) = f p + t f p , f ∆,t> 0. p p k k0 k k1 ∈ We then define an intermediate norm by 1 p − p−1 f := inf (1 + t) Jp t,u(t); Lp(λ) dν(t), k kF (Lp(λ)) Z[0,∞] where the infimum is taken over all representations

f = u(t) dν(t) Z[0,∞] with convergence in Σ. It is straightforward to see that the method F so deﬁned is exact; in [17] it is moreover shown that it is of power p. More precisely, it is there proved that f p = f(x) p h(w(x)) dµ(x), k kF (Lp(λ)) | | ZX where 1 − 1 (1 + t) p 1 − p−1 (6.11) h(λ) = 1 dν(t), λ w(X). p−1 ∈ Z[0,∞] (1 + tλ) Let us denote by (p) the totality of functions h representable in the form (6.11). We thus have thatJ (p) (p). In view of our preceding remarks, we conclude J ⊂ I that if all weighted Lp-couples have the Kp property, then necessarily (p) (p). Note that (2) = (2) by Theorem 6.6. J ⊂ K J K Appendix: The complex method is quadratic Let S = z C; 0 Re z 1 . Fix a Hilbert couple and let be the set of functions S{ ∈Σ which≤ are bounded≤ } and continuous in SH, analyticF in the interior of S, and which→ maps the line j + iR into for j = 0, 1. Fix 0 <θ< 1. The Hj norm in the complex interpolation space C is deﬁned by θ H ( ) x = inf f ; f(θ)= x . ∗ k kCθ( H ) {k kF } N i Let denote the set of polynomials f = 1 aiz where ai ∆. We endow with theP inner product ∈ P P

f,g M = f(j + it),g(j + it) j Pj (θ,t) dt, h i θ R h i j=0,1 X Z where P , P is the Poisson kernel for S, { 0 1} e−πt sin θπ Pj (θ,t)= . sin2 θπ + (cos θπ ( 1)je−πt)2 − − 38 YACIN AMEUR

Let Mθ be the completion of with this inner product. It is easy to see that the elements of M are analytic inP the interior of S, and that evaluation map f f(θ) θ 7→ is continuous on Mθ. Let Nθ be the kernel of this functional and deﬁne a Hilbert space by Hθ = M /N . Hθ θ θ We denote the norm in by . Hθ k·kθ Proposition A.1. C = with equality of norms. θ H Hθ Proof. Let f . By the Calderón lemma in [7, Lemma 4.3.2], we have the estimate ∈ F

log f(θ) C ( H ) log f(j + it) j Pj (θ,t) dt. k k θ ≤ R k k j=0,1 X Z Applying Jensen’s inequality, this gives that

2 1/2 f(θ) C ( H ) ( f(j + it) j Pj (θ,t) dt) = f M . k k θ ≤ R k k k k θ j=0,1 X Z Hence θ Cθ( ) and θ. On the other hand, for f one has H ⊂ H k·kCθ ( H ) ≤k·k ∈P the estimates f(θ) f sup f(j + it) ; t R, j =0, 1 = f , k kθ ≤k kMθ ≤ {k kj ∈ } k kF whence Cθ( ) θ and . H ⊂ H k·kCθ( H ) ≥ k·kθ It is well known that the method Cθ is of exponent θ (see e.g. [7]). We have shown that Cθ is an exact quadratic interpolation method of exponent θ. Complex interpolation with derivatives. In [15, pp. 421-422], Fan considers the more general complex interpolation method Cθ(n) for the n:th derivative. This 1 (n) means that in (*), one consider representations x = n! f (θ) where f ; the complex method C is thus the special case C . It is shown in [15] that, for∈ Fn 1, θ θ(0) ≥ the Cθ(n)-method is represented, up to equivalence of norms, by the quasi-power θ θ(1−θ) n function h(λ) = λ /(1 + n log λ ) . The complex method with derivatives was introduced by Schechter [37];| for more| details on that method, we refer to the list of references in [15]. .

