INTERPOLATION BETWEEN HILBERT SPACES YACIN AMEUR Abstract. This note comprises a synthesis of certain results in the theory of exact interpolation between Hilbert spaces. In particular, we examine various characterizations of interpolation spaces and their relations to a number of results in operator-theory and in function-theory. 1. Interpolation theoretic notions 1.1. Interpolation norms. When X, Y are normed spaces, we use the symbol (X; Y ) to denote the totality of bounded linear maps T : X Y with the operator normL → T = sup T x ; x 1 . k k L(X;Y ) {k kY k kX ≤ } When X = Y we simply write (X). L Consider a pair of Hilbert spaces = ( 0, 1) which is regular in the sense that is dense in as well as in H . WeH assumeH that the pair is compatible, i.e., H0 ∩H1 H0 H1 both i are embedded in some common Hausdorff topological vector space . WeH define the K-functional (1) for the couple by M H 2 2 K (t, x)= K t, x ; = inf x0 0 + t x1 1 , t> 0, x . H x=x0+x1{k k k k } ∈M The sum of the spaces 0 and 1 is defined to be the space consisting of all x H 2 H ∈M such that the quantity x Σ := K (1, x) is finite; we denote this space by the symbols k k Σ=Σ( )= + . H H0 H1 We shall soon see that Σ is a Hilbert space (see Lemma 1.1). The intersection ∆ = ∆( )= H H0 ∩ H1 arXiv:1401.6090v5 [math.FA] 29 Jun 2019 is a Hilbert space under the norm x 2 := x 2 + x 2. k k∆ k k0 k k1 A map T : Σ( ) Σ( ) is called a couple map from to if the restriction of H → K H K T to maps boundedly into for i =0, 1. We use the notations T ( ; ) Hi Hi Ki ∈ L H K or T : to denote that T is a couple map. It is easy to check that ( ; ) is a BanachH → K space, when equipped with the norm L H K (1.1) T := max T . k k L( H ; K ) j=0,1{k k L( Hj ;Kj ) } If T 1 we speak of a contraction from to . k k L( H ; K ) ≤ H K A Banach space X such that ∆ X Σ (continuous inclusions) is called ⊂ ⊂ intermediate with respect to the pair . H 1 More precisely, this is the quadratic version of the classical Peetre K-functional. 1 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 interme- diate 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).