Boundary values and Cowen-Douglas curvature

John E. McCarthy ^{∗} Washington University, St. Louis, Missouri 63130, U.S.A.

April 11 1995

Abstract We define a metric in terms of the Cowen-Douglas curvature for an operator T in B_{1}(Ω). Any boundary point of Ω that is a finite distance, with respect to this metric, from the eigenvalues in the interior is itself an eigenvalue of T. If T is represented as the adjoint of multiplication by the coordinate function on some holomorphic Hilbert space on Ω, this gives a condition under which functions in the space have limits along a path going to the boundary of Ω.

0 Introduction

Let Ω be an open connected set in ℂ, and H be a Hilbert space of functions holomorphic on Ω with the property that evaluation at each point of Ω is a continuous functional on H. It can be shown that there must exist a kernel function k(z, w) on Ω × Ω, holomorphic in z and conjugate holomorphic in w, with the property that, for each w, k(., w) := k_{w}(.) is in H and ⟨f, k_{w}⟩ = f(w) for all f in H [2]. We shall call H a holomorphic Hilbert space on Ω. Define a metric ρ(w) on Ω by

[ρ(w)]^{2} = \frac{∂^{2} log ∥k_{w}∥^{2}}{∂w∂w̄} .  (0.1)

We address the connection between the growth of ρ near the boundary of Ω and the existence of limits for functions in H along paths tending to ∂Ω.

The definition (0.1) is motivated by the work of M. Cowen and R. Douglas in [5]. Let Ω^{∗} := {w̄ : w ∈ Ω}; then the Cowen-Douglas class B_{1}(Ω^{∗}) consists of those bounded linear operators T acting on a separable Hilbert space K that satisfy (i) Each point of Ω^{∗} is an eigenvalue of multiplicity one for T. (ii) Ran(T − w) = K for all w in Ω^{∗}. (iii) ⋁_{w∈Ω^{∗}} Ker(T − w) = K. Any such operator T can be represented as the adjoint of multiplication by the coordinate

∗The author was partially supported by the National Science Foundation grant DMS 9301508.

Mathematics Subject Classification 47B38, 31A20

function on some holomorphic Hilbert space on Ω (this was pointed out in [5], and a detailed proof can be found in [10]). Many of the commonly studied cyclic operators are the adjoints of operators in this class (for example, weighted shifts, or pure cyclic subnormal operators [15]). Cowen and Douglas proved that the metric ρ(w) is a complete unitary invariant for operators in B_{1}(Ω^{∗}), i.e. two such operators are unitarily equivalent if and only if the corresponding metrics are equal everywhere on Ω. Actually they considered K_{T} (w) = −[ρ(w)]^{2}, which is the curvature of the Hermitian holomorphic vector bundle over Ω^{∗} whose fibre over each point w is Ker(T −w), and proved the theorem using techniques of differential geometry; but for our purposes we want to consider ρ as a metric. It can be calculated as in (0.1) for k_{w} any anti-holomorphic cross-section of the bundle.

Even if Ω is a maximal domain of holomorphy for H, it is possible for all functions in H to tend to a limit along certain paths approaching ∂Ω (in operator theoretic terms, this implies that M_{z}^{∗} also has eigenvalues on ∂Ω). For example, in [8] it is shown that if Ω is the unit disk 𝔻(0, 1) minus a sequence of little disks 𝔻(a^{n}, r_{n}), for some 0 < a < 1, and H is the Bergman space on Ω, i.e. all analytic functions f for which ∫_{Ω} |f(z)|^{2}dz ∧ dz̄ < ∞, then lim_{r→0^{−}} f(r) exists for all f in H if and only if ∑_{n=0}^{∞} _{\frac{1}{a^{2n}log \frac{1}{rn}}} < ∞.

Let γ : [0, 1] → \overline{Ω} be a simple arc, with γ([0, 1)) contained in Ω, and γ(1) a point on the boundary of Ω. We are interested in the question of when

lim_{s→1^{−}} f(γ(s))  (0.2)

exists for every f in H, and how this is related to the metric ρ given by (0.1).

As we are considering H to be a space of holomorphic functions, evaluating the derivative of a function is also a continuous linear functional. Define the function k_{w}^{1} by ⟨f, k_{w}^{1}⟩ = f′(w). Then Equation (0.1) becomes, after a routine calculation,

[ρ(w)]^{2} = \frac{∥k_{w}∥^{2}∥k_{w}^{1}∥^{2} − |⟨k_{w}^{1}, k_{w}⟩|^{2}}{∥k_{w}∥^{4}} .  (0.3)

This is often easier to calculate.

The Hardy space for the unit disk, H^{2}, is the Hilbert space of analytic functions on the disk for which the norm, given by

∥ ∑_{n=0}^{∞} a_{n}z^{n}∥_{H^{2}}^{2} = ∑_{n=0}^{∞} |a_{n}|^{2}

is finite. The Bergman space L_{a}^{2}(D) is the space with norm given by

∥ ∑_{n=0}^{∞} a_{n}z^{n}∥_{L_{a}^{2}(D)}^{2} = ∑_{n=0}^{∞} \frac{|a_{n}|^{2}}{n + 1}.

A calculation shows that for the Hardy space, the metric is

ρ_{H^{2}}(w) = \frac{1}{(1 − |w|^{2})},

and for the Bergman space it is

ρ_{L_{a}^{2}(D)}(w) = \frac{\sqrt{2}}{(1 − |w|^{2})}.