References

[1] Agler, J., McCarthy, J. E., Young, N., Operator monotone functions and Löwner functions of several variables, Ann. Math. 176 (2012), 1783–1826. [2] Ameur, Y., The Calderón problem for Hilbert couples, Ark. Mat. 41 (2003), 203–231. [3] Ameur, Y., A new proof of Donoghue’s interpolation theorem, Journal of Function Spaces and Applications 3 (2004), 253–265. [4] Ameur, Y., A note on a theorem of Sparr, Math. Scand. 94 (2004), 155–160. [5] Ameur, Y., Cwikel, M., On the K-divisibility constant for some special, ﬁnite-dimensional Banach couples, J. Math. Anal. Appl. 360 (2009), 130–155. [6] Aronszajn, N., Donoghue, W., On exponential representations of functions, J. Analyse Math. 5 (1956-57), 321–388. [7] Bergh, J., Löfström, J., Interpolation spaces, an introduction. Springer 1976. [8] Brudnyi, Y. A., Krugljak, N. Y., Interpolation functors and interpolation spaces, North Holland 1991. [9] Calderón, A. P., Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. INTERPOLATION BETWEEN HILBERT SPACES 39

[10] Calderón, A. P., Spaces between L1 and L∞ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299. [11] Cwikel, M., Monotonicity properties of interpolation spaces, Ark. Mat. 14 (1976), 213–236. [12] Donoghue, W., Monotone matrix functions and analytic continuation, Springer 1974. [13] Donoghue, W., The interpolation of quadratic norms, Acta Math. 118 (1967), 251–270. [14] Donoghue, W., The theorems of Loewner and Pick, Israel J. Math. 4 (1966), 153–170. [15] Fan, M., Quadratic interpolation and some operator inequalities, Journal of Mathematical Inequalities 5 (2011), 413–427. [16] Foiaş, C., Ong, S. C., Rosenthal, P., An interpolation theorem and operator ranges, Integral Equations Operator Theory 10 (1987), 802–811. [17] Foiaş, C., Lions, J. L., Sur certains théorèmes d’interpolation, Acta Sci. Math. 22 (1961), 269–282. [18] Halmos, P. R., Quadratic interpolation, J. Operator Theory 7 (1982), 303–305. [19] Hansen, F., An operator inequality, Math. Ann. 246 (1980), 249–250. [20] Hansen, F., Selfadjoint means and operator monotone functions, Math. Ann. 256 (1981), 29–35. [21] Heinävaara, O., Local characterizations for the matrix monotonicity and convexity of ﬁxed order, Proc. Amer. math. Soc. 146 (2018), 3791-3799. [22] Korányi, A., On some classes of analytic functions of several variables, Trans. Amer. Math. Soc. 101 (1961), 520–554. [23] Kraus, F., Über konvexe Matrixfunktionen, Math. Z. 41 (1936), 18–42. [24] Lions, J. L., Espaces intermédiaires entre espaces Hilbertiens et applications, Bull. Math. de la Soc. Sci. Math. Phys. de la R. P. R. 2 (1958), 419–432. [25] Lions, J. L., Magenes, E., Non-homogeneous boundary value problems and applications 1, Springer 1972. [26] Löwner, K., Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177–216. [27] McCarthy, J. E., Geometric interpolation of Hilbert spaces, Ark. Mat. 30 (1992), 321–330. [28] Mityagin, B., An interpolation theorem for modular spaces, Mat. Sbornik 66 (1965), 473– 482. [29] Murphy, G. J., C∗-algebras and operator theory, Academic Press 1991. [30] Ovchinnikov, V. I., The method of orbits in interpolation theory, Math. Rep. 1 (1984), 349–515. [31] Peetre, J., On interpolation functions III, Acta Szeged 30 (1969), 235–239. [32] Peetre, J., Two new interpolation methods based on the duality map, Acta Math. 173 (1979), 73–91. [33] Pusz, W., Woronowicz, S. L., Functional calculus for sesquilinear forms and the puriﬁcation map, Rep. Math. Phys. 8 (1975), 159–170. [34] Rosenblum, M., Rovnyak, J., Hardy classes and operator theory, Dover 1997. [35] Sedaev, A., Description of interpolation spaces for the pair (Lp(a0),Lp(a1)), Soviet Math. Dokl. 14 (1973), 538–541. [36] Sedaev, A., Semenov, E. M., On the possibility of describing interpolation spaces in terms of the K-functional of Peetre, Optimizacja 4 (1971), 98–114. [37] Shechter, M., Complex interpolation, Compositio Math. 18 (1967), 117–147. [38] Sparr, G., A new proof of Löwner’s theorem, Math. Scand 47 (1980), 266–274. [39] Sparr, G., Interpolation of weighted Lp spaces, Studia Math. 62 (1978), 229–271. [40] Uhlmann, A., Relative entropy and the Wigner–Yanase–Dyson–Lieb concavity in an interpolation theory, Commun. Math. Phys. 54 (1977), 21–32. [41] Vasudeva, H., On monotone matrix functions of two variables, Trans. Amer. Math. Soc. 176 (1973), 305–318.

Department of Mathematics, Faculty of Science, Lund University, P.O. Box 118, 221 00 Lund, Sweden E-mail address: [email protected]