ANGULAR DERIVATIVES ON BOUNDED SYMMETRIC DOMAINS

M. MACKEY AND P. MELLON

Abstract. In this paper we generalise the classical Julia–Wolff–Carathéodory theorem to holomorphic functions defined on bounded symmetric domains.

1. Introduction

Throughout, let Hol(A, B) denote the holomorphic functions from A to B, where A and B are domains in a complex Banach space and let Δ denote the open unit disc in C. A classical theorem of complex analysis, known as the Julia–Wolff–Carathéodory theorem, is the following.

Theorem 1.1. Let f ∈ Hol(Δ, Δ) satisfy

α := lim inf_{ζ→1} \frac{1 − |f(ζ)|^{2}}{1 − |ζ|^{2}} < ∞

and f(ζ_{n}) → 1 for some sequence ζ_{n} converging to 1. Then the angular limit of f at 1 exists and equals 1, and the angular limit of the derivative f′ exists at 1 and equals α.

If one transfers this theorem to the right half plane Π ⊂ C by means of the Cayley transform ζ ↦ \frac{1+ζ}{1−ζ} then the statement becomes the following.

Theorem 1.2. Let F ∈ Hol(Π, Π) and let a := inf_{Π} \frac{Re F (z)}{Re z}. Then

a = ∠- lim_{z→∞} \frac{Re F (z)}{Re z} = ∠- lim_{z→∞} \frac{F (z)}{z} = ∠- lim_{z→∞} F′(z).

Generalisations of these results have evolved in two directions. In 1980, Rudin [16] achieved a complete extension for a function f ∈ Hol(B_{n}, B_{m}), where B_{n} denotes the open unit ball in the Euclidean norm of C^{n} (which is a strictly convex bounded symmetric domain). Let ∂B_{n} denote the boundary of B_{n}. Rudin’s result contains the following.

Theorem 1.3. Let e ∈ ∂B_{n} and let F = (f_{1}, …, f_{m}) ∈ Hol(B_{n}, B_{m}) satisfy F (0) = 0 and

α := lim inf_{z→e} \frac{1 − ∥F (z)∥^{2}}{1 − ∥z∥^{2}} < ∞.

Date: August 1, 2000.

1991 Mathematics Subject Classification. 17C65, 46G20.

This work comprises part of the Ph.D. thesis [13] of the first author who gratefully acknowledges the support of Forbairt Basic Research grant SC/97/614.

If (z_{k})_{k} ⊂ B satisfies lim_{k} z_{k} = e,

lim_{k} \frac{1 − ∥F (z_{k})∥^{2}}{1 − ∥z_{k}∥^{2}} = α

and lim_{k} F (z_{k}) = e′ ∈ ∂B_{m}, then the angular limit of F (z) as z → e is e′ and the restricted angular limit F′_{1}(z)e is αe′, where F_{1}(z) = ⟨F (z), e′⟩.

The Hilbert ball is a prototype for two important classes of domain, namely the strictly pseudoconvex domains and the bounded symmetric domains. The theory of angular limits and angular derivatives for functions on a strictly pseudoconvex domain is well developed (see [1] for a comprehensive account). We concentrate on the second category, that of bounded symmetric domains.

Fan [6], in 1986, proved the following Julia–Wolff–Carathéodory theorem for operator valued holomorphic functions on Δ. H denotes a complex Hilbert space and L(H) is the C^{∗}- algebra of bounded linear operators from H to itself. Π_{L(H)} := {T ∈ L(H) : Re T > 0} is a generalised half-plane in L(H).

Theorem 1.4. Let F ∈ Hol(Π, Π_{L(H)}). Suppose that there exists A = A^{∗} ∈ L(H) with

\frac{Re F (z)}{Re z} > A

for all z ∈ Π and such that

inf_{Π} ∥\frac{Re F (z)}{Re z} − A∥ = 0.

Then

A = ∠- lim_{z→∞} \frac{Re F (z)}{Re z} = ∠- lim_{z→∞} \frac{F (z)}{z} = ∠- lim_{z→∞} F′(z).

W�odarczyk [18] generalises Theorem 1.4 by allowing in place of L(H) any J^{∗}-algebra having a non-zero partial isometry. In forthcoming work [14] the authors further extend Theorem 1.4 to the case of JB^{∗}-triples using the concept of Siegel domain in a JB^{∗}-triple in place of Π_{L(H)}. In this paper, we pursue a more general result for holomorphic functions between arbitrary bounded symmetric domains, more in the spirit of Theorem 1.3. Results of this type in the literature are sparse even in finite dimensions, with the case of the polydisc Δ^{n}, that is Hol(Δ^{n}, Δ^{m}), only being recently resolved [2]. A principal reason for this sparsity has been the lack of suitable analogues of either the classical Julia’s lemma or Wolff’s theorem for bounded symmetric domains. These were recently provided in [15] and play a crucial role in achieving angular limit and angular derivative results, in particular motivating the definition of angular approach region for bounded symmetric domains. The existence of angular limits over these regions is then reduced by an extension of the classical Lindelöf–Čirka principle (cf. [16]) to the existence of certain radial limits. A crucial tool throughout is the concept of a Bergman operator on a JB^{∗}-triple.

2. Bounded Symmetric Domains and JB^{∗}-triples

Definition 2.1. A JB^{∗}-triple is a complex Banach space Z with a continuous map {·, ·, ·} : Z^{3} → Z, (x, y, z) → {x, y, z}, which is complex linear and symmetric in x and z, anti-linear in y and satisfies (i) the operator x □x has spectrum in [0, ∞), (ii) exp(i x □x) is both an algebraic automorphism and an isometry, (iii) ∥{x, x, x}∥ = ∥x∥^{3}, for all x ∈ Z, where x □y denotes the linear map z ↦ {x, y, z}.

The equality

{a, b, {x, y, z}} = {{a, b, x}, y, z} − {x, {b, a, y}, z} + {x, y, {a, b, z}} (1)

which ensues from (ii) for all a, b, x, y and z ∈ Z is known as the Jordan triple identity, and provides a weak form of associativity for the triple product. The inequality

∥{x, y, z}∥ ≤ ∥x∥∥y∥∥z∥ (2)

is proved in [7].

Any C^{∗}-algebra, and more generally any J^{∗}-algebra, is a JB^{∗}-triple with triple product given by \{x, y, z\} = \frac{1}{2}(xy^{∗}z + zy^{∗}x) where x^{∗} denotes the usual operator adjoint of x. In particular, every complex Hilbert space is a JB^{∗}-triple whose triple product is given by \{x, y, z\} = \frac{1}{2}(⟨x, y⟩z + ⟨z, y⟩x).

The Bergman operator B(x, y) ∈ L(Z) defined by

B(x, y) = Id −2 x □y + Q_{x}Q_{y}

where Q_{x}(z) = {x, z, x} is an important tool, encoding much of the geometry of Z in the same way that an inner product does for a Hilbert space. On a C^{∗}-algebra, the Bergman operator reduces to B(x, y)z = (1 − xy^{∗})z(1 − y^{∗}x).

An element e ∈ Z for which {e, e, e} = e is called a tripotent and, from (iii) above, a nonzero tripotent has norm one. For example, a tripotent of a C^{∗}-algebra is an element v satisfying v = vv^{∗}v, that is, a partial isometry. Each tripotent induces a splitting of Z, called the Peirce decomposition, into Z = Z_{1} ⊕ Z_{\frac{1}{2}} ⊕ Z_{0} where Z_{k} is the k-eigenspace of e □e, with mutually orthogonal projections P_{k} onto the subspaces Z_{k},

P_{1} = Q_{e}Q_{e},
P_{\frac{1}{2}} = 2 e □e − 2Q_{e}Q_{e},
P_{0} = B(e, e),

satisfying P_{1} + P_{\frac{1}{2}} + P_{0} = Id. Where the need arises, we write P_{j}^{e} rather that P_{j} to highlight the tripotent in question. The tripotent e is called maximal if Z_{0} = {0} and this is the case precisely when e is an extreme point of the unit ball of Z [10]. The tripotent is called unitary if P_{0} = P_{\frac{1}{2}} = 0. We say z ∈ Z is proportional to a tripotent if z = λe for some tripotent e ∈ Z and some λ ∈ C. We point out that since the triple product is continuous, the set of tripotents forms a closed subset of Z.

