A non-commutative Julia Inequality

Abstract

We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphic functions on classical domains in ℂ^d. We look at differentiability at a boundary point for functions that have a certain regularity there.

1 Introduction

The classical Julia inequality asserts that if a holomorphic function ϕ maps the unit disk 𝔻 to itself, and if at some boundary point τ ∈ ∂𝔻 one has

$$\underset{z\to \tau }{\text{lim inf}}\frac{1-{|\phi (z)|}^2}{1-{|z|}^2}=\alpha <\infty ,$$ (1.1)

then there exists ω ∈ ∂𝔻 such that

$$\frac{{|\phi (z)-\omega |}^2}{1-{|\phi (z)|}^2}\le \alpha \frac{{|z-\tau |}^2}{1-{|z|}^2}.$$ (1.2)

The inequality was proved, with an extra regularity hypothesis on φ, by G. Julia in [22], and in the form stated by C. Carathéodory in [16]. D. Sarason found a proof using model theory [33, Chap VI].

Generalizations of Julia’s inequality have been found for functions on the ball by M. Hervé [20], W. Rudin [32, Sec. 8.5] and M. Jury [23], and on the polydisk by K. Wlodarczyk [37], F. Jafari [21] and M. Abate [1]. In the case of the bidisk, a detailed analysis of points for which the analogue of the Julia quotient (1.1) remains bounded (these are called B-points) has been carried out in [7, 8, 10, 15].

It is the purpose of this note to extend Julia’s inequality, and the study of B-points, to non-commutative functions (which we shall define in Subsection 1.1 below) that are bounded on polynomial polyhedra. Our methods rely on the model-theoretic ideas of [8].

Our results are of interest even in the commutative case, because they provide a unified approach to proving boundary versions of the Schwarz-Pick lemma in the Schur-Agler class of various domains, such as the polydisk, or the multipliers of the Drury-Arveson space. The methods also show that at B-points where the function is not analytic, it does have directional derivatives in all directions pointing into the set, and the derivative is a holomorphic (but not necessarily linear) function of the direction. This is explained in Section 6 below.

1.1 Non-commutative Functions

Non-commutative function theory, which originated in the work of J.L. Taylor [34,35], has recently started to flourish [6,11,12,14,17–19,25–28,30,31]. The foundations are developed in the book [24].

The idea is to study functions of non-commuting variables that are generalized non-commuting polynomials in an analogous fashion to thinking of a holomorphic function as a generalization of a polynomial. Our domains are domains of d-tuples of matrices, but they don’t reside in just one dimension. We let 𝕄_n denote the n-by-n complex matrices, and let

$${𝕄}^{[d]}={\cup }_{n=1}^{\infty }{𝕄}_n^d.$$

We shall call a function ϕ defined on a subset of 𝕄^[d] graded if, whenever $x\in {𝕄}_n^d$, then ϕ(x) ∈ 𝕄_n. If $x\in {𝕄}_n^d$ and $y\in {𝕄}_m^d$, we shall let x ⊕ y denote the d-tuple in 𝕄_n+m obtained by direct summing each component. If $x\in {𝕄}_n^d$ and s is an invertible matrix in 𝕄_n, then s⁻¹xs denotes the d-tuple (s⁻¹x¹s, …, s⁻¹x^ds).

Definition 1.3

An nc-function ϕ on a set Ω ⊆ 𝕄^[d] is a graded function that respects direct sums and joint similiarities, i.e.

$$\begin{array}{cc}\hfill \varphi (x\oplus y)& =\varphi (x)\oplus \varphi (y)\hfill \\ \hfill \varphi (s^{-1}\mathit{\text{xs}})≔\varphi (s^{-1}{x}^{1}s,\dots ,s^{-1}{x}^{d}s)& =s^{-1}\varphi (x)s,\hfill \end{array}$$

where the equations are only required to hold when the arguments on both sides are in Ω, and in the second one if $x\in {𝕄}_n^d$, then s is invertible in 𝕄_n.

(We use superscripts for components, since we shall have many sequences indexed by subscripts.) Notice that every non-commutative polynomial is an nc-function on 𝕄^[d]. A particularly nice class of domains on which to study nc-functions are polynomial polyhedra. These are defined in terms of a matrix δ, each of whose entries is a non-commutative polynomial in d variables. Then 𝒢_δ is defined as

$${𝒢}_{\delta }≔{\cup }_{n=1}^{\infty }\{x\in {𝕄}_n^d:\Vert \delta (x)\Vert <1\},$$ (1.4)

where δ(x), a matrix of matrices, is given the operator norm. A primary example is the d-dimensional noncommutative polydisk, which is the set

$$\{x\in {𝕄}^{[d]}:\Vert x\Vert ≔\underset{1\le r\le d}{\text{max}}\Vert x^r\Vert <1\}.$$ (1.5)

For this set, we can take δ to be the diagonal d-by-d matrix with the co-ordinate functions on the diagonal. Another well-studied example (see e.g. [29]) is the non-commutative ball, which we shall take to be the column contractions¹,

$$\{x\in {𝕄}^{[d]}:\sum _{r=1}^d{x}^{r\ast }{x}^r\le I\},$$ (1.6)

which is obtained by letting

$$\delta (x)=\left(\begin{array}{c}\hfill x^1\hfill \\ \hfill x^2\hfill \\ \hfill ⋮\hfill \\ \hfill x^d\hfill \end{array}\right).$$

1.2 Principal Results

Let 𝒢_δ be defined by (1.4). We shall let ℬ_δ denote the topological boundary of 𝒢_δ. If x ∈ ℬ_δ, then ‖δ(x)‖ = 1, but the converse need not hold — e.g. with d = 1, take $\delta (x)=\left(\begin{array}{cc}\hfill x-1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill x+1\hfill \end{array}\right)$. Then 𝒢_δ is empty, but ‖δ(0)‖ = 1.

The distinguished boundary of 𝒢_δ, which we shall denote ℐ_δ, is

$${ℐ}_{\delta }≔\{x\in {ℬ}_{\delta }:\delta (x)\text{ is an isometry}\}.$$ (1.7)

The reader should keep in mind the example of the non-commutative polydisk (1.5), in which case ℐ_δ is 𝒰^[d], the set of d-tuples of unitary matrices in 𝕄^[d], and ℬ_δ is the set of contractive d-tuples such that at least one has norm equal to one. For the column-ball (1.6), the distinguished boundary will agree with ℬ_δ when n = 1, but will be smaller when n > 1.

Definition 1.8

The Schur class of 𝒢_δ, denoted 𝒮(𝒢_δ), is the set of nc functions ϕ on 𝒢_δ such that ‖ϕ(x)‖ < 1 ∀ x ∈ 𝒢_δ.

A B-point for ϕ is a point in the boundary where a certain regularity occurs.

Definition 1.9

Let ϕ ∈ 𝒮(𝒢_δ), and let T ∈ ℬ_δ. Then T is a B-point for ϕ if

$$\underset{j\to \infty }{\text{lim}}\frac{\Vert I-\varphi {(Z_j)}^{\ast }\varphi (Z_j)\Vert }{1-{\Vert \delta (Z_j)\Vert }^2}<\infty$$ (1.10)

for some sequence Z_j converging to T.

Here is our non-commutative Julia inequality, which says that at a B-point, one has a boundary version of the Schwarz-Pick inequality, akin to (1.2).

Theorem 3.6

Suppose ϕ ∈ 𝒮(𝒢_δ) and T ∈ ℬ_δ. If

$$\underset{Z\to T}{\text{lim inf}}\frac{\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert }{1-{\Vert \delta (Z)\Vert }^2}=\alpha ,$$

then there exists W ∈ 𝒰_n such that for all Z in 𝒢_δ

$$\frac{{\Vert \varphi (Z)-W\Vert }^2}{\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert }\le \alpha \left(\frac{{\Vert I-\delta {(T)}^{\ast }\delta (Z)\Vert }^2}{1-{\Vert \delta (Z)\Vert }^2}\right).$$

We note that the noncommutative structure then gives that if T is a B-point, then so is U*TU for any unitary matrix U. Furthermore, given two B-points S and T, then S ⊕ T must also be a B-point.