It is clear that the limit lim_{r→1^{−}} f(r) does not exist for every f in either of these spaces, as they both contain unbounded functions, and functions that oscillate infinitely often as the boundary is approached.

Let σ denote Lebesgue measure on the unit circle, and P be the orthogonal projection from L^{2}(σ) onto H^{2} (so P (∑_{−∞}^{∞} a_{n}z^{n}) = ∑_{0}^{∞} a_{n}z^{n}). For a in L^{∞}(σ), the Toeplitz operator T_{a} : H^{2} → H^{2} is defined by T_{a}(f) = P (af). The space M(a) is defined to be the range of T_{a}, with an inner product that makes T_{a} into a partial isometry, i.e. if f_{1} and f_{2} are orthogonal to the kernel of T_{a}, then

⟨T_{a}f_{1}, T_{a}f_{2}⟩_{M(a)} = ⟨f_{1}, f_{2}⟩_{H^{2}}.

Now if a is actually analytic, then in M(ā) the kernel function is

k_{w}^{M(ā)} = Tā \frac{a(z)}{1 − w̄z} ,

and

∥k_{w}^{M(ā)}∥^{2} = ∫ \frac{|a(z)|^{2}}{|1 − w̄z|^{2}} dσ(z).

Let a(z) = 1 − z. Then lim_{r→1^{−}} f(r) exists for every f in M(ā): indeed the Fourier series of every function in M(ā) actually converges at 1, not just in the sense of Abel. A calculation yields that

[ρ_{M(ā)}(w)]^{2} = \frac{1}{(1 − |w|^{2})^{2}} − \frac{1}{(1 − |w|^{2} + |1 − w|^{2})^{2}}

so as r tends to 1 radially,

ρ_{M(ā)}(r) = \frac{1}{2\sqrt{1 − r}} + o(\frac{1}{\sqrt{1 − r}}).

Notice that instead of growing like the reciprocal of the distance to the boundary, the metric is growing only like the square root of the reciprocal; this leads one to suspect that the existence of limits along some path terminating at 1 may be related to the growth of the metric along that path.

The situation, however, is more complex, for the following reason. Suppose H is a space for which the limit lim_{r→1^{−}} f(r) always exists, and such that, for some function g, this limit is non-zero. Let G(z) = \frac{1}{1−z}, and consider the space H_{1} = GH, with inner product ⟨Gf_{1}, Gf_{2}⟩_{H_{1}} = ⟨f_{1}, f_{2}⟩_{H}. The metric ρ_{H_{1}}(w) = ρ_{H}(w); but Gg(r) is unbounded as r tends to 1. Notice too that if one considers \frac{1}{G}H^{2}, then every function in the space will have a radial limit at 1, but this limit will always be zero.

The question of interest, then, is when a given holomorphic Hilbert space H can be rescaled by some holomorphic G so that in GH limits exist and are not always zero. This is also the right question from the point of view of operator theory, for multiplication by z on H and GH are unitarily equivalent (see Section 1 below). If k_{γ(r)} converge weakly to a non-zero function k_{1}, then k_{1} will be an eigenvector of T, the adjoint of multiplication by z, of eigenvalue \overline{γ(1)}. So the existence of the limit (0.2) implies that T has an eigenvalue on the boundary of Ω^{∗}.

Our principal results are the following:

Theorem 2.2 Let H be a holomorphic Hilbert space on Ω, with kernel function k(z, w), and such that only a finite number of common zeroes of H lie on the path {γ(r) : 0 ≤ r < 1}. Then H can be rescaled by a non-vanishing holomorphic function G so that Λ : f ↦ lim_{r→1^{−}} f(r) is a non-zero bounded linear functional on GH if and only if, for every a in Ω except for at most a discrete set,

lim_{r→1^{−}} lim_{s→1^{−}} \frac{k(γ(r), γ(s))}{k(a, γ(s))k(γ(r), a)}∥k_{a}∥^{2}

exists and is non-zero, and

lim sup_{r→1^{−}} \frac{k(γ(r), γ(r))}{|k(a, γ(r))|^{2}}∥k_{a}∥^{2} < ∞.

Theorem 3.1 Let H be a reproducing kernel Hilbert space of holomorphic functions on Ω, with only finitely many common zeroes on γ. Suppose that ∫_{0}^{1} ρ_{H}(γ(r))|γ′(r)|dr < ∞. Then there exists a holomorphic function G on Ω such that in the space GH the limit lim_{r→1^{−}} f(γ(r)) always exists and is not always zero.

Corollary 3.8 Let T be in the Cowen-Douglas class B_{1}(Ω), and let λ be a point on the boundary of Ω which is a finite distance from a point in the interior with respect to the metric ρ. Then λ is an eigenvalue of T.

In Section 4, we apply these results to the de Branges-Rovnyak spaces H(b). The results in this section are not new, as the problem of existence of boundary limits in these spaces has been solved by Ahern and Clark for b inner, and Sarason in general [1], [13] (see Section 4 for a definition of the spaces, and a discussion). The results do show, however, that our general theorems have non-trivial applications.

In Section 5 we show that the condition of Theorem 3.1 is not necessary for the existence of an eigenvalue of T ∈ B_{1}(Ω) on the boundary of Ω.

1 Unitary Equivalence

In [5], it is proved that operators in B_{1}(Ω^{∗}) are unitarily equivalent if and only if their curvature functions are equal. This result was generalised to m-tuples of operators on domains in ℂ^{n} by R. Curto and N. Salinas, and they gave a more analytic proof; they also described what form the unitary intertwining operators could take [6]. The following result is therefore not new; that (i) and (iv) are equivalent is contained in [5], and that (i) and (iii) are equivalent is contained in [6]. We think it is worthwhile, however, for expository reasons, to include a simple proof that illustrates our ideas.