3. Generalised Horocycles and the angular limit

In Δ the sequence (ξ_{n})_{n} is said to approach a ∈ ∂Δ non-tangentially if ξ_{n} → a and if, for some k > 0, (ξ_{n})_{n} is eventually in the angular region

Δ_{k}(a) = \{ξ ∈ Δ : \frac{|a − ξ|}{1 − |ξ|^{2}} < k\}.

Let f ∈ Hol(Δ, Δ). Then f is said to have angular limit b at a, written ∠-lim_{ξ→a} f(ξ) = b, if f(ξ_{n}) → b whenever ξ_{n} → a non-tangentially.

Let B be a bounded symmetric domain, which we may take to be the open unit ball of a JB^{∗}-triple Z [8]. For e ∈ ∂B and k > 0, we define a generalised angular region by

D_{k}(e) := \{w ∈ B : ∥B(w, e)Q_{e}Q_{e}∥^{\frac{1}{2}} < k(1 − ∥w∥^{2})\}.

Of course, if B = Δ then D_{k}(1) = Δ_{k}(1). We note that e is in the boundary of D_{k}(e) only if B(e, e)Q_{e}Q_{e} = 0, a condition which is equivalent to e being a tripotent. As we wish to consider approach paths to a boundary point e which are contained in the angular region D_{k}(e) (such approach paths will be called non-tangential), we take it that from this point on, e denotes a tripotent of the JB^{∗}-triple Z. The angular region D_{k}(e) can then be rewritten as

D_{k}(e) := \{w ∈ B : ∥B(w, e)|_{Z_{1}}∥^{\frac{1}{2}} < k(1 − ∥w∥^{2})\}

where Z_{1} is the Peirce 1-space of the tripotent e. Note that if Z is a Hilbert space this reduces to D_{k}(e) := {w ∈ B : |1 − ⟨w, e⟩| < k(1 − ∥w∥^{2})}.

Another type of domain which appears naturally in any discussion of angular limits in Δ is the horocycle, which is an open disc in Δ internally tangent to the boundary. The horocycle

E_{λ}(a) = \{ξ ∈ Δ : \frac{|a − ξ|^{2}}{1 − |ξ|^{2}} < λ\}

is a Euclidean disc of radius \frac{λ}{λ+1} internally tangent to ∂Δ at a. In Figure 1, we sketch the angular region Δ_{1.2}(1) and the horocycle E_{\frac{1}{2}} (1) in Δ. Bergman operators are used to generalise

Figure 1. Angular region and horocycle in the unit disc

these horocycles to a bounded symmetric domain B. For a tripotent e in B and λ > 0, define

E_{λ}(e) := {w ∈ B : ∥B_{w}^{−1}B(w, e)Q_{e}Q_{e}∥ < λ}

where B_{w} := B(w, w)^{\frac{1}{2}} is invertible for ∥w∥ < 1 by [8]. We will frequently use the fundamental identity [9] which holds for all x ∈ B,

∥B_{x}^{−1}∥ = \frac{1}{1 − ∥x∥^{2}} . (3)

The next result is a concrete realisation of the horocycles E_{λ}(e). The proof is a distillation of various results and techniques in [15].

Proposition 3.1. Let e ∈ ∂B be a tripotent. For all λ > 0, the horocycle E_{λ}(e) has the form

E_{λ}(e) = (\frac{1}{1 + λ}) e + B(se, se)(B)

for s > 0 satisfying (1 − s^{2})^{2} = \frac{λ}{1+λ}.

Moreover, for any y ∈ B, y ∈ ∂E_{λ_{y}}(e) where λ_{y} := ∥B_{y}^{−1}B(y, e)Q_{e}Q_{e}∥ > 0.

We note that B(se, se) ∈ GL(Z), the group of invertible bounded linear operators on Z. In particular, E_{λ}(e) is a convex domain in B containing e in its boundary.

Proof. It is easy to calculate

B(βe, βe) = P_{0} + (1 − |β|^{2})P_{\frac{1}{2}} + (1 − |β|^{2})^{2}P_{1} (4)

and

B(βe, e) = P_{0} + (1 − β)P_{\frac{1}{2}} + (1 − β)^{2}P_{1} (5)

for all β ∈ C. Choose a sequence (α_{k})_{k}, 0 < α_{k} < 1 with lim_{k} α_{k} = 1. Then

B_{α_{k}e}^{−1} = B(α_{k}e, α_{k}e)^{− \frac{1}{2}}

= (P_{0} + (1 − α_{k}^{2})P_{\frac{1}{2}} + (1 − α_{k}^{2})^{2}P_{1})^{− \frac{1}{2}}

= P_{0} + \frac{1}{\sqrt{1 − α_{k}^{2}}} P_{\frac{1}{2}} + \frac{1}{1 − α_{k}^{2}} P_{1}

and therefore

lim_{k}(1 − α_{k}^{2})B_{α_{k}e}^{−1} = P_{1} = Q_{e}Q_{e}. (6)

Take w ∈ E_{λ}(e), that is, ∥B_{w}^{−1}B(w, e)Q_{e}Q_{e}∥ < λ. Then

∥B_{w}^{−1}B(w, α_{k}e)B_{α_{k}e}^{−1}∥ < \frac{λ}{1 − α_{k}^{2}}

for all k large. In the notation of [15], and using [15, Corollary 3.2], we have that w belongs to the Kobayashi ball D_{α_{k}e,r_{k}} for all k large where r_{k} > 0 satisfies (1 − r_{k}^{2})^{−1} = λ(1 − α_{k}^{2})^{−1}. Proposition 2.3 of [15] then implies that w ∈ D_{α_{k}e,r_{k}} = c_{k} + T_{k}(B) for all k large where

c_{k} = (1 − r_{k}^{2})B_{r_{k}α_{k}e}^{−1}(α_{k}e) ∈ B and T_{k} = r_{k}B_{α_{k}e}B_{r_{k}α_{k}e}^{−1} ∈ GL(Z).

From (4) one can then calculate that

c_{k} = \frac{α_{k}}{λ + α_{k}^{2}} e and T_{k} = B(s_{k}e, s_{k}e)

for s_{k} > 0 satisfying (1 − s_{k}^{2})^{2} = λ(λ + α_{k}^{2})^{−1}. Since c := lim_{k} c_{k} = (1 + λ)^{−1}e and T := lim_{k} T_{k} = B(se, se) ∈ GL(Z) for s > 0 satisfying (1 − s^{2})^{2} = λ(1 + λ)^{−1}, the above shows that w ∈ \overline{c + T (B)}. Therefore, E_{λ}(e) ⊂ \overline{c + T (B)}. One may reverse this argument to show that c + T (B) ⊂ \overline{E_{λ}(e)} and hence E_{λ}(e) = c + T (B). As T ∈ GL(Z), it follows that c + T (B) is a convex domain in B.

T (e) = B(se, se)e = (1 − s^{2})^{2}e = λ(1 + λ)^{−1}e

implies that e ∈ ∂E_{λ}(e).

Fix y ∈ B. Since ∥y∥∥e∥ < 1, B(y, e) is invertible [8] and hence

λ_{y} := ∥B_{y}^{−1}B(y, e)Q_{e}Q_{e}∥ ≥ \frac{∥Q_{e}Q_{e}∥}{∥B(y, e)^{−1}B_{y}∥} = \frac{1}{∥B(y, e)^{−1}B_{y}∥} > 0.