Our second main result, Theorem 4.17, gives a characterization of when a point in ℐ_δ is a B-point, in terms of a realization of a δ nc-model for ϕ. We shall defer an exact statement until Section 4.

Our third result, Theorem 5.3, holds under the assumption that there are a lot of inward directions at T. For now, we shall just give a special case.

Theorem 1.11

Suppose 𝒢_δ is either (1.5) or (1.6). Suppose T ∈ ℐ_δ is a B-point of ϕ. Then

$$\eta (H)=\underset{t\downarrow 0}{\text{lim}}\frac{\varphi (T+\mathit{\text{tH}})-\varphi (T)}{t}$$

exists for all H satisfying T + tH ∈ 𝒢_δ for t small and positive. Moreover η is a holomorphic function of H, which is homogeneous of degree 1.

There is compatibility between the directional derivative at a B-point T, given by the function η in Theorem 1.11, and the directional derivative at a direct sum of m copies of T, which we will denote by T^(m). In particular, if a scalar point is a B-point, then one obtains B-points at all levels.

Corollary 1.12

Suppose 𝒢_δ is either (1.5) or (1.6). Suppose $T\in {ℐ}_{\delta }\cap {𝕄}_1^d$ is a B-point of ϕ. Then

$$\eta (H)=\underset{t\downarrow 0}{\text{lim}}\frac{\varphi (T^{(m)}+\mathit{\text{tH}})-\varphi (T^{(m)})}{t}$$

exists for all m, and for all H satisfying T^(m) + tH ∈ 𝒢_δ for t small and positive. Moreover η is a free function of H, which is homogeneous of degree 1.

Corollary 1.12 follows immediately from the definition of a free function, and the fact that T is a d-tuple of scalars. We leave the details to the interested reader, but emphasize that the noncommutative Julia-Carathéodory really captures a regularity that holds not only within each level $G_{\delta }\cap {𝕄}_n^d$, but also between levels.

In Example 7.1 we consider the function

$$\varphi (Z^1,Z^2)=\frac{1}{2}(Z^1+Z^2)+\frac{1}{2}(Z^1-Z^2){(2-Z^1-Z^2)}^{-1}(Z^1-Z^2)$$

which we show is in the Schur class of (1.4).

2 Background material

2.1 The one variable Julia-Carathéodory theorem

The Julia-Carathéodory Theorem, due to G. Julia [22] in 1920 and C. Carathéodory [16] in 1929, is the following.

Theorem 2.1

Let φ : 𝔻 → 𝔻 be a holomorphic function, and τ ∈ 𝕋. The following are equivalent:

1. ${\text{lim inf}}_{z\to \tau }\frac{1-{|\phi (z)|}^2}{1-{|z|}^2}<\infty$.

2. The quotient $\frac{1-{|\phi (z)|}^2}{1-{|z|}^2}$ has a non-tangential limit as z tends to τ.

3. The function φ has both a non-tangential limit ω ∈ 𝕋 at τ and also an angular derivative η ∈ ℂ, that is the difference quotient

   $$\frac{\phi (z)-\omega }{z-\tau }$$

   has a non-tangential limit η at τ.

4. There exist ω in 𝕋 and η in ℂ so that at τ, φ(z) tends to ω non-tangentially and ϕ′(ζ) tends to η non-tangentially.

Furthermore, if (1.1) holds, then (1.2) does.

On the bidisk, the analogue of (B) does not imply (C); but it is proved in [8] that (C) implies (D). Moreover, it is shown that even when ϕ does not have a holomorphic differential pointing into the bidisk, the one sided derivative exists and is holomorphic in the direction.

2.2 Background on free holomorphic functions

A free holomorphic function is an nc function that is locally bounded with respect to the topology generated by all the sets 𝒢_δ, as δ ranges over all matrices with entries that are free polynomials. These functions are studied in [3], and two principal results are obtained. One is that a bounded function on 𝒢_δ is nc if and only if it is the pointwise limit of a sequence of non-commutative polynomials. The other is that every bounded nc-function on 𝒢_δ has an nc δ-model. Alternative proofs of both these results have been found in [13] and [2]. Before explaining what this is, we need to slightly expand definition 1.3. Let ℰ₁ and ℰ₂ be Hilbert spaces, and let ℒ(ℰ₁, ℰ₂) denote the bounded linear operators from ℰ₁ to ℰ₂. Following [28], we shall write tensor products vertically to enhance readability and condense realization formulas, so $\begin{array}{c}\hfill A\hfill \\ \hfill \otimes \hfill \\ \hfill B\hfill \end{array}$ represents the same object as A⊗B. We shall assume that the domain Ω of any nc function is closed w.r.t. direct sums. Also, we shall for notational convenience assume that δ is a square J-by-J matrix — we can always add rows or columns of zeroes to ensure this. We shall let Ω_n denote $\mathrm{\Omega }\cap {𝕄}_n^d$, and ℐ_n denote the invertible matrices in 𝕄_n.

Definition 2.2

An ℒ(ℰ₁, ℰ₂)-valued nc function F on a set Ω ⊆ 𝕄^[d] is a function satisfying

$$F(x)\in ℒ(\begin{array}{c}\hfill {ℰ}_1\hfill \\ \hfill \otimes \hfill \\ \hfill {ℂ}^n\hfill \end{array},\begin{array}{c}\hfill {ℰ}_2\hfill \\ \hfill \otimes \hfill \\ \hfill {ℂ}^n\hfill \end{array})\forall n,\forall x\in {\mathrm{\Omega }}_n$$

$$F(x\oplus y)=F(x)\oplus F(y)\forall m,n\forall x\in {\mathrm{\Omega }}_n,y\in {\mathrm{\Omega }}_m$$

$$F(s^{-1}\mathit{\text{xs}})=\begin{array}{c}\hfill I_{{ℰ}_2}\hfill \\ \hfill \otimes \hfill \\ \hfill s^{-1}\hfill \end{array}F(x)\begin{array}{c}\hfill I_{{ℰ}_1}\hfill \\ \hfill \otimes \hfill \\ \hfill s\hfill \end{array}\forall n,\forall s\in {ℐ}_n\mathrm{s}.\mathrm{t}.x,s^{-1}\mathit{\text{xs}}\in {\mathrm{\Omega }}_n.$$

Definition 2.3

Let δ be a J-by-J matrix of free polynomials. Let ϕ be an nc-function on 𝒢_δ. A δ nc-model for ϕ is an $ℒ(ℂ,\begin{array}{c}\hfill ℰ\hfill \\ \hfill \otimes \hfill \\ \hfill {ℂ}^J\hfill \end{array})$-valued nc function u on that Ω satisfies

$$I_{{ℂ}^n}-\varphi {(y)}^{\ast }\varphi (x)=u{(y)}^{\ast }\left(\begin{array}{c}\hfill I_{ℰ}\hfill \\ \hfill \otimes \hfill \\ \hfill I_{{ℂ}^J\otimes {ℂ}^n}-\delta {(y)}^{\ast }\delta (x)\hfill \end{array}\right)u(x).$$ (2.4)

Theorem 2.5

[3] An nc function ϕ defined on 𝒢_δ is bounded by 1 in norm if and only if it has a δ nc-model.