Theorem 1.1 Let H and F be holomorphic Hilbert spaces on a planar domain Ω, in neither of which all functions vanish at some point. Suppose that both H and F are invariant under multiplication by the coordinate function, denoted by S_{H} (respectively S_{F}). Suppose, moreover, that H (or F) has the property that if a function f(z) in the space vanishes at some point w in Ω, then (f(z) − f(w))/(z − w) is also in the space.

Let k(z, w) be the kernel function for H, and j(z, w) be the kernel function for F. Then the following are equivalent:

(i) There exists a unitary operator U : H → F such that US_{H}U^{∗} = S_{F}. (ii) There exists a non-vanishing holomorphic function G on Ω such that U(f) := Gf for all f in H defines a unitary operator U : H → F such that US_{H}U^{∗} = S_{F}. (iii) There exists a non-vanishing holomorphic function G on Ω such that

j(z, w) = \overline{G(w)}G(z)k(z, w).

(iv) ρ_{H}(w) = ρ_{F}(w) for all w in Ω.

Proof: (i) ⇒ (ii): By hypothesis, any function in H orthogonal to k_{w} must be in the range of S_{H} − w, so the kernel of (S_{H} − w)^{∗} is precisely the one-dimensional space spanned by k_{w}.

Define G by

Uk_{w} = \frac{1}{\overline{G}(w)}j_{w}.

As U must map the kernel of (S − w)_{H}^{∗} onto the kernel of (S − w)_{F}^{∗}, k_{w} must get mapped to a non-zero multiple of j_{w}, so G is well-defined. Moreover, as

⟨f, k_{w}⟩_{H} = \frac{1}{G(w)}⟨Uf, j_{w}⟩_{F} = \frac{(U(f))(w)}{G(w)},

it follows that U(f) = Gf for all f in H, and hence G is holomorphic.

(ii) ⇒ (iii):

f(w) = ⟨f(z), k_{w}(z)⟩_{H} = ⟨G(z)f(z), G(z)k_{w}(z)⟩_{F}

and

f(w) = ⟨G(z)f(z), \frac{1}{\overline{G(w)}}j_{w}(z)⟩_{F},

so j(z, w) = \overline{G(w)}G(z)k(z, w) as required.

(iii) ⇒ (iv): As G does not vanish, log |G(w)| is harmonic, so (0.1) will be the same for both spaces.

(iv) ⇒ (i): There is a harmonic function h such that

∥j_{w}∥^{2} = e^{2h(w)}∥k_{w}∥^{2}.

We would like to write e^{h(w)} as the modulus of a holomorphic function, but there might be a problem if Ω is not simply connected, so we shall do it locally. Let 𝔻 be a disk contained in Ω, centred at 0. On 𝔻, h has a harmonic conjugate ^{∗}h; let G = e^{h+i^{∗}h}. Define U : H → F by

U(∑_{n=1}^{N} a_{n}k_{w_{n}}) = ∑_{n=1}^{N} a_{n} \frac{1}{\overline{G}(w_{n})}j_{w_{n}}  (1.2)

for w_{1}, …, w_{N} in 𝔻. As Ω is connected, U is densely defined; moreover for w, z in 𝔻

⟨Uk_{w}, Uk_{z}⟩_{F} = ⟨k_{w}, k_{z}⟩_{H}.  (1.3)

(This holds because both j(z, w) and \overline{G(w)}G(z)k(z, w) are holomorphic in z and antiholomorphic in w; as they agree on the diagonal of 𝔻 × 𝔻 by (1.2), they must be identically equal, which implies (1.3)). Therefore U extends by continuity to a unitary operator; and, as ⟨(S_{F}U − US_{H})k_{w}, j_{z}⟩ = 0, U intertwines S_{F} and S_{H} as desired. □

Remark 1: One can also consider spaces that have common zeroes, e.g. zH^{2}. In this case one must modify the definition (0.1) by subtracting off the point masses that one would get at the zeroes, and work with the absolutely continuous part of the Laplacian. One must then modify the preceding theorem to allow G to have zeroes at the common zeroes of F, and poles at the common zeroes of H. (By a common zero of H we mean a point a in Ω such that every function in H vanishes at a (so k_{a} = 0).)

Remark 2. In general, a holomorphic Hilbert space need not be invariant under multiplication by z. If not, (iii) and (iv) of Theorem 1.1 are still equivalent, as is (ii) if it is required merely that multiplication by G is a unitary operator from H onto F. These equivalent conditions seem to be the right notion of equivalence for general holomorphic Hilbert spaces.

2 Rescaling

A space can always be rescaled (i.e. multiplied by a holomorphic function G as in Theorem 1.1) so that the limit (0.2) always exists, by choosing G to vanish sufficiently rapidly. In this section we consider when it can be rescaled so that the limit is non-zero.

We shall use the following theorem of A. Nersejan on approximation by holomorphic functions [9]; if the curve γ is a line segment, this result goes back to T. Carleman [3].

Theorem 2.1 [9] Let D be a proper subdomain of the extended complex plane, and let γ be a continuous injective map from [0, 1] into D̄, with γ([0, 1)) contained in D and γ(1) ∈ ∂D. Let ε(r) be a positive continuous function on [0, 1), and let f(r) be an arbitrary continuous function on [0, 1). Then there exists a function g holomorphic on D such that |f(r) − g(γ(r))| < ε(r) for all r in [0, 1).