It is then clear from the definition that y ∈ ∂E_{λ_{y}}(e).

Note 3.2. For a tripotent e, one can see from the above formulation of E_{λ}(e) that ∥x− \frac{e}{1+λ}∥ < \frac{λ}{1+λ} for x ∈ P_{1}(E_{λ}(e)). In particular, ∥x − e∥ < \frac{2λ}{1+λ} ≤ 2λ for x ∈ P_{1}(E_{λ}(e)).

The following Julia type lemma for a holomorphic function between bounded symmetric domains B and B′, contained in JB^{∗}-triples Z and Z′, is of a type first proved in [15].

Lemma 3.3. Let f : B → B′ be holomorphic. Let e ∈ ∂B be a tripotent. If there exists (z_{k})_{k} ⊂ B such that z_{k} → e and f(z_{k}) → e′ ∈ ∂B′ where each z_{k} and each f(z_{k}) is proportional to a tripotent, and

α := lim inf_{k→∞} \frac{1 − ∥f(z_{k})∥^{2}}{1 − ∥z_{k}∥^{2}} < ∞

then f(E_{λ}(e)) ⊂ E_{αλ}(e′).

Proof. Since the tripotents form a closed set in any JB^{∗}-triple it follows immediately that e′ is also a tripotent. Since z_{k} = α_{k}e_{k} for α_{k} ∈ C and e_{k} a tripotent, we have from (4) that

(1 − ∥z_{k}∥^{2})B_{z_{k}}^{−1} = (1 − |α_{k}|^{2})P_{0}^{e_{k}} + (1 − |α_{k}|^{2})^{\frac{1}{2}} P e_{k}_{\frac{1}{2}} + P_{1}^{e_{k}}

and thus (1 − ∥z_{k}∥^{2})B_{z_{k}}^{−1} → P_{1}^{e} = Q_{e}Q_{e}. Similarly, (1 − ∥f(z_{k})∥^{2})B_{f(z_{k})}^{−1} → P_{1}^{e′} = Q_{e′}Q_{e′}. The following Schwarz–Pick type result holds [15, Corollary 3.3] as a consequence of the Schwarz lemma,

∥B_{f(w)}^{−1}B(f(w), f(z))B_{f(z)}^{−1}∥ ≤ ∥B_{w}^{−1}B(w, z)B_{z}^{−1}∥ (7)

for all z and w in B. In particular this holds for z = z_{k} and so for all k

∥B_{f(w)}^{−1}B(f(w), f(z_{k})) (1 − ∥f(z_{k})∥^{2})B_{f(z_{k})}^{−1}∥

≤ \frac{1 − ∥f(z_{k})∥^{2}}{1 − ∥z_{k}∥^{2}} ∥B_{w}^{−1}B(w, z_{k}) (1 − ∥z_{k}∥^{2})B_{z_{k}}^{−1}∥ .

We take a limit over k to obtain

∥B_{f(w)}^{−1}B(f(w), e′)Q_{e′}Q_{e′}∥ ≤ α∥B_{w}^{−1}B(w, e)Q_{e}Q_{e}∥

for all w ∈ B. That is, f(E_{λ}(e)) ⊂ E_{αλ}(e′).

Note 3.4. The α appearing in the previous result depends of course on the sequence (z_{k})_{k} chosen. We will have occasion later to choose the least possible α and so the following reformulation may be more appropriate.

Corollary 3.5. Let f : B → B′ be holomorphic. Let e ∈ ∂B be a tripotent and let

α := lim inf_{z→e} \frac{1 − ∥f(z)∥^{2}}{1 − ∥z∥^{2}} < ∞.

If there exists a sequence (z_{k})_{k} in B converging to e such that f(z_{k}) → e′ ∈ ∂B′, each z_{k} and each f(z_{k}) is proportional to a tripotent and (z_{k})_{k} satisfies lim inf_{k→∞} \frac{1−∥f(z_{k})∥^{2}}{1−∥z_{k}∥^{2}} = α then f(E_{λ}(e)) ⊂ E_{αλ}(e′) for all λ > 0.

Not surprisingly when B′ = Δ and the hypothesis of the above result is satisfied, the sequence (z_{k})_{k} may be taken to be of the form (r_{k}e)_{k} where each r_{k} ∈ [0, 1). The proof is similar to that of [1, Lemma 3.2].

Lemma 3.6. Let f ∈ Hol(B, Δ). Let e ∈ ∂B be a tripotent and suppose there exists a sequence (z_{k})_{k} ∈ B converging to e each element of which is proportional to a tripotent and for which

lim inf_{k→∞} \frac{1 − |f(z_{k})|^{2}}{1 − ∥z_{k}∥^{2}} = lim inf_{z→e} \frac{1 − |f(z)|^{2}}{1 − ∥z∥^{2}} .

Then

lim inf_{z→e} \frac{1 − |f(z)|^{2}}{1 − ∥z∥^{2}} = lim inf_{ζ→1} \frac{1 − |f(ζe)|^{2}}{1 − |ζ|^{2}} = lim inf_{t→1^{−}} \frac{1 − |f(te)|^{2}}{1 − t^{2}} .

Proof. Clearly,

α := lim inf_{z→e} \frac{1 − |f(z)|^{2}}{1 − ∥z∥^{2}} ≤ lim inf_{ζ→1} \frac{1 − |f(ζe)|^{2}}{1 − |ζ|^{2}} ≤ lim inf_{t→1^{−}} \frac{1 − |f(te)|^{2}}{1 − t^{2}} =: β

and so we may suppose that α < ∞. We need only show therefore that α ≥ β. Corollary 3.5 implies that f(E_{λ}(e)) ⊂ E_{αλ}(e′) for some e′ ∈ ∂Δ, for all λ > 0. Without loss of generality, we may assume that e′ = 1. Therefore for all w ∈ B,

∥B_{f(w)}^{−1}B(f(w), e′)Q_{e′}Q_{e′}∥ ≤ α∥B_{w}^{−1}B(w, e)Q_{e}Q_{e}∥

and, in particular for w = ωe, ω ∈ Δ. Then (4) and (5) imply

\frac{|1 − f(ωe)|^{2}}{1 − |f(ωe)|^{2}} ≤ α\frac{|1 − ω|^{2}}{1 − |ω|^{2}} .

Let t_{k} = \frac{k−1}{k+1}. Then \frac{|1−t_{k}|^{2}}{1−|t_{k}|^{2}} = \frac{1}{k} and so

\frac{|1 − f(t_{k}e)|^{2}}{1 − |f(t_{k}e)|^{2}} ≤ \frac{α}{k}.

That is, f(t_{k}e) ∈ \overline{E_{α/k}(1)}. We use Note 3.2 to write

1 − |f(t_{k}e)| ≤ |1 − f(t_{k}e)| ≤ \frac{2(α/k)}{1 + (α/k)} = \frac{2α}{α + k} .

As 1 − |t_{k}|^{2} = 4k/(k + 1)^{2} we get

β = lim inf_{t→1} \frac{1 − |f(te)|^{2}}{1 − t^{2}} ≤ lim inf_{k→∞} \frac{1 − |f(t_{k}e)|^{2}}{1 − t_{k}^{2}} ≤ lim_{k→∞} \frac{(4α)/(α + k)}{4k/(k + 1)^{2}} = α

as required.

We now obtain the existence of an angular limit where we take as our hypothesis the conclusion of Lemma 3.3.

Theorem 3.7. Let f ∈ Hol(B, B′) and let e ∈ ∂B be a tripotent. Let e′ be an extreme point in B′. If there exists α > 0 such that f(E_{λ}(e)) ⊂ E_{αλ}(e′) for all λ > 0 then

∠- lim_{x→e} f(x) = e′.