To help the reader, let us rewrite Theorem 2.5 in the case that 𝒢_δ is the non-commutative d-polydisk, given by 1.5, and both ℰ₁ and ℰ₂ are one dimensional. Then J = d, and Theorem 2.5 becomes:

Theorem 2.6

Let Ω be the non-commutative d-polydisk, (1.5). The graded function ϕ is in the Schur class of Ω if and only if there are d ℒ(ℂ, ℰ)-valued nc functions u¹, … u^d so that, for all n, for all $x,y\in \mathrm{\Omega }\cap {𝕄}_n^d$, we have

$$I_{{ℂ}^n}-\varphi {(y)}^{\ast }\varphi (x)={\left(\begin{array}{c}\hfill u^1(y)\hfill \\ \hfill u^2(y)\hfill \\ \hfill ⋮\hfill \\ \hfill u^d(y)\hfill \end{array}\right)}^{\ast }\left(\begin{array}{cccc}\hfill I-\begin{array}{c}\hfill I_{ℰ}\hfill \\ \hfill \otimes \hfill \\ \hfill {(y^1)}^{\ast }{x}^1\hfill \end{array}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill I-\begin{array}{c}\hfill I_{ℰ}\hfill \\ \hfill \otimes \hfill \\ \hfill {(y^2)}^{\ast }{x}^2\hfill \end{array}\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \begin{array}{c}\hfill I_{ℰ}\hfill \\ \hfill \otimes \hfill \\ \hfill I-{(y^d)}^{\ast }{x}^d\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}\hfill u^1(x)\hfill \\ \hfill u^2(x)\hfill \\ \hfill ⋮\hfill \\ \hfill u^d(x)\hfill \end{array}\right)={\sum }_{j=1}^d{u}^j{(y)}^{\ast }(I_{ℰ\otimes {ℂ}^n}-\begin{array}{c}\hfill I_{ℰ}\hfill \\ \hfill \otimes \hfill \\ \hfill (y^j)\ast x^j\hfill \end{array})u^j(x).$$

3 Julia’s Inequality and Consequences

We shall assume for the remainder of the paper that δ is a J-by-J matrix of non-commutative polynomials, that ϕ ∈ 𝒮(𝒢_δ), and u is a δ nc-model for ϕ, with values in $ℒ(ℂ,\begin{array}{c}\hfill ℰ\hfill \\ \hfill \otimes \hfill \\ \hfill {ℂ}^J\hfill \end{array})$. We shall further assume that T ∈ ℬ_δ is in ${𝕄}_n^d$. We shall let 𝒰_n denote the n-by-n unitaries.

If ϕ is in 𝒮(𝒢_δ), it follows from Theorem 2.5 that

$$\Vert I-\varphi {(x)}^{\ast }\varphi (x)\Vert \ge {\Vert u(x)\Vert }^2(1-{\Vert \delta (x)\Vert }^2).$$

So if (1.10) holds, then {‖u(Z_j)‖} is bounded, and the following definition is non-vacuous.

Definition 3.1

Let T be a B-point for ϕ. Let Y_T = Y_T (u) denote the set of all weak-limits of u(Z_j), where Z_j is a sequence in 𝒢_δ that converges to T and has

$$\frac{\Vert I-\varphi {(Z_j)}^{\ast }\varphi (Z_j)\Vert }{1-{\Vert \delta (Z_j)\Vert }^2}$$ (3.2)

bounded. We shall call Y_T the cluster set of the model u.

Proposition 3.3

Let T be a B-point for ϕ. Then there exists W ∈ 𝒰_n such that for all υ ∈ Y_T, for all Z in 𝒢_δ, we have

$$I-W^{\ast }\varphi (Z)={\upsilon }^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(T)}^{\ast }\delta (Z)\hfill \end{array}\right)u(Z).$$ (3.4)

Moreover, if

$$\underset{j\to \infty }{\text{lim}}\frac{\Vert I-\varphi {(Z_j)}^{\ast }\varphi (Z_j)\Vert }{1-{\Vert \delta (Z_j)\Vert }^2}\le \alpha$$ (3.5)

holds for some sequence, then there exists υ ∈ Y_T with ‖υ‖² ≤ α.

Proof

Suppose (1.10) holds and u(Z_j) tends weakly to υ. Then ‖I − ϕ(Z_j)*ϕ(Z_j)‖ → 0, so by passing to a subsequence we can assume that ϕ(Z_j) tends to some unitary W. Taking the limit in

$$I-\varphi {(Z_j)}^{\ast }\varphi (Z)=u{(Z_j)}^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(Z_j)}^{\ast }\delta (Z)\hfill \end{array}\right)u(Z).$$

we get (3.4). To see that W is unique, if another sequence $Z_j^{\prime }$ tending to T with $u(Z_j^{\prime })$ weakly convergent had $\varphi (Z_j^{\prime })\to W\prime$, then letting $Z=Z_j^{\prime }$ in (3.4) we get

$$I-W^{\ast }\varphi (Z_j^{\prime })={\upsilon }^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(T)}^{\ast }\delta (Z_j^{\prime })\hfill \end{array}\right)u(Z_j^{\prime }).$$

Since ϕ(Z_j)′ and $\delta (Z_j^{\prime })$ act on finite dimensional spaces, we can pass to a subsequence so that they converge in norm, and $u(Z_j^{\prime })$ converges weakly. In the limit, we get

$$I-W^{\ast }W\prime =0,$$

so W = W′.

For the latter part, note

$$(1-{\Vert \delta (Z_j)\Vert }^2)u{(Z_j)}^{\ast }u(Z_j)\le u{(Z_j)}^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill 1-\delta {(Z_j)}^{\ast }\delta (Z_j)\hfill \end{array}\right)u(Z_j)=I-\varphi {(Z_j)}^{\ast }\varphi (Z_j),$$

so

$${\Vert u(Z_j)\Vert }^2\le \frac{\Vert 1-\varphi {(Z_j)}^{\ast }\varphi (Z_j)\Vert }{1-{\Vert \delta (Z_j)\Vert }^2}.$$

Taking υ to be any weak cluster point of u(Z_j), we get ‖υ‖² ≤ α.

If T is a B-point for ϕ, we shall let ϕ(T) denote the matrix W that satisifies (3.4).

Here is the nc Julia inequality.

Theorem 3.6

Suppose T ∈ ℬ_δ, and

$$\underset{{𝒢}_{\delta }\ni Z\to T}{\text{lim inf}}\frac{\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert }{1-{\Vert \delta (Z)\Vert }^2}=\alpha .$$

Then there exists W ∈ 𝒰_n such that for all Z in 𝒢_δ

$$\frac{{\Vert \varphi (Z)-W\Vert }^2}{\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert }\le \alpha \left(\frac{{\Vert I-\delta {(T)}^{\ast }\delta (Z)\Vert }^2}{1-{\Vert \delta (Z)\Vert }^2}\right).$$ (3.7)

Proof

By Proposition 3.3, we can choose υ in Y_T with ‖υ‖² ≤ α. From (3.4) we have

$$I-W^{\ast }\varphi (Z)={\upsilon }^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(T)}^{\ast }\delta (Z)\hfill \end{array}\right)u(Z),$$

so

$${\Vert \varphi (Z)-W\Vert }^2={\Vert {\upsilon }^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(T)}^{\ast }\delta (Z)\hfill \end{array}\right)u(Z)\Vert }^2\le \alpha {\Vert u(Z)\Vert }^2{\Vert I-\delta {(T)}^{\ast }\delta (Z)\Vert }^2.$$ (3.8)

Now

$$\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert =\Vert u{(Z)}^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(Z)}^{\ast }\delta (Z)\hfill \end{array}\right)u(Z)\Vert \ge \Vert u{(Z)\Vert }^2(1-{\Vert \delta (Z)\Vert }^2).$$ (3.9)

Combining (3.8) and (3.9), we get

$${\Vert \varphi (Z)-W\Vert }^2\le \alpha \left(\frac{{\Vert I-\delta {(T)}^{\ast }\delta (Z)\Vert }^2}{1-{\Vert \delta (Z)\Vert }^2}\right)\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert ,$$

which yields (3.7).