We can give necessary and sufficient for non-zero boundary limits to exist in a rescaled space:

Theorem 2.2 Let H be a holomorphic Hilbert space on Ω, with kernel function k(z, w), and such that only a finite number of common zeroes of H lie on the path {γ(r) : 0 ≤ r < 1}. Then H can be rescaled by a non-vanishing holomorphic function G so that Λ : f ↦ lim_{r→1^{−}} f(γ(r)) is a non-zero bounded linear functional on GH if and only if, for every a in Ω except for at most a discrete set,

lim_{r→1^{−}} lim_{s→1^{−}} \frac{k(γ(r), γ(s))}{k(a, γ(s))k(γ(r), a)}∥k_{a}∥^{2}  (2.3)

exists and is non-zero, and

lim sup_{r→1^{−}} \frac{k(γ(r), γ(r))}{|k(a, γ(r))|^{2}}∥k_{a}∥^{2} < ∞.  (2.4)

Proof: (Necessity) Suppose there does exist a holomorphic function G such that in the space GH the kernel functions j_{γ(r)} converge weakly to a non-zero function, j_{1}. Choose a so that ⟨j_{a}, j_{1}⟩ ≠ 0 (this function of a can vanish on at most a discrete set). Then

lim_{r→1^{−}} lim_{s→1^{−}} \frac{k(γ(r), γ(s))}{k(a, γ(s))k(γ(r), a)}∥k_{a}∥^{2} = lim_{r→1^{−}} lim_{s→1^{−}} \frac{j(γ(r), γ(s))}{j(a, γ(s))j(γ(r), a)}∥j_{a}∥^{2}
= \frac{∥j_{1}∥^{2}}{|⟨j_{a}, j_{1}⟩|^{2}} ∥j_{a}∥^{2}
≠ 0.

Moreover,

lim sup_{r→1^{−}} \frac{k(γ(r), γ(r))}{|k(a, γ(r))|^{2}}∥k_{a}∥^{2} = lim sup_{r→1^{−}} \frac{j(γ(r), γ(r))}{|j(a, γ(r))|^{2}}∥j_{a}∥^{2}
= \frac{∥j_{a}∥^{2}}{|⟨j_{a}, j_{1}⟩|^{2}} lim sup_{r→1^{−}} ∥j_{γ(r)}∥^{2}
< ∞.

(Sufficiency) Define G_{1}(z) = \frac{∥k_{a}∥}{k(z,a)}. Then

∥j_{γ(r)}∥^{2} = \frac{∥k_{a}∥^{2}k(γ(r), γ(r))}{|k(a, γ(r))|^{2}},

so from (2.4) it follows that {∥j_{γ(r)}∥^{2}} is bounded. Let j_{1} be the weak-limit of some sequence {j_{γ(s_{n})}} with s_{n} increasing to 1 (such a j_{1} exists because the unit ball of G_{1}H is weakly compact). Now, j_{1} is well-defined, because it follows from (2.3) that, for r sufficiently close to 1,

lim_{s→1^{−}}⟨j_{γ(r)}, j_{γ(s)}⟩

exists and so

⟨j_{γ(r)}, j_{1}⟩

does not depend on the choice of {s_{n}}, and linear combinations of the j_{γ(r)} are dense in G_{1}H. Moreover, the fact that (2.3) is non-zero means that j_{1} is non-zero.

The problem is that G_{1} may not be holomorphic, because k(z, a) can have zeroes in Ω. Let H_{1} be a holomorphic function on Ω which has zeroes only at the poles of G_{1}, and with the same multiplicities, so that G_{1}H_{1} is a non-vanishing holomorphic function on Ω (such an H_{1} exists by Weierstrass’ theorem). The existence of the limit (2.3) means that there exists some δ > 0 such that for 1 − δ < r < 1, k(γ(r), a) ≠ 0 (here we are using the fact that there are no common zeroes of H on the tail-end of γ). Therefore H_{1} is non-zero on the end of the path, so we can apply Theorem 2.1 to find a holomorphic function H_{2} with |H_{1}(γ(r))H_{2}(γ(r))−1| < 1−r for r close to 1. Define G to be G_{1}H_{1}H_{2}. This is holomorphic on Ω, and as G(γ(r)) is close to G_{1}(γ(r)), the same argument that proved limits exist in G_{1}H applies to GH. □

Remark 1. If there is a point a such that lim_{r→1^{−}} k(γ(r), a) exists and is not zero, then rescaling has no effect: either limits already exist before rescaling, or non-zero limits can never be attained by rescaling.

Remark 2. The quantities in (2.3) and (2.4) are invariant under the equivalence relation of Theorem 1.1, and therefore are expressible in terms of the metric ρ_{H}. It would be nice to have a version of Theorem 2.2 in which necessary and sufficient conditions are given explicitly in terms of the metric.

Remark 3. One might think that condition (2.3) should imply (2.4), for j_{1} is defined by its inner product with j_{γ(r)} for r close to 1, linear combinations of which are dense; and then ∥j_{1}∥^{2} is

lim_{r→1^{−}} lim_{s→1^{−}} j(γ(r), γ(s))

which is the value of (2.3). This presupposes, however, that the norm of j_{1} as a linear functional is finite, and it need not be, as the following example shows.

Let

k(z, w) = \frac{(1 − w̄)^{\frac{1}{3}} (1 − z)^{\frac{1}{3}}}{1 − w̄z} + 1.