Proof. Fix an angular region D_{k}(e) and let ε > 0. If w ∈ D_{k}(e) then

∥B_{w}^{−1}B(w, e)Q_{e}Q_{e}∥ ≤ ∥B_{w}^{−1}∥∥B(w, e)Q_{e}Q_{e}∥ = \frac{∥B(w, e)Q_{e}Q_{e}∥}{1 − ∥w∥^{2}} < k^{2}(1 − ∥w∥^{2}).

It follows that if (w_{n})_{n} ⊂ D_{k}(e) converges to e then there exists n_{ε} such that w_{n} ∈ E_{ε}(e) for all n ≥ n_{ε} and thus, by hypothesis, we have f(w_{n}) ∈ E_{αε}(e′) for all n ≥ n_{ε}. In other words,

∥B_{f(w_{n})}^{−1}B(f(w_{n}), e′)Q_{e′}Q_{e′}∥ < αε for all n ≥ n_{ε}.

Since ε is arbitrary, ∥B_{f(w_{n})}^{−1}B(f(w_{n}), e′)Q_{e′}Q_{e′}∥ → 0. This implies, by Lemma 3.8 below, that f(w_{n}) → e′. Since k is arbitrary this implies that ∠-lim_{x→e} f(x) = e′.

Lemma 3.8. Let e be an extreme point of B. Let (x_{n})_{n} ⊂ B satisfy

lim_{n} ∥B_{x_{n}}^{−1}B(x_{n}, e)Q_{e}Q_{e}∥ = 0.

Then lim_{n} x_{n} = e.

Proof. We consider the domains E_{γ_{n}}(e) for γ_{n} = ∥B_{x_{n}}^{−1}B(x_{n}, e)Q_{e}Q_{e}∥. We know from Proposition 3.1 that x_{n} and e are elements of ∂E_{γ_{n}}(e). It follows from [15, Proposition 3.15] that E_{γ_{n}}(e) has the alternative description

E_{γ_{n}}(e) = \{z ∈ B : ∥\frac{1}{t_{n}} P_{\frac{1}{2}} (z) + \frac{P_{1}(z) − e}{t_{n}^{2}} + e∥ < 1\}

where t_{n} = \sqrt{\frac{γ_{n}}{1+γ_{n}}}. This gives

∥\frac{P_{\frac{1}{2}} (x_{n})}{t_{n}} + \frac{P_{1}(x_{n}) − e}{t_{n}^{2}} + e∥ = 1 (8)

for all n. Since t_{n} → 0, one sees easily that P_{1}(x_{n}) → e. Now consider the Peirce reflection g = exp(2πi e □e) which, by definition of a JB^{∗}-triple, is a linear isometry. It is not difficult to see that g acts as the identity Id on Z_{1} and as − Id on Z_{\frac{1}{2}} . Then from (8)

∥\frac{−P_{\frac{1}{2}} (x_{n})}{t_{n}} + \frac{P_{1}(x_{n}) − e}{t_{n}^{2}} + e∥ = 1

and this implies that ∥\frac{P_{\frac{1}{2}} (x_{n})}{t_{n}} ∥ < 1. As t_{n} → 0, we must have P_{\frac{1}{2}} (x_{n}) → 0 and so lim_{n} x_{n} =

lim_{n} P_{1}(x_{n}) + lim_{n} P_{\frac{1}{2}} (x_{n}) = e.

The following technical lemma generalises [6, Proposition 2.1] and [16, Lemma 8.5.5(i)]. It provides useful information about the geometry of the angular regions D_{k}(e) and is used several times in the sequel.

Lemma 3.9. Suppose 1 < β < α and ∥b∥ = 1, so that D_{β}(b) ⫋ D_{α}(b). There exists δ > 0 such that for x ∈ D_{β}(b) and ∥y∥ ≤ 1,

|λ| ≤ δ∥B(x, b)Q_{b}Q_{b}∥^{\frac{1}{2}} ⟹ x + λy ∈ D_{α}(b).

Proof. Let δ = (1/β − 1/α)/10 and x ∈ D_{β}(b). Of course, δ < \frac{1}{10} so |λ| ≤ δ∥B(x, b)Q_{b}Q_{b}∥^{\frac{1}{2}} implies that |λ| < 1 because from (2) ∥B(u, v)∥ ≤ (1 + ∥u∥∥v∥)^{2} for any u and v. We have

∥x + λy∥^{2} ≤ ∥x∥^{2} + 2∥x∥|λ| + |λ|^{2}
≤ ∥x∥^{2} + 3|λ|.

A standard identity for Bergman operators [12, JP34] lets us write

B(x + λy, b)Q_{b}Q_{b} = B(λy, b^{x})B(x, b)Q_{b}Q_{b}

where b^{x} = B(x, b)^{−1}(x − Q_{x}b) denotes the quasi-inverse of b with respect to x (see [5, 12]). Thus

∥B(x + λy, b)Q_{b}Q_{b}∥ ≤ ∥B(x, b)Q_{b}Q_{b}∥(1 + |λ|∥b^{x}∥)^{2}. (9)

Since ∥b∥∥x∥ < 1, it follows easily from the series expansion of b^{x} = ∑_{j=0}^{∞}( b □x)^{j}b (see [12]) that

∥b^{x}∥ ≤ \frac{∥b∥}{1 − ∥b∥∥x∥} = \frac{1}{1 − ∥x∥} = \frac{1 + ∥x∥}{1 − ∥x∥^{2}} < \frac{2}{1 − ∥x∥^{2}} . (10)

Now we use the fact that x ∈ D_{β}(b) to write ∥b^{x}∥ < 2(1 − ∥x∥^{2})^{−1} < 2β∥B(x, b)Q_{b}Q_{b}∥^{− \frac{1}{2}} . Therefore

∥B(x + λy, b)Q_{b}Q_{b}∥^{\frac{1}{2}} ≤ ∥B(x, b)Q_{b}Q_{b}∥^{\frac{1}{2}} (1 + |λ|\frac{2β}{∥B(x, b)Q_{b}Q_{b}∥^{\frac{1}{2}}} )

≤ ∥B(x, b)Q_{b}Q_{b}∥^{\frac{1}{2}} + 2|λ|β. (11)

Since \frac{β}{α} < 1, we have

∥x + λy∥^{2} + \frac{1}{α}∥B(x + λy, b)Q_{b}Q_{b}∥^{\frac{1}{2}} ≤ ∥x∥^{2} + 3|λ| + \frac{1}{α}∥B(x, b)Q_{b}Q_{b}∥^{\frac{1}{2}} + 2|λ|

which gives

∥x + λy∥^{2} + \frac{1}{α}∥B(x + λy, b)Q_{b}Q_{b}∥^{\frac{1}{2}} ≤ ∥x∥^{2} + (5δ + \frac{1}{α})∥B(x, b)Q_{b}Q_{b}∥^{\frac{1}{2}} .

Finally, as 5δ + \frac{1}{α} < \frac{1}{β} and x ∈ D_{β}(b), we can write

∥x + λy∥^{2} + \frac{1}{α}∥B(x + λy, b)Q_{b}Q_{b}∥^{\frac{1}{2}} ≤ ∥x∥^{2} + \frac{1}{β} ∥B(x, b)Q_{b}Q_{b}∥^{\frac{1}{2}}
< 1.

That is, x + λy ∈ D_{α}(b).

The following estimates are required in the sequel.

Lemma 3.10. Let Z be a JB^{∗}-triple and let e ∈ Z be a tripotent. Then,

∥x − e∥^{2} ≤ ∥B(x, e)Q_{e}Q_{e}∥ ≤ 3∥x − e∥^{2}

for any x ∈ Z_{1}.

Proof. The space Z_{1} = P_{1}(Z) is a JB^{∗}-algebra with respect to the product x ∘ y := {x, e, y} and involution x^{∗} := {e, x, e} (cf. [17]). The Bergman operator B(x, e)|_{Z_{1}} may be represented as 2L_{e−x}^{2} − L_{(e−x)^{2}} where L_{x}(y) := x ∘ y = {x, e, y}. Thus the right hand inequality is immediate using (2).