If T ∈ ℐ_δ, a non-tangential approach region is a region of the form

$$\{Z:\Vert \delta (Z)-\delta (T)\Vert \le c(1-{\Vert \delta (Z)\Vert }^2)\}.$$ (3.10)

A corollary of Julia’s lemma is that a function’s behavior is controlled non-tangentially at a B-point on the distinguished boundary.

Proposition 3.11

If T ∈ ℐ_δ is a B-point for ϕ, then

$$\frac{\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert }{1-{\Vert \delta (Z)\Vert }^2}$$

is bounded on all sets that approach T non-tangentially.

Proof

Suppose υ and W are such that (3.4) holds:

$$I-W^{\ast }\varphi (Z)={\upsilon }^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(T)}^{\ast }\delta (Z)\hfill \end{array}\right)u(Z).$$

Fix c, and let S be the non-tangential approach region

$$S=\{Z:\Vert \delta (T)-\delta (Z)\Vert \le c(1-{\Vert \delta (Z)\Vert }^2)\}.$$

By (3.7), we have for Z in S that

$${\Vert \varphi (Z)-W\Vert }^2\le \alpha \left(\frac{{\Vert I-\delta {(T)}^{\ast }\delta (Z)\Vert }^2}{1-{\Vert \delta (Z)\Vert }^2}\right)\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert .$$ (3.12)

Now

$$\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert =\Vert W^{\ast }W-\varphi {(Z)}^{\ast }\varphi (Z)\Vert =\Vert W^{\ast }(W-\varphi (Z))-(\varphi {(Z)}^{\ast }-W^{\ast })\varphi (Z)\Vert \le \Vert W-\varphi (Z)\Vert (\Vert W\Vert +\Vert \varphi (Z)\Vert )\le 2\Vert W-\varphi (Z)\Vert .$$ (3.13)

Squaring and using (3.12), we get

$${\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert }^2\le 4\alpha \left(\frac{{\Vert I-\delta {(T)}^{\ast }\delta (Z)\Vert }^2}{1-{\Vert \delta (Z)\Vert }^2}\right)\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert .$$ (3.14)

Since T ∈ ℐ_δ, we have δ(T) is an isometry, so

$$\Vert \delta (T)-\delta (Z)\Vert =\Vert I-\delta {(T)}^{\ast }\delta (Z)\Vert .$$

Therefore if Z ∈ S, the expression in parantheses on the right-hand side of (3.14) is bounded by c²(1 − ‖δ(Z)‖²), so we get

$$\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert \le 4\alpha c^2(1-{\Vert \delta (Z)\Vert }^2),$$

as required.

4 Models and B-points

In this section we shall study how being a B-point is related to properties of the δ nc model. In the case of the bidisk, these results are in [8].

Proposition 4.1

Let T ∈ ℐ_δ, and suppose Z_j in 𝒢_δ converges to T non-tangentially in the region (3.10). The following are equivalent:

1. ‖I − ϕ(Z_j)*ϕ(Z_j)‖ ≤ M ‖I − δ(Z_j)*δ(Z_j)‖.

2. ‖I − ϕ(Z_j)*ϕ(Z_j)‖ ≤ M′ (1 − ‖δ(Z_j)‖²).

3. ‖u(Z_j)‖² ≤ M″ for some u satisfying (2.4).

4. ‖u(Z_j)‖² ≤ M″ for every u satisfying (2.4).

Proof

(i) ⇒ (iv)

$$(1-{\Vert \delta (Z_j)\Vert }^2)u{(Z_j)}^{\ast }u(Z_j)\le u{(Z_j)}^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(Z_j)}^{\ast }\delta (Z_j)\hfill \end{array}\right)u(Z_j)=I-\varphi {(Z_j)}^{\ast }\varphi (Z_j)\le M\Vert I-\delta {(Z_j)}^{\ast }\delta (Z_j)\Vert I.$$

Therefore

$$(1-{\Vert \delta (Z_j)\Vert }^2){\Vert u(Z_j)\Vert }^2\le M\Vert I-\delta {(Z_j)}^{\ast }\delta (Z_j)\Vert .$$

But

$$\Vert I-\delta {(Z_j)}^{\ast }\delta (Z_j)\Vert =\Vert \delta {(T)}^{\ast }\delta (T)-\delta {(Z_j)}^{\ast }\delta (Z_j)\Vert =\Vert \delta {(T)}^{\ast }(\delta (T)-\delta (Z_j))+(\delta {(T)}^{\ast }-\delta {(Z_j)}^{\ast })\delta (Z_j)\Vert \le (1+\Vert \delta (Z_j)\Vert )(\Vert \delta (T)-\delta (Z_j)\Vert )\le c(1+\Vert \delta (Z_j)\Vert )(1-{\Vert \delta (Z_j)\Vert }^2).$$

Therefore we get

$${\Vert u(Z_j)\Vert }^2\le \mathit{\text{Mc}}(1+\Vert \delta (Z_j)\Vert )\le 2\mathit{\text{Mc}}.$$

(iii) ⇒ (i) Since

$$I-\varphi {(Z_j)}^{\ast }\varphi (Z_j)=u{(Z_j)}^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(Z_j)}^{\ast }\delta (Z_j)\hfill \end{array}\right)u(Z_j),$$

taking norms we get

$$\Vert I-\varphi {(Z_j)}^{\ast }\varphi (Z_j)\Vert \le M″\Vert I-\delta {(Z_j)}^{\ast }\delta (Z_j)\Vert .$$

(i) ⇔ (ii) In the region (3.10), we have

$$1-{\Vert \delta (Z)\Vert }^2\le \Vert I-\delta {(Z)}^{\ast }\delta (Z)\Vert \le 2c(1-{\Vert \delta (Z)\Vert }^2).$$

We can now give a different characterization of B-points that are on the distinguished boundary.

Corollary 4.2

Let T ∈ ℐ_δ, and let u be a δ nc-model for ϕ. Suppose: (NT) There exists some sequence in 𝒢_δ that approaches T non-tangentially.

Then The following are equivalent:

1. T is a B-point of ϕ.

2. u(Z_j) is bounded on some sequence Z_j that approaches T non-tangentially.

3. u(Z) is bounded on every set that approaches T non-tangentially.

4. $\frac{\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert }{1-{\Vert \delta (Z)\Vert }^2}$ is bounded on every set that approaches T non-tangentially.

Proof

(i) ⇒ (iv) by Proposition 3.11, and (iii) ⇒ (ii) is trivial.

(iv) ⇒ (iii) and (ii) ⇒ (i) both follow from Proposition 4.1, and the observation that the proof shows that all the constants M, M′, M″ are comparable once the aperture of the non-tangential approach region is fixed.

Remark

Condition (NT) is very mild. It will hold if Γ(T) (see Def. 4.6 below) is non-empty.

