Clark Measures on Bounded Symmetric Domains

Given a bounded symmetric domain $D$, we study (positive) pluriharmonic functions on $D$ and investigate a possible analogue of the family of Clark measures associated with a holomorphic function from $D$ into the unit disc in $\mathbb C$.

= iIm α + ϕ(0) α − ϕ(0) for every z ∈ D. The measures µ α , α ∈ T are called Clark (or Aleksandrov-Clark) measures, and were introduced by D. N. Clark in [13] in order to study the restricted shift operator.Their main properties were then established by A. B. Aleksandrov (cf., e.g., [2,3,4,5,6]).Among these, µ α is singular (with respect to the normalized Haar measure β T on T) if and only if ϕ is inner, that is, ϕ has radial limits of modulus 1 at almost every point in T. In addition, in the sense of integrals of positive measures.This is known as Aleksandrov's disintegration theorem, since its first formulation was only concerned with inner functions, in which case the above integral of measures is indeed a disintegration of β T relative to its pseudo-image measure β T under ϕ (more precisely, under the boundary value function associated with ϕ).Cf., e.g., [12,28] for an account of the classical theory of Clark measures.
Clark measures proved to be a valuable tool in the study of both the theory of holomorphic functions in the unit disc and in the study of contractions.It is then natural to investigate whether they may be extended to more general settings.If U denotes the unit ball in C n , then two different possible analogues of Clark measures were introduced.On the one hand, inspired by the link with the study of contractions, M. T. Jury and then R. T. W. Martin (cf.[15,16,17]) developed a theory of operator-valued Clark measures which may be used to study row contractions.These measures are in some sense linked to the choice of the Drury-Arveson space as the 'correct' generalization of the Hardy space on D. On the other hand, more recently A. B. Aleksandrov and E. Doubtsov (cf.[7,8]) introduced another possible extension of Clark measures associated with holomorphic functions ϕ : U → D, defined as the positive Radon measures µ α on ∂U such that for every z ∈ U , where P denotes the Poisson integral operator (on ∂U ).It is important to notice that, whereas in the classical case every non-zero positive Radon measure on T is the Clark measure associated with some holomorphic function ϕ : D → D, in this more general situation this is no longer the case.In fact, the Poisson integral Pµ α must be a positive pluriharmonic function (and for this reason measures such as µ α are sometimes called 'pluriharmonic').This property dictates some severe restrictions on µ α (essentially since the Poisson kernel is far from being pluriharmonic in each variable).For example, as shown in [7], if π denotes the canonical projection of ∂U onto the projective space P n−1 and β denotes the image of the normalized measure β on ∂U under π, then β is a pseudo-image measure of µ α under π (that is, a subset N of P n−1 is β-negligible if and only if π −1 (N ) is µ α -negligible), and µ α has a disintegration relative to β.
In this paper we shall generalize further this second approach to Clark measures to the case of bounded symmetric domains (of which D and U are the simplest examples).In fact, we shall fix a circular (hence convex, cf.[24,Corollary 4.6]) bounded symmetric domain D ⊆ C n , that is, a (circular convex) connected open subset of C n such that for every z ∈ D there is a (unique) holomorphic involution having z as an isolated (actually, unique) fixed point.It is known that D is then homogeneous (that is, that the group of its biholomorphism acts transitively on D), but converse is false (even though every homogeneous bounded domain which has some 'canonical' convex realization is necessarily symmetric, cf.[18]).The reason why we chose to deal with bounded symmetric domains is twofolod: one the one hand, the Poisson integral operator on a bounded symmetric domain D is well studied, and enjoys several nice properties which allow, for example, to ensure that the correspondence between inner functions and singular Clark measures be preserved; on the other hand, every bounded symmetric domain has a circular convex realization, and this allows to perform the disintegration trick introduced in [1] which allows to extend several results from positive measures on T to positive pluriharmonic measures on the Šilov boundary bD of D. Given a holomorphic function ϕ : D → D and α ∈ T, we shall then define the Clark measure µ α (on bD) so that Re α + ϕ α − ϕ = Pµ α on D, where P denotes the Poisson integral operator on bD.Denote by β the unique normalized positive Radon measure on bD which is invariant under all linear automorphisms of D. As for the case of the unit ball U , denoting by π : bD → bD the canonical projection of bD onto its quotient under the action (α, ζ) → αζ of the torus, the image β of β under π is a pseudo-image measure of µ α under π.In addition, µ α has a vaguely continuous disintegration (µ α,ξ ) ξ∈ bD relative to β, the µ α,ξ being nothing but the classical Clark measures (on π −1 (ξ)) associated with the restriction of ϕ to the disc Dπ −1 (ξ).As before, this allows to show that µ α is absolutely continuous with respect to the Hasudorff measure H m−1 , where m denotes the (real) dimension of the (real) analytic manifold bD.
One may then prove the analogues of several properties of the classical Clark measures.We mention, though, that the results in [7,8] are sharper, since in the case of the unit ball U more tools are available, such as Henkin's and Cole-Range's theorem, whose validity on more general symmetric domains is unknown to us.As a matter of fact, both Henkin's and Cole-Range's theorem do not hold in any reducible bounded symmetric domain which admits non-trivial inner functions (cf.Proposition 5.6).
Here is a plan of the paper.In Section 2, we review some basic properties of the Poisson integral operator on bD and prove some basic properties of pluriharmonic measures.In Section 3, we prove some results on the disintegration of pluriharmonic measures, as well as some consequences.In Section 4, we introduce and study the analogue of Clark measures described above.In particular, we shall consider the connection with possible analogues of the classical model and de Branges-Rovnyak spaces.In Section 5, we summarize some implications between the theory of Henkin measures and Clark measures.In Section 6, we state the analogue of the characterization of the essential norm for composition operators; we provide no proofs since they amount to an almost word-by-word transcription of [7,Section 6].In Section 7, we briefly review the classical correspondence between Clark measures and angular derivatives, making use of the recent results proved in [25].Finally, in the appendix we provide some background information on the disintegration of (not necessarily positive) measures.