For any JB^{∗}-algebra A, we have that ∥2y(y^{∗}y) − y^{2}y^{∗}∥ = ∥y∥^{3} for all y ∈ A. Therefore, for x ∈ Z_{1},

∥B(x, e)|_{Z_{1}}∥ ≥ ∥[2L_{e−x}^{2} − L_{(e−x)^{2}}]\frac{(e − x)^{∗}}{∥e − x∥} ∥
= ∥2(e − x)((e − x)^{∗}(e − x)) − (e − x)^{2}(e − x)^{∗}∥/∥e − x∥
= ∥e − x∥^{3}/∥e − x∥ = ∥e − x∥^{2}.

4. The Lindelof Principle

The Lindelöf principle [11] allows one to deduce the existence of an angular limit at b ∈ ∂Δ for a bounded function on Δ from the existence of a limit along any one approach curve to b. The following result is a version of this principle for functions mapping Δ to a JB^{∗}-triple Z. The proof, which we include for completeness, is a very slight modification of the classical one provided in [16]. For a bounded symmetric domain B and b ∈ ∂B, we define a b-curve to be a continuous curve σ : [0, 1) → B such that σ(t) → b as t → 1.

Theorem 4.1. Let f : Δ → Z be holomorphic and bounded and γ : [0, 1) → Δ be a 1-curve. If lim_{t→1^{−}} f(γ(t)) = l ∈ Z then f has angular limit l at 1.

Proof. Without loss of generality, ∥f∥ = 1 and l = 0. Let Σ denote the strip {z ∈ C : |Re z| < 1} and let φ : Δ → Σ be a conformal mapping for which φ(0) = 0. Let Γ = φ ∘ γ and F = f ∘φ^{−1}. Then Im (Γ(t)) → ∞ and F (Γ(t)) → 0 as t → 1. In this setting, an angular limit of f(t) as t → 1 in Δ is equivalent to a uniform limit of F (x + iy) as y → ∞ in the strip {x + iy ∈ C : |x| ≤ 1 − δ} for any δ > 0.

Given δ ∈ (0, 1) we have to show therefore that F (x + iy) → 0 uniformly as y → ∞ for |x| ≤ 1 − δ. If we fix ε ∈ (0, 1), and choose y_{0} > Im Γ(0) such that ∥F (Γ(t))∥ < ε for Im Γ(t) ≥ y_{0}, the proof will follow from the statement that

∥F (x + iy_{0})∥ ≤ ε^{δ/6} if |x| ≤ 1 − δ. (12)

To show (12) we can assume, by a vertical translation of Σ, that y_{0} = 0. Choosing t_{0} with Im Γ(t_{0}) = 0 and Im Γ(t) > 0 for all 1 > t > t_{0}, we let E = \{Γ(t) : t_{0} ≤ t < 1\} and \overline{E} be the reflection of E in the x-axis. Let x_{0} = Γ(t_{0}) ∈ IR. Suppose x_{0} ≤ x ≤ 1 − δ. Then we can define, for η ∈ (0, \frac{1}{3}) and z ∈ Σ,

G_{η}(z) = \{F (z), F (\overline{z}), F (z)\}\frac{ε^{(1+z)/2}}{1 + η(1 + z)}. (13)

G_{η} is a holomorphic function and is bounded on Σ.

Since ∥F (z)∥ < ε on E and ∥F (\overline{z})∥ < ε on \overline{E}, we have that ∥\{F (z), F (\overline{z}), F (z)\}∥ < ε on E ∪ \overline{E}. It follows that ∥G_{η}(z)∥ < ε on E ∪ \overline{E}. On the right hand boundary of Σ, ∥G_{η}(z)∥ < ε and for |Im z| sufficiently large we also have ∥G_{η}(z)∥ < ε. Now we apply the maximum principle in the component of the strip bounded by Re z = 1, E ∪ \overline{E} and ±∞ to get that ∥G_{η}(z)∥ < ε in this component. In particular,

∥F (x)∥^{3} \frac{ε^{(1+x)/2}}{1 + η(1 + x)} = ∥\{F (x), F (x), F (x)\}\frac{ε^{(1+x)/2}}{1 + η(1 + x)}∥
= ∥G_{η}(x)∥ < ε

and so ∥F (x)∥^{3} < ε.ε^{−(1+x)/2}(1+η(1+x)). We let η → 0 to get ∥F (x)∥^{3} ≤ ε^{(1−x)/2} < ε^{δ/2} and so ∥F (x)∥ < ε^{δ/6} for x_{0} ≤ x < 1 − δ. On the other hand, if x_{0} > x > −1 + δ then we simply replace 1 + z in equation (13) by 1 − z and repeat the argument to conclude that (12) holds.

Note. We can slightly change the statement of this theorem to the following: Let f : Δ → Z be holomorphic and bounded in the angular Δ_{k}(1). Let α > β and γ : [0, 1) → Δ be a 1-curve contained in Δ_{α}(1) . If lim_{t→1^{−}} f(γ(t)) = l ∈ Z then f has the limit l at 1 along any approach curve in Δ_{β}(1).

Čirka [4] extended Lindelöf’s result to functions of several variables and we show that this can be improved to include functions defined on a bounded symmetric domain B. For b ∈ ∂B, a key tool required is a continuous linear projection π = π_{b} : B → bΔ = {ζb : ζ ∈ Δ} of the bounded symmetric domain onto the one dimensional subspace containing b which satisfies π(D_{k}(b)) ⊂ D_{k}(b) for all k > 0 and π(b) = b. Following the terminology of [2], we will call π_{b} a projective device at b. Throughout the remainder of this paper, we restrict attention to a boundary point b of a bounded symmetric domain B for which a projective device π at b exists. In many situations, there is a canonical choice for the projective device. For example, if B is the n-dimensional Hilbert ball then one can take π to be the orthogonal projection of B onto bΔ. A canonical choice for π when B = Δ^{n} is given [2] by π(x) = \frac{1}{d}⟨x, b̂⟩b where d is the cardinality of the set {j : |b_{j}| = 1}, b = (b_{1}, …, b_{n}) and b̂ = (b′_{1}, …, b′_{n}) where

b′_{j} = \begin{cases}b_{j}, & |b_{j}| = 1 \\ 0, & |b_{j}| < 1 \end{cases} .

Here ⟨·, ·⟩ denotes the usual inner product on C^{n}.

In the following Γ : [0, 1) → B will be an e-curve for some tripotent e. An e-curve is called non-tangential if it lies eventually in D_{k}(e) for some k > 0. Clearly, if Γ is a non-tangential e-curve then the projection of Γ under π = π_{e}, denoted γ, is also a non-tangential e-curve since π(D_{k}(e)) ⊂ D_{k}(e) for all k > 0. The proof of the following theorem is a modification of that of Čirka’s result given in [16].

Theorem 4.2. Let e be a tripotent in Z having projective device π_{e} and let Γ be a nontangential e-curve. Let γ = π_{e} ∘ Γ. Let f: B → Z′ be holomorphic and bounded in every angular region D_{k}(e). If

lim_{t→1^{−}} \frac{1 − ∥γ(t)∥}{∥Γ(t) − γ(t)∥} = ∞

then

lim_{t→1^{−}} f(Γ(t)) − f(γ(t)) = 0.

Proof. Define R(t) := \frac{1−∥γ(t)∥}{∥Γ(t)−γ(t)∥}. Fix λ ∈ C and choose t_{0} such that R(t) > \sqrt{R(t)} > |λ| for t > t_{0}. Then for t > t_{0},

C_{λ}(t) := λΓ(t) + (1 − λ)γ(t)

is an element of B. This is due to the fact that ∥λ(Γ(t) − γ(t)) + γ(t)∥ < R∥Γ(t) − γ(t)∥ + ∥γ(t)∥ = 1. As Γ is non-tangential, there exists k > 0 such that for t sufficiently close to 1, say t > t_{1} ≥ t_{0}, Γ(t) ∈ D_{k}(e) and so γ(t) ∈ D_{k}(e).