If u is a δ nc model for ϕ, then by [3, Cor. 8.2], there is an isometry (which is called a realization of the model)

$$\left(\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill C\hfill & \hfill D\hfill \end{array}\right):ℂ\oplus \begin{array}{c}\hfill ℰ\hfill \\ \hfill \otimes \hfill \\ \hfill {ℂ}^J\hfill \end{array}\to ℂ\oplus \begin{array}{c}\hfill ℰ\hfill \\ \hfill \otimes \hfill \\ \hfill {ℂ}^J\hfill \end{array}$$ (4.3)

so that for $x\in {𝒢}_{\delta }\cap {𝕄}_n^d$,

$$\left[\begin{array}{cccc}\hfill I_{ℰ}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \otimes \hfill & \hfill \hfill & \hfill D\hfill & \hfill I_{ℰ}\hfill \\ \hfill I_{{ℂ}^J}\hfill & \hfill -\hfill & \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill \otimes \hfill & \hfill \hfill & \hfill I_{{ℂ}^n}\hfill & \hfill \delta (x)\hfill \\ \hfill I_{{ℂ}^n}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\right]u(x)=\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I_{{ℂ}^n}\hfill \end{array},$$ (4.4)

and

$$\varphi (x)=\begin{array}{c}\hfill A\hfill \\ \hfill \otimes \hfill \\ \hfill I_{{ℂ}^n}\hfill \end{array}+\begin{array}{c}\hfill B\hfill \\ \hfill \otimes \hfill \\ \hfill I_{{ℂ}^n}\hfill \end{array}\begin{array}{c}\hfill I_{ℰ}\hfill \\ \hfill \otimes \hfill \\ \hfill \delta (x)\hfill \end{array}u(x)=\begin{array}{c}\hfill A\hfill \\ \hfill \otimes \hfill \\ \hfill I_{{ℂ}^n}\hfill \end{array}+\begin{array}{c}\hfill B\hfill \\ \hfill \otimes \hfill \\ \hfill I_{{ℂ}^n}\hfill \end{array}\begin{array}{c}\hfill I_{ℰ}\hfill \\ \hfill \otimes \hfill \\ \hfill \delta (x)\hfill \end{array}{[I-\begin{array}{c}\hfill D\hfill \\ \hfill \otimes \hfill \\ \hfill I_{{ℂ}^n}\hfill \end{array}\begin{array}{c}\hfill I_{ℰ}\hfill \\ \hfill \otimes \hfill \\ \hfill \delta (x)\hfill \end{array}]}^{-1}\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I_{{ℂ}^n}\hfill \end{array}.$$ (4.5)

For T ∈ ℐ_δ, the inward directions for T are those H such that T + tH is inside 𝒢_δ for t small and positive. Formally, if $T\in {𝕄}_n^d$, and $H\in {𝕄}_n^d$, let

$$\nabla \delta (T)[H]=\underset{t\to 0}{\text{lim}}\frac{1}{t}[\delta (T+\mathit{\text{tH}})-\delta (T)]$$

denote the derivative of δ at T in the direction H. If A is a self-adjoint matrix, we write A < 0 to mean A is negative definite.

Definition 4.6

Let $T\in {𝕄}_n^d$ be in ℐ_δ. The inward set of T is the set

$$\mathrm{\Gamma }(T)=\{H\in {𝕄}_n^d:\Vert H\Vert \le 1,\text{ and Re }[\delta {(T)}^{\ast }\nabla \delta (T)[H]]<0\}.$$

The transverse inward set of T is the subset of Γ(T) defined by

$$\mathrm{\Delta }(T)=\{H\in {𝕄}_n^d:\Vert H\Vert \le 1,\text{ and }\delta {(T)}^{\ast }\nabla \delta (T)[H]<0\}.$$

We have the following elementary result.

Lemma 4.7

Let H ∈ Γ(T). Then there exists ε > 0 such that

$$T+\mathit{\text{tH}}\in {𝒢}_{\delta }\forall 0<t<\epsilon .$$

Moreover, T + tH approaches T non-tangentially as t ↓ 0.

Proof

Let V = δ(T), an isometry since T ∈ ℐ_δ. Then

$$\text{Re }[V^{\ast }\nabla \delta (T)[H]]\le -\beta I,$$

so

$$I-\delta {(T+\mathit{\text{tH}})}^{\ast }\delta (T+\mathit{\text{tH}})=-2\text{Re }[{\mathit{\text{tV}}}^{\ast }\nabla \delta (T)[H]]+O(t^2)\ge 2\beta \mathit{\text{tI}}+O(t^2).$$ (4.8)

This yields the first assertion, and the second follows from this and the mean value theorem, which implies that

$$\Vert \delta (T+\mathit{\text{tH}})-\delta (T)\Vert =O(t).$$

For the rest of this section, we shall make the following assumption:

• (A1)
  The set Δ(T) is non-empty, so there exists $K\in {𝕄}_n^d$ and β > 0 so that

  $$\delta {(T)}^{\ast }\nabla \delta (T)[K]\le -\beta I.$$ (4.9)

Lemma 4.10

Let T ∈ ℐ_δ, and u be a δ nc model for ϕ. Suppose that T is a B-point for ϕ, and that K ∈ Δ(T) satisfies (4.9). Let Z_j = T + t_jK, where 0 < t_j < 1 and t_j → 0. If u(Z_j) converges weakly to υ, then u(Z_j) converges in norm to υ.

Proof

We have

$$I-W^{\ast }\varphi (Z)={\upsilon }^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(T)}^{\ast }\delta (Z)\hfill \end{array}\right)u(Z)$$ (4.11)

$$I-\varphi {(Z)}^{\ast }\varphi (Z)=u{(Z)}^{\ast }\left(\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill I-\delta {(Z)}^{\ast }\delta (Z)\hfill \end{array}\right)u(Z).$$ (4.12)

Let V = δ(T) and X = ∇δ(T)[K]. Let Z = T + tK, so δ(Z) = V + tX + O(t²), and recall that V*V = I since T ∈ ℐ_δ. So the lower parts of the terms in parentheses on the right-hand sides of (4.11) and (4.12) are, respectively, −tV*X + O(t²) and −2tV*X + O(t²). From (4.12) we get

$$(u{(Z)}^{\ast }-{\upsilon }^{\ast })\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill -2{\mathit{\text{tV}}}^{\ast }X\hfill \end{array}u(Z)=I-\varphi {(Z)}^{\ast }\varphi (Z)-{\upsilon }^{\ast }\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill -2{\mathit{\text{tV}}}^{\ast }X\hfill \end{array}u(Z)+O(t^2).$$

Take the real part, and subtract and add twice the real part of (4.11) to get

$$\text{Re }[(u{(Z)}^{\ast }-{\upsilon }^{\ast })\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill -2{\mathit{\text{tV}}}^{\ast }X\hfill \end{array}u(Z)]=I-\varphi {(Z)}^{\ast }\varphi (Z)-2\text{Re }(I-W^{\ast }\varphi (Z))+O(t^2)=-(I-\varphi {(Z)}^{\ast }W)(I-W^{\ast }\varphi (Z))+O(t^2).$$

Therefore

$$\Vert \text{Re }[(u{(Z)}^{\ast }-{\upsilon }^{\ast })\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill V^{\ast }X\hfill \end{array}u(Z)]\Vert \le \frac{{\Vert \varphi (Z)-W\Vert }^2}{2t}+O(t).$$ (4.13)

As Z_j → T within a non-tangential approach region, by Theorem 3.6 and Proposition 4.1 (iii) ⇒ (i), we get some constant M so that

$${\Vert \varphi (Z_j)-W\Vert }^2\le M{\Vert \delta (T)-\delta (Z_j)\Vert }^2=M{\Vert X\Vert }^2{t}^2+O(t^3).$$

Therefore the right-hand side of (4.13) is O(t), and we conclude that, since V*X ≤ − βI,

$$\underset{j\to \infty }{\text{lim}}{\Vert u(Z_j)-\upsilon \Vert }^2\le \frac{1}{\beta }\underset{j\to \infty }{\text{lim}}(u{(Z_j)}^{\ast }-{\upsilon }^{\ast })\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill V^{\ast }X\hfill \end{array}(u(Z_j)-\upsilon )=\frac{1}{\beta }\underset{j\to \infty }{\text{lim}}\text{ Re }[(u{(Z_j)}^{\ast }-{\upsilon }^{\ast })\begin{array}{c}\hfill I\hfill \\ \hfill \otimes \hfill \\ \hfill V^{\ast }X\hfill \end{array}u(Z_j)]=0,$$

so

$$\underset{j\to \infty }{\text{lim}}{\Vert \upsilon -u(Z_j)\Vert }^2=0,$$

as desired.