This is positive definite, so is the reproducing kernel for some holomorphic Hilbert space on the unit disk [2]. Letting a = 0, γ(r) = r, then (2.3) becomes

lim_{r→1^{−}} lim_{s→1^{−}}  2 (\frac{(1 − r)^{\frac{1}{3}} (1 − s)^{\frac{1}{3}}}{1 − rs} + 1) (\frac{1}{(1 − s)^{\frac{1}{3}} + 1}) (\frac{1}{(1 − r)^{\frac{1}{3}} + 1})
= lim_{r→1^{−}} 2 (\frac{1}{(1 − r)^{\frac{1}{3}} + 1})
= 2 ;

whereas (2.4) becomes

lim_{r→1^{−}} 2 (\frac{(1 − r)^{\frac{2}{3}}}{1 − r^{2}} + 1) (\frac{1}{(1 − r)^{\frac{1}{3}} + 1})^{2} = ∞ .

3 Main Theorem

Here is our principal result.

Theorem 3.1 Let H be a reproducing kernel Hilbert space of holomorphic functions on Ω, and assume there are only a finite number of common zeroes of H on the path γ. If ∫_{0}^{1} ρ(γ(r))|γ′(r)|dr < ∞, then there exists a holomorphic function G on Ω such that in the space GH the limit lim_{r→1^{−}} f(γ(r)) always exists and is not always zero.

Proof: (i) Suppose first that there exists c > 0 such that the angle between k_{γ(r)} and k_{γ(r)}^{1} is greater than c, for r ∈ [0, 1). This extra hypothesis means that ( ∥k_{γ(r)}^{1}∥/∥k_{γ(r)}∥ )|γ′(r)| is integrable, and then we can prove the theorem without rescaling.

For simplicity, assume γ is parametrized by arc-length, so |γ′(r)| is identically 1. Choose a < 1 so that

∫_{a}^{1} \frac{∥k_{γ(r)}^{1}∥}{∥k_{γ(r)}∥}dr ≤ \frac{1}{2}.

Then

k_{γ(r)}(ζ) = ∫_{γ(a)}^{ζ}⟨k_{γ(r)}, k_{w}^{1}⟩dw + k_{γ(r)}(γ(a)),

so

k_{γ(r)}(γ(r)) ≤ ∥k_{γ(r)}∥(∫_{γ(a)}^{γ(r)} ∥k_{w}^{1}∥dw + \frac{|k_{γ(r)}(γ(a))|}{∥k_{γ(r)}∥}),

and

1 ≤ ∫_{a}^{r} \frac{∥k_{γ(s)}^{1}∥}{∥k_{γ(s)}∥} \frac{∥k_{γ(s)}∥}{∥k_{γ(r)}∥}ds + \frac{|⟨k_{γ(r)}, k_{γ(a)}⟩|}{∥k_{γ(r)}∥^{2}}.  (3.2)

Suppose that {∥k_{γ(r)}∥} is not bounded. Choose r_{n} tending to 1 so that ∥k_{γ(r_{n})}∥ ≥ sup{∥k_{γ(w)}∥ : 0 ≤ w < r_{n}}. It follows from (3.2) that

1 ≤ ∫_{a}^{r_{n}} \frac{∥k_{γ(s)}^{1}∥}{∥k_{γ(s)}∥}ds + \frac{∥k_{γ(a)}∥}{∥k_{γ(r_{n})}∥}
≤ \frac{1}{2} + \frac{∥k_{γ(a)}∥}{∥k_{γ(r_{n})}∥},

and hence that

∥k_{γ(r_{n})}∥ ≤ 2∥k_{γ(a)}∥.

As a is fixed, this yields a contradiction, so {∥k_{γ(r)}∥ : r ∈ [0, 1)} is bounded.

It follows that for any sequence of points {r_{n}} there is a subsequence {r_{n_{k}}} such that the functions k_{γ(r_{n_{k}})} converge weakly; we must show that if for two different sequences {r_{n}} and {s_{n}} both converging to 1, the kernel functions k_{γ(r_{n})} and k_{γ(s_{n})} both converge weakly, then they converge to the same limit. Deny. Then there is some function f such that, for n large, |f(γ(r_{n})) − f(γ(s_{n}))| ≥ 1. For convenience, assume r_{n} < s_{n}; then

1 ≤ | ∫_{r_{n}}^{s_{n}}⟨f, k_{γ(r)}^{1}⟩dr| ≤ ∥f∥ ∫_{r_{n}}^{s_{n}} ∥k_{γ(r)}^{1}∥dr.  (3.3)

But because {∥k_{γ(r)}∥} is bounded, ∥k_{γ(r)}^{1}∥ is integrable, so the right-hand side of (3.3) tends to zero as r_{n} tends to 1, yielding a contradiction.

It remains to show that k_{γ(r)} does not tend weakly to zero. Suppose it does. Then

0 = ∫_{r}^{1}⟨k_{γ(r)}, k_{γ(s)}^{1}⟩ds + ∥k_{γ(r)}∥^{2},  (3.4)

so

∥k_{γ(r)}∥ ≤ ∫_{r}^{1} ∥k_{γ(s)}^{1}∥ds.

As ∥k_{γ(r)}^{1}∥ is integrable, it follows that k_{γ(r)} tends to zero in norm.

Now, choose r_{n} tending to 1 so that ∥k_{γ(r_{n})}∥ ≥ sup{∥k_{γ(w)}∥ : r_{n} ≤ w < 1}. Then (3.4) applied to r_{n} gives

1 ≤ ∫_{r_{n}}^{1} \frac{∥k_{γ(s)}^{1}∥}{∥k_{γ(s)}∥} \frac{∥k_{γ(s)}∥}{∥k_{γ(r_{n})}∥}ds
≤ ∫_{r_{n}}^{1} \frac{∥k_{γ(s)}^{1}∥}{∥k_{γ(s)}∥}ds,

yielding a contradiction. Therefore the weak-limit of k_{γ(r)} is non-zero, as desired.