The author would like to thank M. M. Peloso for several discussions on the subject.

Poisson Integrals
Notation 2.1.We denote by D a bounded, circular, 1 and convex symmetric domain in a complex space C n , and by bD its Šilov boundary, that is, the smallest closed subset of ∂D such that sup D |f | = max bD |f | for every f ∈ Hol(D) which extends by continuity to the closure Cl(D) of D. We denote by K the (compact) group of linear automorphisms of D, so that K acts transitively on bD (cf., e.g., [24, Corollary 4.9, Theorem 5.3, and Theorem 6.5]).We assume that K ⊆ U (n) and that bD ⊆ ∂B(0, 1).We denote by D the unit disc in C, and by T its boundary, that is, the torus.We denote by β T the normalized Haar measure on T. Remark 2.2.Notice that, under the above assumptions, D ⊆ B(0, 1) (cf., e.g., [24,Theorem 6.5]).Definition 2.3.We denote by β the unique K-invariant probability measure on bD. 2 In addition, we denote by bD the quotient of bD by the canonical action of the torus T, by π : bD → bD the canonical projection, and by β := π * (β) the image of β under π.
For every ξ ∈ bD, define We denote by β ξ the T-invariant normalized measured on π −1 (ξ), that is, the image of β T under the mapping T ∋ α → αζ ∈ bD, for every ζ ∈ π −1 (ξ).We shall also interpret β ξ as a Radon measure on bD, supported in π −1 (ξ).Notation 2.4.As a general rule, we shall use the letters z and ζ to denote elements of D and bD, respectively, ξ to denote an element of bD, ρ to denote an element of [0, 1), and w, α to denote elements of D and T, respectively.We hope that these conventions may help the reader navigate through the manuscript.
Proof.Observe that the mapping ξ → β ξ is vaguely continuous, so that is a well defined probability Radon measure.Now, for every k ∈ K, 2 Observe that the Hausdorff measure H m (on bD, relative to the Euclidean distance), where m is the (real) dimension of the real analytic manifold bD, is since β is K-invariant.Thus, β ′ is K-invariant.Since K acts transitively on bD, and since both β and β ′ are probabily measures, this shows that β = β ′ .Notation 2.6.We denote by P R the space of polynomial mappings from the real space underlying C n into C.In addition, we denote by P C and P C the spaces of holomorphic and anti-holomorphic polynomials on C n , respectively.
In addition, we shall denote by M 1 (bD) the space of Radon measures on bD, endowed with the norm µ → |µ|(bD).
Remark 2.7.The canonical image of P R is dense in C(bD) by the Stone-Weierstrass theorem, hence also in L 2 (β).
Recall that a reproducing kernel Hilbert space of holomorphic functions on D is a Hilbert space H which embeds continuously into the space of holomorphic functions Hol(D) on D (endowed with the topology of compact convergence).In this case, point evaluations are continuous, so that there is a function k : D×D → C such that k( • , z) ∈ H and f (z) = f |k( • , z) H for every z ∈ D and for every f ∈ H.It turns out that k(z, z ′ ) = k(z ′ , z) for every z, z ′ ∈ D and that the set of k( • , z), as z runs through D, is total (that is, generates a dense vector subspace) in H. Furthermore, if (e j ) is an orthonormal basis of H, then k = j e j ⊗ e j , with locally uniform convergence on D × D. Definition 2.8.We denote by C the Cauchy-Szegő kernel of D, that is, the reproducing kernel of the Hardy space and define the Poisson-Szegő kernel as3 Define, for every (Radon) measure µ on bD, , and H(f • β), respectively.In order to simplify the notation, we shall also simply write Cµ, Pµ, and Hµ instead of C(µ), P(µ), and H(µ), respectively.Remark 2.9.When D = D, one has α + w α − w for every w ∈ D and for every α ∈ T. In this case, Pµ = Re Hµ for every real measure µ on T.
Lemma 2.10.The following hold: (1) if P, Q ∈ P C are homogeneous with different degrees, then Proof.
(1) Let h and k be the degrees of P and Q.Then, for every θ ∈ T, , thanks to the K-invariance of β and to the circularity of D. Therefore, P |Q L 2 (β) = 0.
Definition 2.11.Define C (k) , for every k ∈ N, as the reproducing kernel of the space of homogeneous holomorphic polynomials of degree k, endowed with the scalar product induced by H 2 (D).We define C (k) (µ), for every Radon measure µ on bD, accordingly.
Remark 2.12.Notice that for every z, z ′ ∈ D, with locally uniform convergence on D × Cl(D).When D is irreducible, this follows from [14, Theorem 3.8 and the following Remark 1]; the general case may be deduced from the irreducible one.
Definition 2.13.Given a function f on D, we define 1).In addition, we define f 1 as the pointwise limit (where it exists) of the f ρ for ρ → 1 − .Remark 2.14.Take f ∈ C(D).Then, the domain of f 1 is a Borel subset of bD and f 1 is Borel measurable thereon.
Proof.The domain B of f 1 is the set is closed for every j, h ∈ N, this proves that B is a Borel subset of bD.Then, f 1 is the pointwise limit of the sequence of the restrictions of the (6) for every Radon measure µ on bD, (Pµ) 1 is defined almost everywhere and (Pµ) 1 •β is the absolutely continuous part of µ with respect to β.
Proof.Observe that, when D is irreducible, [14, Theorem 3.8 and the following Remark 1] shows that there is a sesqui-holomorphic polynomial P : C n × C n → C such that C 2 = P −1 on D × D, and such that P vanishes nowhere on D × Cl(D) (here we are using the fact that 2n/r is an integer if D is irreducible, where r denotes the rank of D).This proves (1) and ( 2) in this case.The general case then follows.Then, (3) to (5) In addition, for every h, k ∈ N and for every whence the result when P ∈ P C , since C(z ′ , 0) = 1.When P ∈ P C , applying a conjugate leads to the same conclusion, whence the first assertion.Now, take P, Q ∈ P C and h, k ∈ N, and let us prove that bD where one must interpret In a similar way (or taking conjugates), one shows that bD when k > h, and C (h−k) (P ) = 0 when k < h, by Lemma 2.10, so that the second assertion follows.