Lemma 4.14

Under the assumptions of Lemma 4.10, there exists a unique u_T such that

$$u_T\perp \text{ker }[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]$$ (4.15)

and

$$[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]u_T=\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I\hfill \end{array}.$$ (4.16)

Proof

By Corollary 4.2, as t decreases to 0, the vectors u(T + tK) stay bounded; so there is some sequence t_j so that u(T + t_jK) converges weakly, Choose a sequence r_j increasing to 1 so that u(r_jT) converges weakly, and hence, by Lemma 4.10, also in norm, to a vector υ. Writing Z_j = T + t_jK,

$$[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (Z_j)\hfill \end{array}]u(Z_j)=\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I\hfill \end{array},$$

so taking the limit we get

$$[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]\upsilon =\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I\hfill \end{array}.$$

Since $\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I\hfill \end{array}$ is in the range of $[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]$, there exists a a unique vector u_T satisfying (4.16) and (4.15).

We can now give a characterization of B-points, for homogeneous δ’s, in terms of realizations.

Theorem 4.17

Let ϕ ∈ 𝒮(𝒢_δ), let u be a δ nc model for ϕ, and let $\left(\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill C\hfill & \hfill D\hfill \end{array}\right)$ be a realization as in (4.3 – 4.5). Let T ∈ ℑ_δ, and assume that (A1) holds. Then T is a B-point for ϕ if and only if

$$\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I\hfill \end{array}\in \text{Ran }[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}].$$

Proof

If T is a B-point, then the inclusion follows from Lemma 4.14. Conversely, suppose

$$[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]\upsilon =\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I\hfill \end{array}$$

for some vector υ. By (4.4), for any Z in 𝒢_δ we have

$$u(Z)={[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (Z)\hfill \end{array}]}^{-1}\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I\hfill \end{array}={[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (Z)\hfill \end{array}]}^{-1}[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]\upsilon =\upsilon -{[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (Z)\hfill \end{array}]}^{-1}\left[\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)-\delta (Z)\hfill \end{array}\right]\upsilon .$$

Then

$$\Vert u(Z)\Vert \le \Vert \upsilon \Vert (1+\frac{\Vert \delta (Z)-\delta (T)\Vert }{1-\Vert \delta (Z)\Vert }).\le \Vert \upsilon \Vert (1+\frac{2\Vert \delta (Z)-\delta (T)\Vert }{1-{\Vert \delta (Z)\Vert }^2}).$$ (4.18)

Now let Z approach T non-tangentially (such as along T + tK), and the right-hand side of (4.18) stays bounded; therefore T is a B-point by Corollary 4.2.

If δ is homogeneous, then the norm of the vector u_T introduced in Lemma 4.14 is lim_r↑1 ‖u(rT)‖. We shall only prove it when δ is homogeneous of order 1, though the argument can be modified for any positive homogeneity.

Proposition 4.19

Assume that δ(rZ) = rδ(Z). Let T ∈ ℑ_δ be a B-point for ϕ. Then u_T satisfies

$${\Vert u_T\Vert }^2=\underset{r\uparrow 1}{\text{lim}}{\Vert u(\mathit{\text{rT}})\Vert }^2$$ (4.20)

$$=\underset{{𝒢}_{\delta }\ni Z\to T}{\text{lim inf}}\frac{\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert }{1-{\Vert \delta (Z)\Vert }^2}$$ (4.21)

Proof

Let r_j be a sequence increasing to 1 so that u(r_jT) converge weakly, and hence by Lemma 4.10 in norm, to some vector υ. By continuity,

$$[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]\upsilon =\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I\hfill \end{array},$$

so

$$u(r_{j}T)={[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (r_{j}T)\hfill \end{array}]}^{-1}\begin{array}{c}\hfill C\hfill \\ \hfill \otimes \hfill \\ \hfill I\hfill \end{array}={[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill r_j\delta (T)\hfill \end{array}]}^{-1}[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]\upsilon =[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]{[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill r_j\delta (T)\hfill \end{array}]}^{-1}\upsilon .$$

As $\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}$ is a contraction, we have that

$$\text{ker }[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]\perp \text{ ran }[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}].$$

So each u(r_jT) is perpendicular to ker $[I-\begin{array}{cc}\hfill D\hfill & \hfill I\hfill \\ \hfill \otimes \hfill & \hfill \otimes \hfill \\ \hfill I\hfill & \hfill \delta (T)\hfill \end{array}]$ and hence υ is also. Therefore υ is the vector u_T from Lemma 4.14, and (4.20) holds.

To prove (4.21), we need to show that

$$\underset{{𝒢}_{\delta }\in Z\to T}{\text{lim inf}}\frac{\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert }{1-{\Vert \delta (Z)\Vert }^2}=\underset{r\uparrow 1}{\text{lim}}\frac{\Vert I-\varphi {(\mathit{\text{rT}})}^{\ast }\varphi (\mathit{\text{rT}})\Vert }{1-{\Vert \delta (\mathit{\text{rT}})\Vert }^2}$$ (4.22)

Let α denote the left-hand side of (4.22). By Theorem 3.6, we have

$$\frac{{\Vert \varphi (\mathit{\text{rT}})-W\Vert }^2}{\Vert I-\varphi {(\mathit{\text{rT}})}^{\ast }\varphi (\mathit{\text{rT}})\Vert }\le \alpha \left(\frac{{\Vert I-\delta {(T)}^{\ast }\delta (\mathit{\text{rT}})\Vert }^2}{1-{\Vert \delta (\mathit{\text{rT}})\Vert }^2}\right)=\alpha \frac{1-r}{1+r}.$$

As in (3.13), we have ‖I − ϕ(rT)*ϕ(rT)‖ ≤ 2‖W − ϕ(rT)‖. So we have

$$\underset{r\uparrow 1}{\text{lim inf}}\frac{{\Vert I-\varphi {(\mathit{\text{rT}})}^{\ast }\varphi (\mathit{\text{rT}})\Vert }^2}{{(1-{\Vert \delta (\mathit{\text{rT}})\Vert }^2)}^2}\le \underset{r\uparrow 1}{\text{lim inf}}\frac{4{\Vert W-\varphi (\mathit{\text{rT}})\Vert }^2}{{(1-r^2)}^2}\le \underset{r\uparrow 1}{\text{lim inf}}\frac{4}{{(1-r^2)}^2}\alpha \frac{1-r}{1+r}\Vert I-\varphi {(\mathit{\text{rT}})}^{\ast }\varphi (\mathit{\text{rT}})\Vert$$

Dividing by

$$\frac{\Vert I-\varphi {(\mathit{\text{rT}})}^{\ast }\varphi (\mathit{\text{rT}})\Vert }{1-r^2}$$

we get

$$\underset{r\uparrow 1}{\text{lim inf}}\frac{\Vert I-\varphi {(\mathit{\text{rT}})}^{\ast }\varphi (\mathit{\text{rT}})\Vert }{1-{\Vert \delta (\mathit{\text{rT}})\Vert }^2}\le \underset{r\uparrow 1}{\text{lim inf}}\frac{4\alpha }{{(1+r)}^2}=\alpha .$$

So we have proved that (4.22) holds.

5 Derivatives at B-points

Let T be a B-point of ϕ in ℐ_δ, and W = ϕ(T). We will keep these fixed for the remainder of the section. Let us make the following assumption:

• (A) The complex span of Δ(T) is all of ${𝕄}_n^d$.

This condition ensures that we have a full set of transverse directions pointing into G_δ.. Assumption (A) is equivalent to the following two conditions holding:

• (A1) The set Δ(T) is non-empty.