Fix k′ > k. Notice that ∥B(γ(t), e)Q_{e}Q_{e}∥ ≥ ∥γ(t) − e∥^{2} by Lemma 3.10 and so from Lemma 3.9, there exists δ > 0 such that

|μ| < δ∥e − γ(t)∥ ⟹ γ(t) + μy ∈ D_{k′}(e) for ∥y∥ = 1. (14)

Also, for t > t_{1} we have |λ| < \sqrt{R(t)}, and so

∥λ(Γ(t) − γ(t))∥ < \sqrt{R(t)}\frac{1 − ∥γ(t)∥}{R(t)}

= \frac{1 − ∥γ(t)∥}{\sqrt{R(t)}}. (15)

Since \sqrt{R(t)} → ∞ there exists t_{2} ≥ t_{1} such that for t > t_{2} we have δ\sqrt{R(t)} > 1 ≥ \frac{1−∥γ(t)∥}{∥e−γ(t)∥} and so

\frac{1 − ∥γ∥}{\sqrt{R(t)}} < δ∥e − γ(t)∥.

From (15), this gives,

∥λ(Γ(t) − γ(t))∥ < δ∥e − γ(t)∥

for all t > t_{2}. This suggests a choice of μ in (14). Taking y = \frac{λ(Γ−γ)}{∥λ(Γ−γ)∥}(t) and μ = ∥λ(Γ(t) − γ(t))∥ we conclude that γ + λ(Γ − γ) ∈ D_{k′}(e) whenever t > t_{2}. In other words, C_{λ}(t) ∈ D_{k′}(e) for all t > t_{2}.

The rest of the proof proceeds as in the classical case [16] which we now recall. Define the holomorphic function g_{t} from the open disc of radius \sqrt{R(t)} in C to the ball of radius ∥f∥_{D_{k′}} in Z′ by

g_{t}(λ) := f(λΓ(t) + (1 − λ)γ(t)).

Of course g_{t}(1) = f(Γ(t)) and g_{t}(0) = f(γ(t)). The map h_{t}(λ) := g_{t}(λ) − g_{t}(0) is a holomorphic function from the complex disc of radius \sqrt{R(t)} to the ball of radius 2∥f∥_{D_{k′}(e)} which maps 0 to 0. The Schwarz Lemma then gives

∥h_{t}(λ)∥ ≤ \frac{2∥f∥_{D_{k′}}}{\sqrt{R(t)}} |λ|

and, in particular, ∥h_{t}(1)∥ ≤ 2∥f∥_{D_{k′}(e)}/\sqrt{R(t)} for all t ≥ t_{2}. Since R(t) → ∞ we have that lim_{t→1^{−}} h_{t}(1) = 0 and from this it follows that lim_{t→1^{−}} f(Γ(t)) − f(γ(t)) = 0.

The terminology given below is modelled on that of Rudin [16] although our definitions do not agree exactly with those made in the Hilbert space case. (When working in a Hilbert space one may square the quotient factor occurring in (16) below. This allows a larger collection of restricted curves.) Our definition seems a more appropriate one for bounded symmetric domains.

Definition 4.3. We say a b-curve Γ is special if

lim_{t→1^{−}} \frac{1 − ∥γ(t)∥}{∥Γ(t) − γ(t)∥} = ∞, (16)

where γ = π_{b} ∘ Γ. Further, we say a b-curve Γ is restricted if it is special and γ([0, 1)) is eventually contained in bΔ_{k}(1) for some k > 0. In other words, Γ is a restricted curve if it is special and the projection of Γ onto the disc bΔ has non-tangential approach to b.

We say that f : B → Z′ has restricted angular limit l at b if

lim_{t→1^{−}} f(Γ(t)) = l

for every restricted b-curve Γ. For this we write R-lim_{z→b} f(z) = l.

It is not obvious from the definition, but any restricted b-curve is non-tangential if b is a tripotent.

Lemma 4.4. Let e be a tripotent and let Γ be a restricted e-curve. Then Γ is a non-tangential e-curve.

Proof. Let Γ be a restricted e-curve. Then lim_{t→1^{−}} \frac{∥Γ(t)−γ(t)∥}{1−∥γ(t)∥} = 0 and there exists k > 0 and

t_{0} ∈ (0, 1) such that \frac{∥γ(t)−e∥}{1−∥γ(t)∥^{2}} < k for t > t_{0}.

∥B(Γ(t), e)Q_{e}Q_{e}∥ = ∥B(γ + (Γ − γ), e)Q_{e}Q_{e}∥
= ∥B(Γ − γ, e^{γ}) B(γ, e)Q_{e}Q_{e}∥ using [12, JP34]
≤ ∥B(γ, e)Q_{e}Q_{e}∥ ∥B(Γ − γ, e^{γ})∥
≤ ∥B(γ, e)Q_{e}Q_{e}∥(1 + ∥Γ − γ∥∥e^{γ}∥)^{2}

which by Lemma 3.10 and (10) is

≤ 3∥γ − e∥^{2}(1 + ∥Γ − γ∥\frac{∥e∥}{1 − ∥e∥∥γ∥})^{2}

≤ 3∥γ − e∥^{2}(1 + \frac{∥Γ − γ∥}{1 − ∥γ∥})^{2}

< 4∥γ − e∥^{2}

for t > t_{1} ∈ [t_{0}, 1) since \frac{∥Γ(t)−γ(t)∥}{1−∥γ(t)∥} → 0 as t → 1^{−}. Thus, for t > t_{1},

∥B(Γ(t), e)Q_{e}Q_{e}∥^{\frac{1}{2}} < 2∥γ(t) − e∥ < 2k(1 − ∥γ(t)∥^{2}).

Since ∥Γ∥ ≤ ∥Γ − γ∥ + ∥γ∥ we have

1 − ∥Γ(t)∥ ≥ 1 − ∥Γ − γ∥ − ∥γ∥ = 1 − \frac{∥Γ − γ∥}{1 − ∥γ∥}(1 − ∥γ∥) − ∥γ∥

= (1 − ∥γ∥)(1 − \frac{∥Γ − γ∥}{1 − ∥γ∥})

≥ \frac{1}{2}(1 − ∥γ(t)∥)

for t ≥ t_{2} ∈ [t_{1}, 1). Since lim_{t→1^{−}} ∥Γ(t)∥ = lim_{t→1^{−}} ∥γ(t)∥ = 1 we can choose t_{3} ∈ [t_{2}, 1) such that 1+∥Γ(t)∥ ≥ \frac{1}{2}(1+∥γ(t)∥) for t ≥ t_{3}. Thus, for t ≥ t_{3}, 1−∥Γ(t)∥^{2} ≥ \frac{1}{4}(1−∥γ(t)∥^{2}). The resulting inequality

\frac{∥B(Γ(t), b)Q_{e}Q_{e}∥^{\frac{1}{2}}}{1 − ∥Γ(t)∥^{2}} < 8k

for t ≥ t_{3} implies that Γ(t) ∈ D_{8k}(e) for t ≥ t_{3}. That is, Γ is eventually in some angular region at e, and thus Γ is a non-tangential approach curve to e.

The radial limit of a function f at e is the limit of f along the radial path Γ(t) = te, t ∈ [0, 1). Notice that this radial path is restricted and contained in D_{k}(e) for k > 1. We will use the notation lim_{t→1^{−}} f(te) for the radial limit of f at e.

Corollary 4.5. Let e be a tripotent and let f ∈ Hol(B, Z′) be bounded in every angular region D_{k}(e). If the radial limit of f exists at e then the restricted angular limit of f exists at e.