Remark 2.17.Take f ∈ Hol(D).Then, the homogeneous expansion of f converges (to f ) locally uniformly on D.
Proof.Let j P j be the homogeneous expansion of f , so that P j = C (j) (f ) for every j ∈ N. Take ρ ∈ (0, 1).Since f ρ ∈ C(bD), by means of Remark 2.12 we see that j ρ j P j converges to f (ρ • ) locally uniformly on D. The assertion follows by the arbitrariness of ρ.Definition 2.18.Let V be a subset of P R .Then, we define P R ∩V ⊥ := P ∈ P R : ∀Q ∈ V P |Q L 2 (β) = 0 .Proposition 2.19.Let µ be a real measure on bD.Then, the following conditions are equivalent: (1) P(µ) is pluriharmonic; (2) P(µ) = Re H(µ); (3) µ, P = 0 for every Recall that a function is pluriharmonic if its restriction to every complex line is harmonic.In addition, a real-valued function is pluriharmonic if and only if it may be written locally as the real part of a holomorphic function (cf., e.g., [23,Proposition 2.2.3]).Since D is convex, every real-valued pluriharmonic function on D is (globally) the real part of a holomorphic function on D.
Proof.(1) =⇒ (4).Observe that (Pµ) ρ • β converges vaguely to µ as ρ → 1 − by Proposition 2.15, so that it will suffice to prove our assertion for (Pµ) ρ • β, for every ρ ∈ (0, 1).Since P(µ) is pluriharmonic and D is convex, there is a holomorphic function f on D such that P(µ) = Re f .In addition, the homogeneous expansion j P j of f converges locally uniformly to f on D (cf.Remark 2.17).In particular, (Pµ) ρ is the uniform limit of the sequence j N Re (ρ j P j ) of elements of Re P C .
(4) =⇒ (3).It will suffice to prove that P |Q L 2 (β) = 0 for every P ∈ Re P C and for every Q ∈ P R ∩ (P C ∪ P C ) ⊥ = P R ∩ (P C + P C ) ⊥ .Since Re P C ⊆ P C + P C , the assertion follows.
(4) =⇒ (3).It will suffice to observe that P |Q L 2 (β) = 0 for every P ∈ P C and for every for every z ∈ D. In addition, Remark 2.12 implies that uniformly on bD, for every z ∈ D. In addition, it is clear that, for every where for every z ∈ D, whence (2).
Definition 2.21.We say that a complex measure µ on bD is pluriharmonic or holomorphic if P(µ) is pluriharmonic or holomorphic, anti-holomorphic, respectively.
Remark 2.22.Take f ∈ C(bD).Then, the following conditions are equivalent: (1) f • β is holomorphic; (2) there is g ∈ Hol(D) such that g ρ → f uniformly on bD for ρ → 1 − ; (3) f belongs to the closed vector space generated by P C in C(bD); (4) f belongs to the closed vector space generated by the C( • , z), z ∈ D, in C(bD). Proof.
(2) =⇒ (3).It suffices to prove the assertion for the g ρ , ρ ∈ (0, 1).Then, g ρ is the uniform limit of (the restriction to bD of) the homogeneous expansion of g, by Remark 2.17, whence the assertion.
(3) =⇒ (4).Let V be the closed vector subspace of C(bD) generated by the C( • , z), z ∈ D, and observe that V is T-invariant.Since the action of T in C(bD) is continuous, we see that V contains for every z ∈ D and for every k ∈ N (cf.Lemma 2.10).By the reproducing properties of the C (k) , this proves that V contains P C , whence the assertion.
(4) =⇒ (1).It suffices to prove that C( • , z) • β is in the vague closure of P C • β for every z ∈ D, by Proposition 2.20.It suffices to take the homogeneous expansion of C( • , z), thanks to Remark 2.12.
Proposition 2.23.The mapping µ → P(µ) induces a bijection of the space of positive pluriharmonic measures on bD onto the space of positive pluriharmonic functions on D.
Equivalently, the mapping µ → H(µ) induces a bijection of the space of positive pluriharmonic measures on bD onto the space of holomorphic functions on D with positive real part and which are real at 0.
Cf. [22,Theorem 1], where the same theorem is stated for some more specific domains, but extension to this case (or more general ones) is indicated as possible with essentially the same techniques.
Proof.The fact that the mapping µ → P(µ) is one-to-one follows from Proposition 2.15.In addition, the fact that the second assertion follows from the first one is a consequence of Proposition 2.19.In order to conclude, it will then suffice to prove that, if f is a positive pluriharmonic function on D, then there is a positive (necessarily pluriharmonic) measure µ on D such that f = P(µ).Let g be a holomorphic function such that Re g = f .Take ρ ∈ (0, 1) and observe that . Therefore, given an ultrafilter U on (0, 1) which is finer than the filter 'ρ → 1 − ', the f ρ • β converge vaguely to some positive Radon measure µ as ρ runs along U. Since P(f ρ )(z) = f (ρz) for every ρ ∈ (0, 1) and for every z ∈ D, the assertion follows passing to the limit.
Proposition 2.25.The mapping µ → e −H(µ) induces a bijection of the set of positive pluriharmonic measures on bD which are singular with respect to β, and the set of nowhere-vanishing inner functions on D which are > 0 at 0.