• (A2) The complex span of

  $$\sum (T)≔\{H\in {𝕄}_n^d:\delta {(T)}^{\ast }\nabla \delta (T)[H]\text{ is self}-\text{adjoint}\}$$

  is all of ${𝕄}_n^d$.

Let H ∈ Γ(T); we want to show that

$$\underset{t\downarrow 0}{\text{lim}}\frac{1}{t}[\varphi (T+\mathit{\text{tH}})-W]$$

exists and is holomorphic in H.

First, let us sharpen Lemma 4.7.

Lemma 5.1

Let β > 0. Then there exists ε > 0 such that, if

$$\text{Re }[\delta {(T)}^{\ast }\nabla \delta (T)[H]]\le -\beta I$$ (5.2)

then

$$T+\mathit{\text{tH}}\in {𝒢}_{\delta }\forall 0<t<\epsilon .$$

Proof

This follows from (4.8), and the observation that the error term can be bounded by some absolute constant (which depends on δ and its derivatives in a neighborhood of T) times t².

Let

$$U=\{(z,H)\in ℂ\times {𝕄}_n^d:H\in \mathrm{\Gamma }(T),\mathit{\text{zH}}\in \mathrm{\Gamma }(T),T+\mathit{\text{tzH}}\in {𝒢}_{\delta }\forall 0<t\le 1\}.$$

Consider the set of functions

$$\{{\xi }_t(z,H)=\frac{1}{t}[\varphi (T+\mathit{\text{tzH}})-W]:0<t<1\}.$$

These functions are all defined on U, and are locally bounded by Corollary 4.2, and are holomorphic in both z and H. So they form a normal family by Montel’s theorem. Let S = (t_n) be a sequence decreasing to 0 such that

$$\underset{n\to \infty }{\text{lim}}{\xi }_{t_n}(z,H)$$

exists; call this limit η_S(z, H). We wish to show that η_S does not, in fact, depend on the choice of sequence S.

Let K be in Δ(T). Multiplying K by a small positive number if necessary, we can assume that

$$(z,K)\in U\forall z\in \{\text{Re }(z)>0\text{ and }|z|<2\}.$$

Let υ be a unit vector in ℂⁿ, and define the function

$$f(z)=\langle \varphi (T+\mathit{\text{zK}})\upsilon ,\varphi (T)\upsilon \rangle .$$

Then f : 𝔻(1, 1) → 𝔻, where 𝔻(1, 1) is the disk centered at 1 of radius 1. Moreover, 0 is a B-point for f, because, for t ∈ (0, 1),

$$1-|f(t)|\le 2\Vert I-\varphi {(T+\mathit{\text{tK}})}^{\ast }\varphi (T+\mathit{\text{tK}})\Vert$$

and, letting M denote ‖∇δ(T)‖,

$$\Vert I-\delta {(T+\mathit{\text{tK}})}^{\ast }\delta (T+\mathit{\text{tK}})\Vert \le 2\mathit{\text{Mt}}+O(t^2),$$

so

$$\underset{t\downarrow 0}{\text{lim}}\frac{1-|f(t)|}{t}\le 4M\underset{t\downarrow 0}{\text{lim}}\frac{\Vert I-\varphi {(T+\mathit{\text{tK}})}^{\ast }\varphi (T+\mathit{\text{tK}})\Vert }{1-{\Vert \delta (T+\mathit{\text{tK}})\Vert }^2},$$

and the right-hand side is bounded since T is a B-point of ϕ.

So we can apply the one variable Julia-Carathéodory Theorem 2.1 to conclude that

$$\underset{t\downarrow 0}{\text{lim}}\frac{f(t)-f(0)}{t}$$

exists — indeed the limit exists as 0 is approached non-tangentially from within 𝔻(1, 1). Since this holds for every unit vector υ, by polarization we can conclude that

$$\underset{t\downarrow 0}{\text{lim}}\frac{1}{t}[\varphi (T+\mathit{\text{tK}})-W]$$

exists, so every function η_S agrees on points of the form (t, K), and, by holomorphicity, on points in U of the form (z, K), whenever K ∈ Δ(T).

Now, fix some element K ∈ Δ(T) such that (2, K) ∈ U. Then, for some ε > 0, if H is in Σ(T) and ‖H‖ < ε, then K + H is in Δ(T). So all the η_S agree on points in U of the form (t, K + H), with t > 0 and H in Σ(T). By assumption (A2), since η_S is holomorphic in H, we get that in fact η_S is independent of the choice of S.

Thus we have proved:

Theorem 5.3

Suppose T ∈ ℐ_δ is a B-point of ϕ, and assumption (A) holds. Then

$$\eta (H)=\underset{t\downarrow 0}{\text{lim}}\frac{\varphi (T+\mathit{\text{tK}})-\varphi (T)}{t}$$ (5.4)

exists for all H ∈ Γ(T). Moreover η is a holomorphic function of H, homogeneous of degree 1.

6 Deducing the scalar case from the nc theorem

Knowing Theorem 3.6, how could we deduce the classical Julia inequality? We would need to know that any holomorphic function ψ : 𝔻 → 𝔻 could be extended to a function in the Schur class of {x ∈ 𝕄^[1] : ‖x‖ < 1}. This indeed holds, by von Neumann’s inequality [36].

More generally, suppose Ω is a domain in ℂ^d, and ψ : Ω → 𝔻 is holomorphic. If we wish to deduce a Julia inequality using the results of the previous section, first we need to find a matrix γ of polynomials in d commuting variables so that Ω = {z ∈ ℂ^d : ‖γ(z)‖ < 1}. This of course may not be possible, though it is for the polydisk, or any polynomial polyhedron, and for the ball.

We can define G_γ to be the subset of 𝒢_γ consisting of commuting d-tuples of matrices x = (x¹, …, x^d) for which ‖γ(x)‖ < 1. The original function ψ, which is holomorphic and bounded on $G_{\gamma }\cap {𝕄}_1^d$, can be extended to all of G_γ, either by approximating ψ by polynomials, or using the Taylor functional calculus (see [4, 9] for a discussion). Let us define H^∞(G_γ) to be those holomorphic functions ψ so that

$${\Vert \psi \Vert }_{H^{\infty }(G_{\gamma })}≔\text{sup}\{\Vert \psi (x)\Vert :x\in G_{\gamma }\}$$

is finite. (If Ω were the ball, we would get the multiplier algebra of the Drury-Arveson space). By [5], any function in H^∞(G_γ) can be extended to a bounded function on 𝒢_γ of the same norm. So we can deduce the following corollary of Theorem 3.6.

Corollary 6.1

Let $\mathrm{\Omega }=G_{\gamma }\cap {𝕄}_1^d$ be a domain in ℂ^d, and let ψ be a holomorphic function on Ω. Assume that ‖ψ‖_H^∞(Gγ) = 1. Suppose τ ∈ ∂Ω satisfies

$$\underset{z\to \tau }{\text{lim inf}}\frac{1-{|\psi (z)|}^2}{1-{\Vert \gamma (z)\Vert }^2}=\alpha .$$

Then there exists a complex number ω of modulus 1 so that

$$\frac{{|\psi (z)-\omega |}^2}{1-{|\psi (z)|}^2}\le \alpha \left(\frac{{\Vert I-\gamma {(\tau )}^{\ast }\gamma (z)\Vert }^2}{1-{\Vert \gamma (z)\Vert }^2}\right).$$

In the case of the ball, Corollary 6.1 is proved as Theorem 8.5.3 in [32], though with the weaker assumption that ψ is bounded by 1 in the sup-norm, not in the multiplier norm of the Drury-Arveson space. In [37], K. Wlodarczyk obtained a version for the unit ball of any J*-algebra, which includes the polydisk.