Proof. If Γ is any restricted e-curve then, by Theorem 4.2 and Lemma 4.4, f has the same limit along Γ as it does along its projection γ = π_{e} ∘ Γ. Define φ: Δ → Z′ by φ(ζ) = f(ζe). Then φ is bounded in every Δ_{k}(1) and lim_{t→1^{−}} φ(t) exists. Let l be the value of this limit. Theorem 4.1 implies that φ has non-tangential limit l at 1. As γ is a non-tangential e-curve in eΔ, lim_{t→1^{−}} f(γ(t)) may be identified with a non-tangential limit of φ in Δ. As the value of this limit is l, we conclude that the limit of f along the restricted curve Γ is also l.

Corollary 4.5 implies that f has an angular derivative at a tripotent e ∈ ∂B (by which we mean a restricted angular limit of the function z ↦ f′(z)e) if the map z ↦ f′(z)e is, firstly, bounded in every angular region D_{k}(e) and, secondly, has a radial limit at e. A natural simplification is to consider instead the ‘projected’ map z ↦ P′_{1}f′(z)e where P′_{1} is the Peirce 1-projection of e′, where e′ ∈ Z′ is the tripotent that arises as the angular limit of f at e.

In the classical setting, B = B′ = Δ, one shows that the function z ↦ f′(z) is bounded in every angular region Δ_{k}(e) by using the Cauchy integral formaula to write f′(z) as an integral of the incremental ratios \frac{f(z)−e′}{z−e} where e, e′ ∈ ∂Δ and e′ = ∠-lim_{z→e} f(z) and then showing that these incremental ratios are themselves bounded in every angular region. Our approach is essentially the same, using the Banach space version of Cauchy’s integral formula (see, for example, [3]) and a natural analogue for the ratios \frac{f(z)−e′}{z−e}. We take as this analogue

\frac{∥B(f(z), e′)P′_{1}∥}{∥B(z, e)P_{1}∥}

where P_{1} = Q_{e}Q_{e} and P′_{1} = Q_{e′}Q_{e′} are the Peirce 1-projections corresponding to the tripo-

tents e and e′ respectively. (This reduces to |\frac{f(z)−e′}{z−e} |^{2} when B = Δ.) Part of the argument therefore in proving that z ↦ P′_{1}f′(z)e is bounded in every angular region D_{k}(e) is to show that the map

z ↦ \frac{∥B(P′_{1}f(z), e′)P′_{1}∥}{∥B(z, e)P_{1}∥}

is bounded in every D_{k}(e). As we require e′ to be ∠-lim_{z→e} f(z) we adopt here the hypothesis of Theorem 3.7.

Theorem 4.6. Let f ∈ Hol(B, B′). Suppose that e ∈ ∂B and e′ ∈ ∂B′ are tripotents and that for some α > 0,

f(E_{r}(e)) ⊂ E_{αr}(e′)

for all r > 0. Then the following functions are bounded in every angular region D_{k}(e).

(i) z ↦ \frac{∥B(P′_{1}f(z), e′)P′_{1}∥}{∥B(z, e)P_{1}∥}, (ii) z ↦ P′_{1}f′(z)e,

where P_{1} = Q_{e}Q_{e} and P′_{1} = Q_{e′}Q_{e′} are the Peirce 1-projections of the tripotents e and e′ respectively.

Proof. (i) Fix an angular region D_{k}(e) = \{z ∈ B : ∥B(z, e)Q_{e}Q_{e}∥^{\frac{1}{2}} < k(1 − ∥z∥^{2})\}. Let z ∈ D_{k}(e) and let r = k∥B(z, e)Q_{e}Q_{e}∥^{\frac{1}{2}} . Then

∥B(z, e)Q_{e}Q_{e}∥ = \frac{r}{k} ∥B(z, e)Q_{e}Q_{e}∥^{\frac{1}{2}} < r(1 − ∥z∥^{2})

and hence from (9)

∥B_{z}^{−1}B(z, e)Q_{e}Q_{e}∥ ≤ \frac{∥B(z, e)Q_{e}Q_{e}∥}{1 − ∥z∥^{2}} < r,

that is, z ∈ E_{r}(e). By hypothesis therefore, f(z) ∈ E_{αr}(e′) and P′_{1}f(z) ∈ P′_{1}E_{αr}(e′). From Note 3.2, ∥P′_{1}f(z) − e′∥ < 2αr and by Lemma 3.10,

∥B(P′_{1}f(z), e′)Q_{e′}Q_{e′}∥ ≤ 3∥P′_{1}f(z) − e′∥^{2} < 12α^{2}r^{2}.

Thus ∥B(P′_{1}f(z), e′)Q_{e′}Q_{e′}∥ < 12α^{2}r^{2} = 12α^{2}k^{2}∥B(z, e)Q_{e}Q_{e}∥ and we have

\frac{∥B(P′_{1}f(z), e′)P′_{1}∥}{∥B(z, e)P_{1}∥} < 12α^{2}k^{2}

giving (i).

(ii) Fix an angular region D_{k}(e) and fix k′ > k. By Lemma 3.9 choose δ > 0 so that for z ∈ D_{k}(e), |λ| ≤ δ∥B(z, e)Q_{e}Q_{e}∥^{\frac{1}{2}} implies z + λe ∈ D_{k′}(e). Let r = δ∥B(z, e)Q_{e}Q_{e}∥^{\frac{1}{2}} . By the Cauchy integral formula

P′_{1}f′(z)e = \frac{1}{2πi} ∫_{|λ|=r} \frac{P′_{1}f(z + λe)}{λ^{2}}dλ

= \frac{1}{2πi} ∫_{|λ|=r} \frac{P′_{1}f(z + λe) − e′}{λ^{2}} dλ

= \frac{1}{2πi} ∫_{|λ|=r} A(λ) B(λ) C(λ) dλ

where

A(λ) = \frac{P′_{1}f(z + λe) − e′}{∥B(P′_{1}f(z + λe), e′)P′_{1}∥^{\frac{1}{2}}} ,

B(λ) = \frac{∥B(P′_{1}f(z + λe), e′)P′_{1}∥^{\frac{1}{2}}}{∥B(z + λe, e)P_{1}∥^{\frac{1}{2}}} ,

C(λ) = \frac{∥B(z + λe, e)P_{1}∥^{\frac{1}{2}}}{λ^{2}} .

Therefore

∥P′_{1}f′(z)e∥ ≤ ( sup_{|λ|=r} ∥A(λ)∥)( sup_{|λ|=r} B(λ))( sup_{|λ|=r} r|C(λ)|).

Lemma 3.10 implies that sup_{|λ|=r} ∥A(λ)∥ is bounded (that is, uniformly bounded over z ∈ D_{k}(e)). By part (i), sup_{|λ|=r} B(λ) is bounded, since z + λe is contained in the angular region D_{k′}(e). A glance back at (11) in the proof of Lemma 3.9 shows that r|C(λ)| = \frac{1}{|λ|}∥B(z + λe, e)P_{1}∥^{\frac{1}{2}} is bounded by 2k + \frac{1}{δ} and hence (ii) is proved.