Proof.If µ is a positive pluriharmonic measure on bD which is singular with respect to β, then Re H(µ) 0 on D and Re (Hµ) 1 = (Pµ) 1 = 0 β-almost everywhere, thanks to Propositions 2.15 and 2.19.Therefore, Conversely, let ϕ be a nowhere-vanishing inner function on bD which is > 0 at 0. Since D is convex, − log ϕ may be defined uniquely as a holomorphic function on D which is real at 0. Then, 23 shows that there is a positive pluriharmonic measure µ on bD such that H(µ) = − log ϕ.Since Re (− log(ϕ 1 )) = − log|ϕ 1 | = 0 β-almost everywhere, Propositions 2.15 and 2.19 show that µ is singular with respect to β.

Disintegration of Pluriharmonic Measures
Theorem 3.1.Let µ be a non-zero pluriharmonic measure on bD.Then, β is a pseudo-image measure of µ under π.
In addition, if we denote with (µ ξ ) a disintegration of µ relative to β, then the following hold: • for β-almost every ξ ∈ bD and for every z ∈ D ξ , • if µ is holomorphic, then µ ξ is holomorphic (in particular, absolutely continuous with respect to β ξ ) for almost every ξ ∈ bD.
The proof extends [7, Proposition 2.1].Notice that we assume that µ be non-zero only to ensure that β be a pseudo-image measure of µ under π.
Proof.Observe that P(µ) is a pluriharmonic function on D. Consequently, the restriction (Pµ) (ξ) of P(µ) to D ξ is harmonic for every ξ ∈ bD.In addition, by monotone convergence, Proposition 2.15, and Remark 2.5 (since |Pµ| is subharmonic by [23,Corollary 2.1.14]).Then, (Pµ) (ξ) belongs to the harmonic Hardy space H 1 (D ξ ) for almost every ξ ∈ bD, so that there is a measure µ ξ on bD, supported in π −1 (ξ), such that for every z ∈ D ξ .In addition, Proposition 2.15 and the previous remarks show that bD where the last equality follows from the dominated convergence theorem, since bD for β-almost every ξ ∈ bD, and bD by the previous remarks.Now, let N be the set of ζ ∈ bD such that µ π(ζ) is defined and vanishes.Then, (Pµ)(ρζ) = 0 for every ρ ∈ [0, 1) and for every ζ ∈ N .Assume by contradiction that β(N ) > 0. Since (Pµ) ρ is real analytic, this implies that Pµ vanishes on ρbD for every ρ ∈ (0, 1).Since (Pµ)(ρ • ) is holomorphic on a neighbourhood of Cl(D) and since bD is the Šilov boundary of D, this implies that P(µ) vanishes on ρD for every ρ ∈ (0, 1), so that P(µ) = 0 and µ = 0: contradiction.Then, N is β-negligible.The fact that β is a pseudo-image measure of µ under π and that (µ ξ ) is actually a disintegration of µ then follows from Propositions 8.3 and 8.6.
Finally, the fact that, if µ is holomorphic, then almost every µ ξ is holomorphic (hence absolutely continuous with respect to β ξ ) follows from the fact that P(µ) is then holomorphic, so that also (Pµ) (ξ) is holomorphic for every ξ ∈ bD.Corollary 3.2.Let µ be a holomorphic measure on bD.Then, µ is absolutely continuous (with respect to β).
Proof.This follows from Remarks 8.5 and 2.5 and Theorem 3.1.
Corollary 3.3.Let µ be a pluriharmonic measure on bD.Then, µ is absolutely continuous with respect to H m−1 , where m denotes the dimension of the (real) analytic manifold bD. 4n particular, µ cannot have any atoms unless n = 1.Notice that this does not mean that µ has a density with respect to H m−1 , since this latter measure is far from being σ-finite on bD.
Proof.Let E be an H m−1 -negligible subset of bD.Observe first that, by a simple extension of [9, remarks following Theorem 2], one may prove that E × T is H m -negligible (with respect to the distance Consequently, β(π(E)) = 0, so that E is µ-negligible since either µ = 0 or β is a pseudo-image measure of µ under π by Theorem 3.1.Proposition 3.4.Let µ be a non-zero positive pluriharmonic measure on bD.Then, µ has a vaguely continuous disintegration (µ ξ ) relative to its pseudo-image measure β.In addition, for every ξ ∈ bD and for every z ∈ D ξ .
Proof.By Proposition 2.23, for every ζ ∈ bD there is a unique positive measure µ ′ ζ on T such that for every w ∈ D, so that µ ′′ ζ = µ ′′ αζ for every α ∈ T. In addition, if ξ = π(ζ) and µ ξ := µ ′′ ζ , then (µ ξ ) is a disintegration of µ relative to its pseudo-image measure β by Theorem 3.1.It will then suffice to show that the mapping bD ∋ ζ → µ ′′ ζ ∈ M 1 (bD) is vaguely continuous.In other to do that, observe first that The fact that the mapping bD ∋ ζ → (Pµ)(wζ) ∈ C is continuous for every w ∈ D and the density of the vector space generated by the functions is continuous.By the arbitrariness of f , this completes the proof.
The following result is an example of the properties of measures on T which may be transferred to pluriharmonic measures on bD by means of disintegrations.The classical result is konwn as Poltoratski's distribution theorem (cf.[27]).Cf. [7, Proposition 2.12] for the case of the unit ball.Proposition 3.6.Let µ be a pluriharmonic measure on bD.Then, in the vague topology, where µ s denotes the singular part of µ with respect to β.
In particular, lim Proof.We may assume that µ is non-zero.Take a disintegration (µ ξ ) of µ relative to its pseudo-image measure β, by Theorem 3.1.Observe that, by Proposition 3.5, it is clear that (Cµ) 1 is well defined β-almost everywhere, and that for almost every ξ ∈ bD, and for β ξ -almost every ζ ∈ bD, where C ξ denotes the Cauchy integral on π −1 (ξ) (considered as the boundary of D ξ ).Therefore, for every f ∈ C(bD), by Remark 2.5.Now, observe that the mapping ν → (C T ν) , and is continuous by the closed graph theorem (cf.[27, Theorem 1]), where C T denotes the Cauchy integral on T. Consequently, there is a constant C > 0 such that for every y > 0 and for every f ∈ C(bD).Since the function ξ → µ ξ M 1 (bD) is β-integrable, by the dominated convergence theorem and [27, Theorem 1] we see that, for every f ∈ C(bD), where µ s ξ denotes the singular part of µ ξ with respect to β ξ (cf.Remark 8.5).The proof is complete.