Theorem 5.3 also can be applied to the scalar case. Assumptions (A1) and (A2) can be checked in many concrete cases, such as the ball or the polydisk.

Corollary 6.2

Assume Ω, ψ and τ are as in Corollary 6.1, and that (A) holds at τ. Then ψ has a directional derivative in all inward directions at τ, and moreover this directional derivative is a holomorphic function of the direction.

Of course, if ψ were regular at τ, the directional derivative would be a linear function of the direction.

7 Examples

Example 7.1

Suppose

$$\delta (Z)=\left(\begin{array}{ccc}\hfill Z^1\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill Z^d\hfill \end{array}\right)$$

Then T ∈ ℐ_δ if and only if each T^r is an isometry. We have

$$\mathrm{\Delta }(T)=\{\Vert K\Vert \le 1\text{ and }{T}^{r\ast }{K}^r<0,1\le r\le d\},$$

and this is non-empty (e.g. take K = −T). The set Σ(T) is the set of H such that each H^r is T^r times a self-adjoint, so (A2) is also satisfied.

Let d = 2, and consider the scalar rational inner function

$$f(z,w)=\frac{z+w-2\mathit{\text{wz}}}{2-z-w}.$$ (7.2)

This has a B-point at (1, 1). By Andô’s theorem, f is of norm one on G_δ, so by [5], we can extend f to a function of norm one on 𝒢_δ. It is not immediately obvious how to do so.

Claim

The function

$$\varphi (Z^1,Z^2)=\frac{1}{2}(Z^1+Z^2)+\frac{1}{2}(Z^1-Z^2){(2-Z^1-Z^2)}^{-1}(Z^1-Z^2)$$ (7.3)

is in 𝒮(𝒢_δ) and agrees with f on commuting variables.

Let us temporarily accept the claim. For each n, the point (I_n, I_n) is a B-point, because

$$\underset{r\uparrow 1}{\text{lim}}\frac{\Vert I-\varphi {(\mathit{\text{rI}})}^{\ast }\varphi (\mathit{\text{rI}})\Vert }{1-r^2}=\underset{r\uparrow 1}{\text{lim}}\frac{1-r^2}{1-r^2}=1.$$

Theorem 3.6 then says that for ϕ as in (7.3), and Z a pair of contractions,

$$\frac{{\Vert \varphi (Z)-I\Vert }^2}{\Vert I-\varphi {(Z)}^{\ast }\varphi (Z)\Vert }\le \frac{{\text{max}}_{r=1,2}{\Vert I-Z^r\Vert }^2}{1-{\text{max}}_{r=1,2}{\Vert Z^r\Vert }^2}.$$ (7.4)

If Z¹ = Z², we get equality in (7.4).

If we calculate η(H) as in (5.4), we get that for all H with Re (H¹) and Re (H²) negative definite, the derivative of ϕ at (I_n, I_n) in the direction H is

$$\eta (H)=\frac{1}{2}(H^1+H^2)-\frac{1}{2}(H^1-H^2){(H^1+H^2)}^{-1}(H^1-H^2),$$

which is clearly holomorphic and homogeneous of degree 1.

Proof of claim

We shall write down a realization for f, and extend it to non-commutative variables.

Let

(7.5)

Let

$$\delta (z,w)=\left(\begin{array}{cc}\hfill z\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill w\hfill \end{array}\right),$$

and let

$$u(z,w)={(I-D\delta (z,w))}^{-1}C.$$

Since (7.5) is a unitary matrix, the function A + Bδ(z, w)u(z, w) is a rational inner function on 𝔻², which by inspection agrees with f in (7.2). Now, we keep the same unitary, and using (4.3) – (4.5) (where ℰ is just ℂ, and J = 2), we get a formula for ϕ, which, after some algebra, becomes

$$\varphi (Z)=\frac{1}{2}(\begin{array}{cc}\hfill Z^1\hfill & \hfill Z^2\hfill \end{array}){\left(\begin{array}{cc}\hfill I-\frac{1}{2}{Z}^1\hfill & \hfill \frac{1}{2}{Z}^2\hfill \\ \hfill \frac{1}{2}{Z}^1\hfill & \hfill I-\frac{1}{2}{Z}^2\hfill \end{array}\right)}^{-1}\left(\begin{array}{c}\hfill I\hfill \\ \hfill I\hfill \end{array}\right).$$ (7.6)

Inverting the matrix on the right-hand side of (7.6) using the Boltz-Banachiewicz formula does not lead to a nice formula; but if one expands the inverse in a Neumann series, and observes that

$$\left(\begin{array}{cc}\hfill Z^1\hfill & \hfill -Z^2\hfill \\ \hfill -Z^1\hfill & \hfill Z^2\hfill \end{array}\right)\left(\begin{array}{c}\hfill I\hfill \\ \hfill I\hfill \end{array}\right)=(Z^1-Z^2)\left(\begin{array}{c}\hfill I\hfill \\ \hfill -I\hfill \end{array}\right)$$

$$\left(\begin{array}{cc}\hfill Z^1\hfill & \hfill -Z^2\hfill \\ \hfill -Z^1\hfill & \hfill Z^2\hfill \end{array}\right)\left(\begin{array}{c}\hfill A\hfill \\ \hfill -A\hfill \end{array}\right)=(Z^1+Z^2)\left(\begin{array}{c}\hfill A\hfill \\ \hfill -A\hfill \end{array}\right),$$

then for k ≥ 1, we get

$$\frac{1}{2^k}{\left(\begin{array}{cc}\hfill Z^1\hfill & \hfill -Z^2\hfill \\ \hfill -Z^1\hfill & \hfill Z^2\hfill \end{array}\right)}^k\left(\begin{array}{c}\hfill I\hfill \\ \hfill I\hfill \end{array}\right)=\frac{1}{2^k}{(Z^1+Z^2)}^{k-1}(Z^1-Z^2)\left(\begin{array}{c}\hfill I\hfill \\ \hfill -I\hfill \end{array}\right).$$

Then (7.6) becomes

$$\varphi (Z)=\frac{1}{2}(Z^1+Z^2)+\frac{1}{4}(Z^1-Z^2)\left[\sum _{j=0}^{\infty }\frac{1}{2^j}{(Z^1+Z^2)}^j\right](Z^1-Z^2)$$

and summing the Neumann series we get (7.3), as claimed.

Remark

The function

$$\psi (Z)=(Z^1+Z^2-Z^1{Z}^2-Z^2{Z}^1){(2I-Z^1-Z^2)}^{-1}$$

is another nc extension of f, but it is not in 𝒮(𝒢_δ). Indeed, evaluating it on the pair of unitaries

$$Z=\left(\right(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}),(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\left)\right),$$

we get

$$\psi (Z)=\left(\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right),$$

which has norm $1+\sqrt{2}$. So by continuity

$${\Vert \psi \Vert }_{{𝒢}_{\delta }}\ge \underset{0<r<1}{\text{sup}}\Vert \psi (\mathit{\text{rZ}})\Vert >1.$$

Example 7.7

Let Ω be the classical Cartan domain of symmetric J-by-J contractive matrices in $d=\frac{J(J+1)}{2}$ dimensions. There is an obvious embedding δ that takes d numbers and writes them as a J-by-J symmetric matrix, and we can extend this map to matrices, giving 𝒢_δ and the commutative version G_δ. A point is in the distinguished boundary of 𝒢_δ when δ(T) is a symmetric isometry. (A1) holds at every distinguished boundary point, and so does (A2), since Σ(T) is the set of H such that δ(H) can be written as the sum of d(T) times a self-adjoint matrix and (I − δ(T)*δ(T)) times anything. So if ϕ is in the Schur class of 𝒢_δ or G_δ, we can apply both Theorem 3.6 and Theorem 5.3.