(ii) If one rescales by G, so j(z, w) = \overline{G(w)}G(z)k(z, w), one does not change the metric, but one can change the angle between j_{w}^{1} and j_{w}. Letting ξ = \frac{\overline{G′(w)}}{\overline{G(w)}}, a calculation yields

\frac{|⟨j_{w}^{1}, j_{w}⟩|^{2}}{∥j_{w}^{1}∥^{2}∥j_{w}∥^{2}} = \frac{∥k_{w}∥^{2}|ξ|^{2} + 2ℜ(ξ⟨k_{w}, k_{w}^{1}⟩) + |⟨k_{w}^{1}, \frac{k_{w}}{∥k_{w}∥}⟩|^{2}}{∥k_{w}∥^{2}|ξ|^{2} + 2ℜ(ξ⟨k_{w}, k_{w}^{1}⟩) + ∥k_{w}^{1}∥^{2}} .  (3.5)

Let F (ξ) denote the right-hand side of (3.5). Then F is continuous, and vanishes at

ξ_{w} = −\frac{⟨k_{w}^{1}, k_{w}⟩}{∥k_{w}∥^{2}}.

It follows that there is a positive continuous function ε(r) such that if |ξ −ξ_{γ(r)}| < ε(r) then, for w = γ(r), F (ξ) < \frac{1}{2}. By Theorem 2.1, there is a function g holomorphic on ℂ_{∞} \ \{1\} such that

|g(r) + \frac{⟨k_{γ(r)}, k_{γ(r)}^{1}⟩}{∥k_{γ(r)}∥^{2}}| < ε(r)

on [0, 1). Define G(w) = exp ∫_{0}^{w} g(z)dz. Then in GH, the angle between j_{γ(r)} and j_{γ(r)}^{1} is at least \frac{π}{4}, so Case (i) can be applied. □

Remark 1: Equation 0.1 is the standard way of defining the square of the metric associated with a reproducing kernel (see e.g. [7]). The hypothesis that ρ be in L^{1} then says that the path [0, 1) has finite length. The rescaling makes functions Lipschitz with respect to the metric along [0, 1), and the limit of a Lipschitz function along a path of finite length must exist. Then the fact that the functions k_{r} are themselves Lipschitz allows us to prove that they converge to a non-zero function.

Remark 2: If H does have an infinite number of common zeroes on γ accumulating at γ(1), one must rescale by a meromorphic function.

Actually, the kernel functions converge not only weakly, but also in norm:

Corollary 3.6 If ∫_{0}^{1} ρ_{H}(γ(r))|γ′(r)|dr < ∞, then there exists a holomorphic function G on Ω such that in the space GH the kernel functions j_{γ(r)} converge in norm to a non-zero function.

Proof: We can assume, as in the preceding theorem, that the space has been rescaled so that the angle between j_{γ(r)} and j_{γ(r)}^{1} is bounded away from zero, and hence that ∥j_{γ(r)}∥ is bounded and ∥j_{γ(r)}^{1}∥ is integrable. Let j_{1} be the weak limit of j_{γ(r)}; to prove it is also the norm limit, it suffices to prove that lim_{r→1^{−}} ∥j_{γ(r)}∥ = ∥j_{1}∥. But

lim_{r→1^{−}} |∥j_{γ(r)}∥^{2} − ∥j_{1}∥^{2}| = lim_{r→1^{−}} | ∫_{r}^{1}⟨j_{γ(r)}, j_{γ(w)}^{1}⟩dw|
≤ lim_{r→1^{−}} ∥j_{γ(r)}∥ ∫_{r}^{1} ∥j_{γ(w)}^{1}∥dw,  (3.6)

and (3.6) tends to zero, as desired. □

It is possible, in general, that kernel functions can converge weakly but not in norm. Let

k(z, w) = \frac{\sqrt{1 − w̄}\sqrt{1 − z}}{1 − w̄z} + 1.  (3.7)

This is a kernel function for some holomorphic Hilbert space on the disk. By Theorem 2.2, with a = 0, k_{r} converges weakly to k_{1} as r increases to 1. But

∥k_{1}∥^{2} = lim_{r→1^{−}} lim_{s→1^{−}} \frac{\sqrt{1 − r}\sqrt{1 − s}}{1 − rs} + 1 = 1,

and

lim_{r→1^{−}} ∥k_{r}∥^{2} = lim_{r→1^{−}} \frac{|1 − r|}{1 − |r|^{2}} + 1 = \frac{3}{2},

so k_{r} does not converge in norm.

This example also shows that the integrability of the metric is not a necessary condition for the kernel functions to converge weakly to a non-zero function. However the space corresponding to the kernel function (3.7) is not z-invariant (it corresponds to the space \{\sqrt{1 − z}f + λ : f ∈ H^{2}, λ ∈ ℂ\}). In Section 5, an example is given of a z-invariant space that does have a bounded point evaluation on the boundary that is not a finite distance from the interior. (By z-invariant we mean invariant under multiplication by the coordinate function, so it is a model space for some operator in the Cowen-Douglas class).