Clark Measures
Definition 4.1.Take a holomorphic function ϕ : D → D.Then, Proposition 2.23 shows that, for every α ∈ T, there is a unique positive pluriharmonic measure µ α on bD such that We shall also write µ α [ϕ] instead of µ α .In particular, µ is singular with respect to β if and only if ψ is inner.
Proof.By Proposition 2.23, H(µ) is a holomorphic function, is strictly positive at 0, and maps D into the right half-plane The fact that ψ is unique follows from the fact that h • ψ is determined by the relation Re (h • ψ) = P(µ).
Proof.This follows from the equality Proposition 4.6.For every α ∈ T, the absolutely continuous part of µ α (with respect to β) has density In particular, ϕ is an inner function if and only if µ α is singular with respect to β for some (or, equivalently, every) α ∈ T.
Proof.This follows from Proposition 2.15.
By the previous remarks, the second assertion will be established once we prove that for every z ∈ D. To this aim, observe that Proposition 2.15 shows that so that the second assertion follows.Now, assume that ϕ is inner.Then, Proposition 4.7 shows that µ α is concentrated in ϕ −1 1 (α) for every α ∈ T. Since µ α = 0 for every α ∈ T by Lemma 4.4, Proposition 8.6 shows that β T is a pseudo-image measure of β under ϕ 1 , and that (µ α ) is a disintegration of β relative to β T .If, in addition, ϕ(0) = 0, then µ α is a probability measure for every α ∈ T by Lemma 4.4, so that β T is actually the image measure of β under ϕ 1 .Proposition 4.10.Let ψ : D → D be a holomorphic map.Then, for every α ∈ T, Proof.Observe first that the mapping α ′ → µ α ′ [ϕ] is vaguely continuous by Theorem 4.9, so that the positive measure ν := T µ α ′ [ϕ] dµ α [ψ](α ′ ) is well defined (and pluriharmonic by Proposition 2.19).Then, observe that for every z ∈ D. This completes the proof.
Conserning the second assertion, we have to prove that z,z ′ ∈D a z a z ′ C ϕ (z, z ′ ) 0 for every (a z ) ∈ C (D) .However, where a ′ z := (1 − αϕ(z))a z for every z ∈ D. One may also see this fact as a consequence, e.g., of [33, Lemma 3.9].Definition 4.13.We define C ϕ as in Proposition 4.12, and ϕ * (H 2 (D)) as the unique Hilbert space with reproducing kernel C ϕ . 5We also define, for every α ∈ T, Furthermore, we define H 2 (µ α ) as the closure of P C (or, equivalently, of the vector space generated by the C( • , z), z ∈ D, cf.Remark 2.22) in L 2 (µ α ).
The space ϕ * (H 2 (D)) is thus an analogue of the de Branges-Rovnyak spaces, of model spaces when ϕ is inner.Notice that, when D is the unit ball, a different possible analogue of model spaces has been proposed in [7] (alongside the present one).
Proof.Observe that, by Proposition 4.12, 5 Thus, ϕ * (H 2 (D)) is the completion of the vector space generated by the Cϕ( • , z), z ∈ D, with respect to the scalar product for every z, z ′ ∈ D. Therefore, for every z, z ′ ∈ D, thanks to Proposition 4.12.Consequently, C ϕ,α induces an isometry of H 2 (µ α ) onto ϕ * (H 2 (D)).To conclude, observe that, given f ∈ L 2 (µ α ), one has C ϕ,α (f ) = 0 if and only if Proposition 4.17.Denote by σ α the singular part of µ α with respect to β, and take f ∈ ϕ * (H 2 (D)).Then, almost everywhere, where the last equality follows from the fact that ϕ 1 = α σ α -almost everywhere thanks to Proposition 4.7.The same then holds for every f in the vector space V generated by the C ϕ ( • , z), z ∈ D, which is (contained and) dense in ϕ * (H 2 (D)).Now, take an arbitrary f ∈ ϕ * (H 2 (D)).Then, there is a sequence (f (j) ) of elements of V which converges to f in ϕ * (H 2 (D)).Then, f for every α ∈ T. Notice that, since f 1 is Borel measurable and β-almost everywhere defined (and in L 2 (β)), it is in particular µ α -measurable and almost everywhere defined (and in L 2 (µ α )) for β T -almost every α ∈ T, thanks to Theorem 4.9 and [10, Theorem 1 of Chapter V, § 3, No. 3].In addition, f by Fatou's lemma.Consequently, for β T -almost every α ∈ T there is a subsequence of f ϕ,α f (j) σ α -almost everywhere, and C * ϕ,α f (j) converges to C * ϕ,α f in L 2 (µ α ) for every α ∈ T, this proves that C * ϕ,α f = f 1 σ α -almost everywhere for β T -almost every α ∈ T. Now, assume that ϕ is inner and that f 1 ∈ C(bD).Then, Remark 2.22 shows that f 1 belongs to the closure of P C in C(bD), hence to the closure of P C in L 2 (µ α ) for every α ∈ T. In other words, f 1 ∈ H 2 (µ α ) for every α ∈ T. Hence, our assertion will follow in we prove that C ϕ,α f 1 = f for every α ∈ T. By the previous remarks, this holds for β T -almost every α ∈ T. In addition, the mapping is continuous for every z ∈ D, thanks to Theorem 4.9 and Proposition 2.15.Consequently, C ϕ,α f 1 = f for every α ∈ T, as claimed.Definition 4.18.Let µ be a Radon measure on bD.Then, for every p ∈ [1, ∞], we denote by PM p (µ) the space of f ∈ L p (µ) such that f • µ is pluriharmonic.