The above result, combined with Corollary 4.5 applied to z ↦ P′_{1}f′(z)e, allows us to conclude the existence of a restricted angular limit of P′_{1}f′(z)e from the existence of a radial limit. To examine this radial limit,

lim_{t→1^{−}} P′_{1}f′(te)e

we again use Cauchy’s integral formula to write

P′_{1}f′(te)e = \frac{1}{2πi} ∫_{|λ|=r} \frac{P′_{1}f((t + λ)e) − e′}{λ^{2}} dλ

= \frac{1}{2πi} ∫_{|λ|=r} (\frac{P_{1}f((t + λ)e) − e}{1 − (t + λ)}) (\frac{1 − (t + λ)}{λ}) \frac{dλ}{λ}

= \frac{1}{2π} ∫_{−π}^{π} \frac{P_{1}f((t + re^{iθ})e) − e}{t + re^{iθ} − 1} (1 − \frac{1 − t}{re^{iθ}}) dθ

Fix 1 < k < k′. As te ∈ D_{k}(e) we again use Lemma 3.9 to choose δ > 0 such that

|λ| ≤ δ∥B(te, e)Q_{e}Q_{e}∥^{\frac{1}{2}}

implies (t + λ)e ∈ D_{k′}(e). In particular, |λ| ≤ δ(1 − t) implies that (t + λ)e ∈ D_{k′}(e). If we take r = δ(1 − t) then

P′_{1}f′(te)e = \frac{1}{2π} ∫_{−π}^{π} \frac{P′_{1}f((t + re^{iθ})e) − e′}{t + re^{iθ} − 1} (1 − \frac{e^{−iθ}}{δ})dθ. (17)

We know from part (i) of Theorem 4.6 that the first factor in this integral is bounded since (t + re^{iθ})e lies in the angular region D_{k′}(e). Moreover, if the radial limit of these incremental ratios exists and equals l, that is if

lim_{t→1^{−}} \frac{P′_{1}f(te) − e′}{t − 1} = l (18)

then we apply Theorem 4.1 to the function

g(ζ) = \frac{P′_{1}f(ζe) − e′}{ζ − 1}

for ζ ∈ Δ to get that the angular limit

∠- lim_{z→1} \frac{P′_{1}f(ζe) − e′}{ζ − 1}

exists and equals l. Hence, the integral (17) above has the limit l as t ∈ [0, 1) tends to 1. We have proved the following.

Theorem 4.7. Let f ∈ Hol(B, B′) and let e ∈ ∂B, e′ ∈ ∂B′ be tripotents. Suppose that for some α > 0, f(E_{r}(e)) ⊂ E_{αr}(e′) for all r > 0. If the radial limit

lim_{t→1^{−}} \frac{e′ − P′_{1}f(te)}{1 − t} = l

then the restricted angular limit, R-lim_{z→e} P′_{1}f′(z)e = l.

We now turn our attention to proving that the limit in (18) above does indeed exist in a reasonably general setting.

Theorem 4.8. Let f ∈ Hol(B, Δ) and let e ∈ ∂B be a tripotent such that

α := lim inf_{z→e} \frac{1 − |f(z)|^{2}}{1 − ∥z∥^{2}} < ∞.

Suppose there exists a sequence (z_{k})_{k} in B converging to e, each element of which is proportional to a tripotent and which satisfies lim inf_{k} \frac{1−|f(z_{k})|^{2}}{1−∥z_{k}∥^{2}} = α and f(z_{k}) → e′ ∈ Δ. Then the radial limit

lim_{t→1^{−}} \frac{e′ − f(te)}{1 − t} = αe′

and hence R-lim_{z→e} f′(z)e = αe′.

Proof. Define φ ∈ Hol(Δ, Δ) by φ(ζ) = f(ζe). Lemma 3.6 implies that

lim inf_{ζ→1} \frac{1 − |φ(ζ)|^{2}}{1 − |ζ|^{2}} = α

and so we can apply the one dimensional result (Theorem 1.1) to φ to get ∠-lim_{ζ→1} φ′(ζ) = αe′. In particular,

∠- lim_{ζ→1} \frac{e′ − φ(ζ)}{1 − ζ} = αe′

which gives the result.

If we add the condition that f(0) = 0 then the above result can be extended to include the case where the range of f is a Hilbert space.

Theorem 4.9. Let f ∈ Hol(B, B′), where B′ is the open unit ball of a Hilbert space, satisfy f(0) = 0. Let

α := lim inf_{z→e} \frac{1 − ∥f(z)∥^{2}}{1 − ∥z∥^{2}} < ∞.

Suppose there exists a sequence (z_{k})_{k} in B, each element of which is proportional to a tripotent and which satisfies lim inf_{k} \frac{1−∥f(z_{k})∥^{2}}{1−∥z_{k}∥^{2}} = α and f(z_{k}) → e′ where e′ ∈ ∂B′ is a tripotent. Then the radial limit

lim_{t→1^{−}} \frac{e′ − P′_{1}f(te)}{1 − t} = αe′

and hence R-lim_{z→e} P′_{1}f(z)e = αe′.

Proof. Lemma 3.6 implies that

α = lim inf_{z→e} \frac{1 − ∥f(z)∥^{2}}{1 − ∥z∥^{2}} = lim inf_{t→1^{−}} \frac{1 − ∥f(te)∥^{2}}{1 − t^{2}} .

Let c = \frac{1}{2}(1 − t). From Proposition 3.1, te ∈ ∂E_{\frac{c}{1−c}} (e) and so by Lemma 3.3 f(te) ∈

\overline{E_{α \frac{c}{1−c}} (e′)} and thus (recall Note 3.2) ∥P′_{1}f(te)−e′∥ ≤ \frac{2αc}{1−(1−α)c}. Since f(0) = 0, the Schwarz lemma implies that α ≥ 1 and thus ∥P′_{1}f(te) − e′∥ ≤ 2αc.

We now have that

1 − ∥f(te)∥^{2} ≤ 1 − ∥P′_{1}f(te)∥^{2}
≤ (1 + ∥P′_{1}f(te)∥)∥P′_{1}f(te) − e′∥
≤ 2αc(1 + t) = α(1 − t^{2})

giving

\frac{1 − ∥f(te)∥^{2}}{1 − t^{2}} ≤ \frac{1 − ∥P′_{1}f(te)∥^{2}}{1 − t^{2}} ≤ α.

Hence, lim inf_{t→1^{−}} \frac{1−∥P′_{1}f(te)∥^{2}}{1−t^{2}} ≤ α. Thus,

α = lim inf_{z→e} \frac{1 − ∥f(z)∥^{2}}{1 − ∥z∥^{2}} ≤ lim inf_{z→e} \frac{1 − ∥P′_{1}f(z)∥^{2}}{1 − ∥z∥^{2}}

≤ lim inf_{t→1^{−}} \frac{1 − ∥P′_{1}f(te)∥^{2}}{1 − t^{2}} ≤ α.

We conclude that

lim inf_{z→e} \frac{1 − ∥P′_{1}f(z)∥^{2}}{1 − ∥z∥^{2}} = α.

As P′_{1} is nothing but the orthogonal projection of B′ onto e′, P′_{1}f may be identified with a holomorphic function from B to Δ and the rest of the proof proceeds as in Theorem 4.8.

For example, we have the following.

Corollary 4.10. Let f ∈ Hol(B, Δ) satisfy f(0) = 0, where B is the open unit ball of a C^{∗}-algebra with identity 1. If

α := lim inf_{z→1} \frac{1 − |f(z)|^{2}}{1 − ∥z∥^{2}} < ∞

and there exists a sequence z_{k} → 1 such that each z_{k} is a scalar multiple of a partial isometry, lim_{k→∞} \frac{1−∥f(z_{k})∥^{2}}{1−∥z_{k}∥^{2}} = α and f(z_{k}) → 1 then the angular limit ∠-lim_{z→1} f(z) = 1 and the restricted angular limit R-lim_{z→1} f′(z)1 = α.

In particular, if B is the n-dimensional polydisc we obtain the following, which is contained in [2].

Corollary 4.11. Let f ∈ Hol(Δ^{n}, Δ) satisfy f(0) = 0. If

α := lim inf_{z→1=(1,... ,1)} \frac{1 − |f(z)|^{2}}{1 − ∥z∥^{2}} < ∞

and there exists a sequence z_{k} → 1 such that each z_{k} is a scalar multiple of an extreme point, lim_{k→∞} \frac{1−∥f(z_{k})∥^{2}}{1−∥z_{k}∥^{2}} = α and f(z_{k}) → 1 then the angular limit ∠-lim_{z→1} f(z) = 1 and the restricted angular limit R-lim_{z→1} f′(z)1 = α.

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Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland.

E-mail address: Michael.Mackey@ucd.ie
E-mail address: Pauline.Mellon@ucd.ie