Theorem 3.1 can be reformulated in the terminology of operator theory. The metric ρ enables one to define the distance between any two points a, b in Ω̄ as the infimum, over all paths γ : [0, 1] → Ω̄ which map 0 to a, 1 to b, and (0, 1) into Ω, of

∫_{0}^{1} ρ(γ(r))|γ′(r)|dr.

Then Theorem 3.1 gives:

Corollary 3.8 Let T be in the Cowen-Douglas class B_{1}(Ω), and let λ be a point on the boundary of Ω which is a finite distance from a point in the interior with respect to the metric ρ. Then λ is an eigenvalue of T.

Proof: Represent T as the adjoint of multiplication by the coordinate function on some holomorphic Hilbert space H, and choose a path γ of finite length (with respect to ρ) that maps [0, 1) into Ω and maps 1 to λ. By Corollary 3.6, in some space GH, the functions j_{γ(r)} converge in norm to a non-zero function j_{1}. As (S_{GH} − γ(r))^{∗}j_{γ(r)} = 0, it follows that (S_{GH} − λ)^{∗}j_{1} = 0. Then (1/G)j_{1} is an eigenvector of T with eigenvalue λ. □

This can be contrasted with the result of Cowen and Douglas (in the proof of Proposition 1.29 in [5]) that if ∂Ω is C^{1} and \overline{Ω} is a spectral set for T, then

ρ(w) ≥ \frac{C}{dist(w, ∂Ω)}.

Note that if \overline{Ω} is a spectral set for T, then no boundary point of Ω can be an eigenvalue; this is explained in [4].

4 An application: de Branges-Rovnyak Spaces

For b a function that is holomorphic in the unit disk and with modulus bounded by 1, the de Branges-Rovnyak space H(b) is defined as Ran(1 − T_{b}T_{b}^{∗})^{1/2}, with an inner product which makes (1 − T_{b}T_{b}^{∗})^{1/2} an isometry from H^{2} onto H(b). For more information about these spaces, see the book by D. Sarason [14], or his papers [12, 11]. When the limit lim_{r→1^{−}} f(r) exists for every function f in H(b) was investigated by P. Ahern and D. Clark, in the case that b is inner (|b| = 1 σ − a.e., so H(b) is in this case just H^{2} ⊖ bH^{2}, a backward shift invariant subspace of H^{2}), and by Sarason in general [1], [13]. The necessary and sufficient condition they found is that b have an angular derivative in the sense of Carathéodory at the point 1, which is equivalent to requiring that

lim_{r→1^{−}} \frac{1 − |b(r)|}{1 − r} = c < ∞.  (4.1)

In H(b) the kernel function

k_{H(b)}(z, w) = \frac{1 − \overline{b(w)}b(z)}{1 − \overline{w}z}.

Let us apply Theorem 2.2, taking a = 0, so k_{0}(z) = 1 − \overline{b(0)}b(z). As |b(0)| < 1 except in the trivial case, and condition (4.1) implies that lim_{r→1^{−}} b(r) exists and has unit modulus [14, p. 48], rescaling has no effect; so from Theorem 2.2 limits exist if and only if

lim_{r→1^{−}} lim_{s→1^{−}} \frac{1 − \overline{b(r)}b(s)}{1 − r̄s}

exists, and

lim sup_{r→1^{−}} \frac{1 − |b(r)|^{2}}{1 − r^{2}} < ∞.

The existence of these limits can be fairly easily shown to be equivalent to the existence of the limit (4.1).

Now consider the Bergman space version of H(b). Let A be area measure on the disk; the Bergman space L_{a}^{2}(D) is the set of holomorphic f for which ∫ |f|^{2}dA < ∞. Let P′ be the orthogonal projection from L^{2}(A) onto L_{a}^{2}(D), and define T′_{m} : L_{a}^{2}(D) → L_{a}^{2}(D) by T′_{m}(f) = P′(mf). Let H′(b) be Ran(1 − T′_{b}T′_{b}^{∗})^{1/2}, with an inner product which makes (1 − T′_{b}T′_{b}^{∗})^{1/2} an isometry. The kernel for H′(b) is

k(z, w) = \frac{1 − \overline{b(w)}b(z)}{(1 − \overline{w}z)^{2}}.

Again letting a = 0, we see by Schwarz’ Lemma that the space can never be rescaled to allow non-vanishing boundary limits.

Now consider Theorem 3.1 applied to the spaces H(b). The metric for H(b) is given by

[ρ_{H(b)}(w)]^{2} = \frac{1}{1 − |w|^{2}} \frac{1}{1 − |b(w)|^{2}} \{\frac{1 − |b(w)|^{2}}{1 − |w|^{2}} − \frac{1 − |w|^{2}}{1 − |b(w)|^{2}} |b′(w)|^{2}\}  (4.2)

If b has an angular derivative in the sense of Carathéodory at 1, then

lim_{w→1^{−}} |b′(w)| = lim_{w→1^{−}} \frac{1 − |b(w)|^{2}}{1 − |w|^{2}} ,

so the term in braces in (4.2) tends to zero. If b is in C^{2}[0, 1], the term in braces vanishes to at least first order, so

ρ_{H(b)}(w) = O(\frac{1}{\sqrt{1 − |w|^{2}}} )

and is clearly integrable. (Without some extra assumption on b, such as being C^{2} on the radius, it is hard to estimate the growth of ρ.)

5 An Example

We give an example of a naturally occurring z-invariant space in which lim_{r→1^{−}} f(r) exists for every f, but for which ∫ 1_{\frac{1}{2}} ρ(r)dr = ∞.