In addition, for every α ∈ T we define ϕ * (H 2 ) as the space of f ∈ H for every f ∈ L 1 (µ α ) (cf.Corollary 4.11).In addition, ψ extends to a holomorphic mapping on a neighbourhood of Cl(D), so that the mapping (α, w) → on D, for every α ∈ T.
The following result is a simple consequence of [26].
(2)-( 3) Observe that Proposition 4.6 and Lemma 4.21 show that -almost everywhere and in L 2 (µ α,ξ ) for β-almost every ξ ∈ bD (with some abuse of notation).This is sufficient to prove that f ρ converges to f 1 pointwise µ α -almost everywhere, as well as (3).For what concerns the remainder of (2), observe that Lemma 4.21 shows that there is a constant C > 0 such that for β-almost every ξ ∈ bD (where the first equality follows from the fact that ϕ| D ξ is inner for β-almost every and the latter space in turn embeds isometrically into H 2 (D ξ )).The convergence of f ρ to f 1 in L 2 (µ α ) then follows by means of the dominated convergence theorem.
(1) This follows from the fact that, by Proposition 3.5, for β-almost every ξ ∈ bD and for every z ∈ D ξ .

Henkin Measures
Definition 5.1.Denote by A(D) the space of continuous functions on D ∪ bD whose restriction to D is holomorphic.We denote by A(D) ⊥ the polar of A(D) in M 1 (bD), that is, the space of Radon measures µ on bD such that bD f dµ = 0 for every f ∈ A(D).
A Radon measure µ on bD is a Henkin measure (or an A(D) ⊥ -measure) if for every bounded sequence (f (j) ) in A(D) which converges pointwise (hence locally uniformly) to 0 on D.
A positive Radon measure µ on bD is a representing measure if bD f dµ = f (0) for every f ∈ A(D).
A Radon measure µ on bD is totally singular if it is singular with respect to every representing measure on bD.
Notice that every representing measure µ is a probability measure, since µ M 1 (bD) = bD 1 dµ = 1.In addition, total singularity does not change if one considers representing measure associated with different points in D, as in the case of the unit ball (cf.[29, 9.1.3]).
Remark 5.2.Take z 1 , z 2 ∈ D and a positive Radon measure µ 1 on bD such that bD f dµ = f (z 1 ) for every f ∈ A(D).Then, there is a positive Radon measure µ 2 on bD such that bD f dµ 2 = f (z 2 ) for every f ∈ A(D) and such that µ 1 is absolutely continuous with respect to µ 2 .
Proof.By Proposition 2.15, there is c > 0 such that cP(z Finally, it is clear that µ 1 is absolutely continuous with respect to µ 2 , since µ 1 µ 2 /c.Remark 5.3.For every ρ ∈ (0, 1) and for every z, z ′ ∈ Cl(D), In addition, for every ρ ∈ (0, 1) and for every ζ, ζ ′ ∈ bD, Proof.By Remark 2.12, when z, z ′ ∈ D, whence the first assertion by Proposition 2.15.For the second assertion, observe that where the second equality follows from the fact that there is k ∈ K such that ζ = kζ ′ , and C(ρkζ ′ , ρkζ ′ ) = C(ρζ ′ , ρζ ′ ) since k is linear and C is k×k-invariant (as composition with k induces an isometry of H 2 (D)).
Proposition 5.4.Let µ be a Henkin measure on bD, and take ε > 0.Then, there are f ∈ L 1 (β) and This is the so-called Valskii decomposition.The proof is formally identical to that of [29, Theorem 9.2.1] and will not be repeated (one has to make use of Remark 5.3).Definition 5.5.We say that D satisfies property (H) if evey Radon measure which is absolutely continuous with respect to a Henkin measure is itself a Henkin measure.
We say that D satisfies property (CL) if every Henkin measure is absolutely continuous with respect to a representing measure.
Property (H) means that 'Henkin's theorem' holds on D, whereas property (CL) means that 'Cole-Range theorem' holds on D. Proposition 5.6.Assume that D satisfies property (H).Then, the following hold: (1) every positive pluriharmonic measure on bD which is singular to β is totally singular; (2) D satisfies property (CL); (3) if ϕ is a non-trivial inner function on D, then ) for every α ∈ T; (4) either D is irreducible or admits no non-trivial inner functions.
Notice, in addition, that every irreducible bounded symmetric domain which is either of tube type or of rank 1 admits non-trivial inner functions, thanks to [21,Lemma 2.3] and [1].
(4) Assume by contradiction that D is reducible and admits non-trivial inner functions.Then, using [20, Theorem 3.3 and its Corollary] as in the proof of [21,Theorem 3.3 (iii)], we see that there are two non-trivial (circular, convex) symmetric domains D 1 , D 2 such that (up to a rotation) D = D 1 × D 2 , and such that there is a non-trivial inner function ψ on D 1 .Let pr j : D → D j be the canonical projection for j = 1, 2. Observe that ψ • pr 1 is an inner function on D, so that we may consider the positive pluriharmonic measures µ 1 [ψ] on bD 1 and µ 1 [ψ • pr 1 ] on bD.Observe that where β bD 2 denotes the normalized invariant measure on bD 2 .Then, take a holomorphic polynomial P on D 2 such that P (0) = 0, and observe that ) and we may take P so that where σ α denotes the singular part of µ α [ϕ] with respect to β, while and Cf. [7,Theorem 6.4].The proof is formally identical to that of the cited reference (for the continuity, one has to observe that, if f ∈ H 2 (D) and g is a harmonic majorant of |f | 2 , then g • ϕ is a pluriharmonic majorant of |f • ϕ| 2 , so that f • ϕ ∈ H 2 (D); harmonicity, in this context, does not seem to suffice). 6Here, Cϕ  With the notation of Jordan triple systems, B(z, z is the orthogonal projector onto the Peirce space Z 1 (ζ) associated with the maximal tripotent ζ.In addition, h(0)(v, w) = Tr D(v, w), while k(z, z ′ ) = (H n (D) det B(z, z ′ )) −1 (cf.[24,Theorem 2.10], where some slightly different notation is used; we follow the conventions of [25]).Definition 7.2.Let Γ : [0, 1) → D be a continuous curve, and take ζ ∈ bD.Set γ := Γ|ζ , so that γζ is the orthogonal projection of Γ on Cζ.The curve Γ is said to be a then Γ is said to be special.Finally, if one assumes further that γ is non-tangential (in D), then Γ is said to be restricted.
This terminology follows [25] (combined with [11]).Notice that, when D is the unit ball of C n (endowed with the usual scalar product), then one may considerably relax (2) and simply require lim Proof.Observe first that, by [12, Theorems 1.7.10 and 9.2.1 (and the proof of the latter one)], .

Appendix: Disintegration
Definition 8.1.Let X, Y be two locally compact spaces with a countable base, and let µ and ν be complex Radon measures on X and Y , respectively.Let p : X → Y be a µ-measurable map.Then, we say that ν is a (resp.weak) pseudo-image measure of µ under p if a subset N of Y is ν-negligible if and only if (resp.only if) p −1 (N ) is µ-negligible.
We denote by M + (X) the space of positive Radon measures on X.
One may consider a more general setting, but the statements and proofs then become more complicated.
Remark 8.2.Notice that, since X has a countable base, there are µ-measurable functions f on X such that f • µ is a positive bounded measure and f (x) = 0 for every x ∈ X.For example, if h : X → T is a density of µ with respect to |µ| and (K j ) is an increasing sequence of compact subsets of X whose union is X, then one may take f = h j 2 −j |µ|(Kj)+1 χ Kj .Then, µ and f • µ have the same negligible sets.In addition, the image measure p * (f • µ) is a pseudo-image measure of µ under p in the sense of Definition 8.1, thanks to [10, Corollary 2 to Proposition 2 of Chapter V, § 6, No. 2].Then, saying that ν is a pseudo-image measure of µ under p means that the Radon measures ν and p * (f • µ) have the same negligible sets, that is, are equivalent, whereas saying that ν is a weak pseudo-image measure of µ under p means that p * (f • µ) is absolutely continuous with respect to ν. 7   Proposition 8.3.Let X, Y be two locally compact spaces with a countable base, and let µ and ν be two complex Radon measures on X and Y , respectively.Let p : X → Y be a µ-measurable map, and assume that ν is a weak pseudo-image measure of µ under p.Then, there is a family (µ y ) y∈Y of complex Radon measures on X such that the following hold: (1) for ν-almost every y ∈ Y , the measure µ y is concentrated on p −1 (y); (2) for every f ∈ L 1 (X), the function f is µ y -integrable for ν-almost every y ∈ Y , the mapping y → X f dµ y is ν-integrable, and f dµ y dν(y).
(3) for every positive Borel measurable function f on X, the positive function y → X f d|µ y | is Borel measurable, and f d|µ y | d|ν|(y). 7The same assertion may be stated replacing p * (f • µ) with the not necessarily Radon measure p * (|µ|).One the one hand, (4) if we define A := { y ∈ Y : µ y = 0 }, then A is a Borel subset of Y and χ A • ν is a pseudo-image measure of µ under p.Furthermore, if ν ′ is a weak pseudo-image measure of µ under p and (µ ′ y ) is a family of complex Radon measures on X such that (1) and (2) hold with ν and (µ y ) replaced by ν ′ and (µ ′ y ), respectively, then there are a ν ′ -measurable subset A ′ of Y such that χ A ′ • ν ′ is a pseudo-image measure of µ under p, and a locally ν-integrable complex function g on Y such that χ A ′ • ν ′ = g • ν, and µ y = g(y)µ ′ y for χ A • ν-almost every y ∈ Y .
Notice that if p is µ-proper (that is, if p * (|µ|) is a Radon measure) and ν = p * (µ), then µ y is a finite measure and µ y (X) = 1 for ν-almost every y ∈ Y . 8 In addition, observe that the function g in the second part of the statement necessarily vanishes ν-almost everywhere on the complement of A, so that the last assertion may be equivalently stated as 'µ y = g(y)µ ′ y for ν-almost every y ∈ Y .'We chose to use χ A • ν in order to stress the fact that uniqueness is essentially related to pseudo-image measures (and not simply weak pseudo-image measures).y for every y ∈ j K j and |µ y | = 0 otherwise, it is clear that the mapping y → X f d|µ y | is Borel measurable for every positive f ∈ C c (X), so that the assertion for general Borel measurable functions follows by approximation.Therefore, also (3) follows.Finally, (4) follows from the fact that A = y ∈ Y : X 1 d|µ y | = 0 , so that A is Borel measurable by (3) and χ A • |ν| = χ B • |ν| by the previous remarks.
Step II.Now, assume that ν ′ and (µ ′ y ) satisfy the assumptions of the statement.The existence of A ′ may be proved as the existence of B in step I.Then, χ A • ν ′ and χ A ′ • ν ′ have the same negligible sets, so that the existence of g follows (cf.[10,Corollary 4 to Theorem 2 of Chapter V, § 5, No. 5]).In addition, g(y) = 0 for ν-almost every y ∈ A. Next, observe that we may find a countable dense subset D of C c (X).Then, for every bounded ν-measurable function f 2 on Y and for every f 1 ∈ D, the function X f 1 dµ ′ y dν(y).By the arbitrariness of f 2 , this proves that X f 1 dµ y = g(y) X f 1 dµ ′ y for every f 1 ∈ D and for ν-almost every y ∈ Y .By the arbitrariness of f 1 , this proves that µ y = g(y)µ ′ y for ν-almost every y ∈ Y .Definition 8.4.Let X, Y be two locally compact spaces with a countable base, and let µ and ν be two complex Radon measures on X and Y , respectively.Let p : X → Y be a µ-measurable map, and assume ν is 8 Notice that, in general, |ν| = p * (|µ|), so that the |µy| need not be probability measures.
a weak pseudo-image measure of µ under p.Then, we say that a family (µ y ) y∈Y of complex Radon measures on X is a disintegration of µ relative to ν if (1) and (2) of Proposition 8.3 hold.