Example 5.1

Let Ω be the disk 𝔻(1, 1) minus the disks 𝔻(a_{n}, r_{n}), where a_{n} = 1+\frac{1}{2^{n}}, and r_{n} = exp(−(n^{2}4^{n})). Let H be the Bergman space for Ω. It follows from [8] that the norm of the functional f ↦ lim_{r→1^{−}} f(r) is comparable to

∑_{n=1}^{∞} \frac{1}{(1 − a_{n})^{2} log \frac{1}{r_{n}}} = ∑_{n=1}^{∞} \frac{1}{4^{−n}4^{n}n^{2}} < ∞,

and so

lim sup_{x→0^{+}} ∥k_{(1−x)}∥ < ∞ .  (5.2)

Lemma 5.3 For x small and positive,

∥k_{(1−x)}^{1}∥ ≥ C \frac{1}{x log \frac{1}{x}} .

Proof: Choose a positive integer m so that x is in the interval [2^{−(m+1)}, 2^{−m}]. Let f(z) = \frac{1}{z−a_{m}}. Then

∥f∥ ≈ \sqrt{log \frac{1}{r_{m}}} = m2^{m}.

Therefore

∥k_{(1−x)}^{1}∥ ≥ \frac{|f′(1 − x)|}{∥f∥}
≥ C \frac{1}{(2^{−m} + x)^{2}} \frac{1}{m2^{m}}
≥ C \frac{1}{x log \frac{1}{x}} ,

as desired. □ Now, we claim that ρ(r) is not integrable on [\frac{1}{2}, 1]. Indeed we claim more:

lim inf_{x→0^{+}} [x log \frac{1}{x}] ρ(1 − x) > 0.  (5.4)

For suppose (5.4) is false, and there is a sequence x_{n} decreasing to zero for which the left-hand side of (5.4) tends to zero. For each x, let η_{x} be the number with modulus

|η_{x}| = \frac{|⟨k_{(1−x)}, k_{(1−x)}^{1}⟩|}{∥k_{(1−x)}^{1}∥^{2}}

and argument chosen so that ⟨k_{(1−x)}, η_{x}k_{(1−x)}^{1}⟩ is positive. Let l_{x} be the vector k_{(1−x)} − η_{x}k_{(1−x)}^{1}, so

∥l_{x}∥ = \frac{∥k_{(1−x)}∥^{2}}{∥k_{(1−x)}^{1}∥} ρ(1 − x) ≤ Cx log \frac{1}{x}ρ(1 − x)

by (5.2) and (5.4). Take the inner product of l_{x_{n}} with \frac{1}{z−a_{m}}. Then

|\frac{1}{x_{n} + 2^{−m}} − η_{x_{n}} \frac{1}{(x_{n} + 2^{−m})^{2}} | = |⟨\frac{1}{z − a_{m}} , l_{x_{n}}⟩|
≤ C(m2^{m})x_{n} log \frac{1}{x_{n}} ρ(1 − x_{n})  (5.5)

by Lemma 5.3. Letting n tend to infinity in (5.5), the right-hand side tends to zero, so

lim_{n→∞} η_{x_{n}} = 2^{−m}.

Now do the same thing with a different m to get a contradiction. □

We do not know if k_{(1−x)} converges in norm in this example. Note that if r_{n} is reduced to r_{n} = exp(−(n^{2+ε}4^{n})) for any positive ϵ, then ρ is integrable.

Question 5.6 Let H be a z-invariant holomorphic Hilbert space on Ω. Suppose ρ(1 − x) ≥ \frac{c}{x}. Can k_{(1−x)} converge in norm to a non-zero k_{1}?

References

[1] P. Ahern and D. Clark. Radial limits and invariant subspaces. Amer. J. Math., 92:332– 342, 1970.

[2] N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68:337–404, 1950.

[3] T. Carleman. Sur un théorème de Weierstrass. Arkiv. Mat. Astron. Fys., 20(4):1–5, 1927.

[4] D.N. Clark and G. Misra. On weighted shifts, curvature and similarity. J. London Math. Soc., 31:357–368, 1985.

[5] M.J. Cowen and R.G. Douglas. Complex geometry and operator theory. Acta Math., 141:187–261, 1978.

[6] R. Curto and N. Salinas. Generalized Bergman kernels and the Cowen-Douglas theory. Amer. J. Math., 106(2):447–488, 1984.

[7] S. Krantz. Function theory of several complex variables. Wiley, New York, 1982.

[8] J.E. McCarthy and L. Yang. Bounded point evaluations on the boundaries of L regions. Indiana Math. J., 43(3):857–883, 1994.

[9] A.A. Nersejan. Carleman sets. Izv. Akad. Nauk. Armjan. SSR Ser. Mat., 6:465–471, 1971.

[10] S. Richter. Invariant subspaces in Banach spaces of analytic functions. Trans. Amer. Math. Soc., 304:585–616, 1987.

[11] D. Sarason. Doubly shift-invariant subspaces in H^{2}. J. Operator Theory, 16:75–97, 1986.

[12] D. Sarason. Shift-invariant spaces from the Brangesian point of view. In The Bieberbach Conjecture—Proceedings of the Symposium on the Occasion of the Proof, pages 153– 186, Providence RI, 1986. American Mathematical Society.

[13] D. Sarason. Angular derivatives via Hilbert space. Complex Variables, 10:1–10, 1988.

[14] D. Sarason. Sub-Hardy Hilbert spaces in the unit disk. University of Arkansas Lecture Notes, Wiley, New York, 1994.

[15] J.E. Thomson. Approximation in the mean by polynomials. Annals of Math., 133:477– 507, 1991.