We say that (µ y ) is a disintegration of µ under p if, in addition, ν = p * (µ) (that is, if p * (|µ|) is a Radon measure and µ y (X) = 1 for ν-almost every y ∈ Y ).Remark 8.5.Let X, Y be two locally compact spaces with a countable base, µ 1 , µ 2 two complex Radon measures on X, ν a complex Radon measures on Y , and p : X → Y a µ 1 -and µ 2 -measurable map such that ν is a weak pseudo-image measure of both µ 1 and µ 2 under p. 9 Let (µ j,y ) be a disintegration of µ j relative to its ν, for j = 1, 2. In addition, let µ 1,y = µ a 1,y + µ s 1,y be the Lebesgue decomposition of µ 1,y with respect to µ 2,y (so that µ a 1,y is absolutely continuous and µ s 1,y is singular with respect to µ 2,y ).Analogously, let µ 1 = µ a 1 + µ s 1 be the Lebesgue decomposition of µ 1 with respect to µ 2 .Then, (µ a 1,y ) and (µ s 1,y ) are disintegrations of µ a 1 and µ s 1 , respectively, relative to their common weak pseudo-image measure ν.Proof.Take a Borel subset B of X and a locally µ 2 -integrable Borel function f on X such that µ a 1 = f • µ 2 and µ s 1 = χ B • µ 1 , so that |µ 2 |(B) = 0. Since both µ a 1 and µ s 1 are absolutely continuous with respect to µ 1 , it is clear that ν is a weak pseudo-image measure of µ a for every bounded µ-measurable function g on X, so that (f • µ 2,y ) is a disintegration of µ a 1 relative to ν.In a similar way one proves that (χ B • µ 1,y ) is a disintegration of µ s 1 relative to ν.Therefore, Proposition 8. ) is a disintegration of µ s 1 relative to ν.Finally, observe that χ A • ν is a weak pseudo-image measure of µ a 1 under p, so that f • µ 2,y = 0 for ν-almost every y ∈ Y \ A. Thus, (µ a 1 ) is a disintegration of µ a 1 relative to ν. Proposition 8.6.Let X, Y be two locally compact spaces with a countable base, µ a complex Radon measure on X, and ν a complex Radon measures on Y .Let (µ y ) y∈Y be a family of complex Radon measures on X such that Y |µ y |(K) d|ν|(y) < +∞ for every compact subset K of X, and such that, for every f ∈ C c (X), the mapping y → X f dµ y is ν-integrable and f dµ y dν(y).
Assume, in addition, that there is a Borel measurable map p : X → Y such that µ y is concentrated on p −1 (y) for ν-almost every y ∈ Y .Then, ν is a weak pseudo-image measure of µ under p and (µ y ) is a disintegration of µ relative to ν.This is based on [7, Section 2.2].
Proof.Let F be the set of µ-integrable functions f on X such that f is µ y -integrable for ν-almost every y ∈ Y , the mapping y → X f dµ y is ν-integrable, and (4) holds, so that C c (X) ⊆ F by assumption.Observe that, if (f j ) is a bounded sequence of elements of F which are supported in a fixed compact subset K of X and converge pointwise to some f on X, then f ∈ F by dominated convergence, thanks to (3).Thus, by transfinite induction, F contains all bounded Borel measurable functions with compact support. 9Notice that, if ν 1 and ν 2 are positive weak pseudo-image measures of µ 1 and µ 2 under p, respectively, then one may choose ν = ν 1 + ν 2 .In other words, this assumption is not restrictive.

Definition 2 . 24 .
By an inner function on D we mean a bounded holomorphic function ϕ : D → C with boundary values of modulus 1 almost everywhere on bD (that is, ϕ 1 (ζ) ∈ T for β-almost every ζ ∈ bD).

Remark 4 . 19 .
Let µ be a Radon measure on bD.Then, PM p (µ) is a closed subspace of L p (µ) for everyp ∈ [1, ∞].Proof.It suffices to observe that, by Proposition 2.19, PM p (µ) is the set of f ∈ L p (µ) such that bD f P dµ = 0 for every P ∈ P R ∩ (P C ∪ P C ) ⊥ .Remark 4.20.Take α ∈ T and a biholomorphism ψ of D. Then,

6. Composition Operators Proposition 6 . 1 .
Let ϕ : D → D be a holomorphic function, and define L (H 2 (D);H 2 (D)),e denotes the essential norm of Cϕ, that is, inf T Cϕ + T L (H 2 (D);H 2 (D)) , where T runs through the set of compact operators from H 2 (D) into H 2 (D).

7. Angular Derivatives Definition 7 . 1 .
Let k be the reproducing kernel of the Bergman space A 2 (D) = Hol(D) ∩ L 2 (D), where D is endowed with the Hausdorff measure H n , and denote by h the Bergman metric, so that h(z)(v, w) = ∂ v ∂ w log k(z, z) for every z ∈ D and for every v, w ∈ C n .Denote by B the Bergman operator, that, the unique sesquiholomorphic polynomial map fromC n × C n into L(C n ) such that h(0)(v, w) = h(z)(B(z, z)v, w)for every z ∈ D and for every v, w ∈ C n .For every ζ ∈ bD and for every a > 0, define the angular regionD a (ζ) = z ∈ D : B(z, ζ)P (ζ, ζ) ′1/2 < a(1 − z 2 ) ,where • denotes the norm on C n whose open unit ball is D, A ′ = max z∈D Az for every A ∈ L(C n ), and P is the homogeneous component of degree 4 of B.