Hilbert Spaces on the Unit Disc

Before we discuss some typical examples of Hilbert spaces on the unit disc, we show that manyL²-spaces of holomorphic functions have continuous point evaluations.

Proposition 5.2.5. Let Ω⊆C be an open subset and ρ: Ω→ R^×+ be a mea-surable function such that every pointp∈Ωhas a neighborhood on which ρ is bounded below by someε >0. Then

H:=n

f ∈ O(Ω) : Z

Ω

|f(z)|²ρ(z)dz <∞o is a Hilbert space with respect to the scalar product

hf, gi:=

Z

Ω

f(z)g(z)ρ(z)dz,

and the inclusion H → O(Ω) is continuous with respect to the topology of uni-form convergence on compact subsets sets on O(Ω). In particular, the point evaluations onHare continuous.

Proof. Letp∈Ω and r >0 such that the closed disc D:=B_≤r(p) of radiusr is contained in Ω. Then the compactness ofD implies thatρis bounded below by someε >0 onD. Forf ∈ O(Ω) we obtain from the mean value property

Now the Cauchy–Schwarz inequality inL²(D, dz) leads to

|f(p)|²≤ 1

Applying this to smaller discs B_s(q) ⊆ B_r(p), we find for q ∈ B_r−s(p) the estimate

|f(q)|²≤ 1 επs²kfk².

This estimate proves that every Cauchy sequence (f_n)_n∈NinHis a Cauchy sequence onB_r−s(p) with respect to the sup-norm, hence uniformly convergent to a continuous holomorphic function f:B_r−s(p) → C. Therefore the limit f(z) := lim_n→∞fn(z) exists pointwise on Ω andf is holomorphic because it is holomorphic in a neighborhood of anyp∈Ω.

Since each compact subsetB⊆Ω can be covered by finitely many discs on which the sequence converges uniformly, it converges uniformly on B, so that we obtain Con-vergence Theorem, so thatf ∈ H.

To see thatfn→f holds inH, letδ >0 andkfn−fmk ≤δform, n≥n0. consider for a realm >1 the Hilbert space

Hm={f ∈ O(D) :kfkm<∞}, scalar multiplication, we obtain with Proposition 2.3.8 a continuous unitary representation ofTonHm, given by (t.f)(w) :=f(tw).

Further, Proposition 5.2.5 implies thatHm is a reproducing kernel Hilbert space and Proposition 5.1.6 shows that its kernel K^m isT-invariant, i.e.,

K^m(tz, tw) =K^m(z, w) for z, w∈ D, t∈T.

From the Fundamental Theorem on Unitary Representations of Compact Groups (Theorem 4.2.20) we now derive thatHm is an orthogonal direct sum

of the T-eigenspaces Hm,n, corresponding to the characters χ_n(t) := tⁿ. If

do not extend holomorphically toD.

To see which monomials are contained inHm, we calculate kpnk²_m:

whenevern >0 andk >−1. Proceeding further, we obtainI0,k= 1 and thus

In,k = n!

. We know already from above that the orthogonal family (pn)n∈N0 is complete. With Theorem 3.1.3(iii), we thus obtain for the reproducing kernel

K^m(z, w) = Bergman space ofD. It is the space of square integrable holomorphic functions onD.

One also obtains an interesting “limit space” for m= 1. This can be done as follows. OnO(D) we consider

kfk²:= lim

To evaluate this expression, letf(z) =P∞

n=0a_nzⁿ denote the Taylor series of f about 0 which converges uniformly on each compact subset ofD. Hence we can interchange integration and summation and obtain

1

Applying the Monotone Convergence Theorem to the sequences (|an|²r²ⁿ)_n∈N∈

`<sup>1</sup>(N0), we see that (|a<sub>n</sub>|<sup>2</sup>)<sub>n∈N</sub>∈`¹(N0) if and only ifkfk<∞, and that in this casekfk²=P∞

n=0|a_n|².Therefore

H1:={f ∈ O(D) :kfk<∞} ∼=`²(N0,C)

is a Hilbert space and the polynomials form a dense subspace ofH₁. Moreover, the monomialsp_n(z) =zⁿform an orthonormal basis ofH₁. Note thatkp_nk= 1 is exactly the limit obtained for generalm >1 ifmtends to 1. We put

K¹(z, w) =

This proves thatH1has continuous point evaluations and that its reproducing kernel is given byK¹.

The space H1 is called theHardy space of Dand K¹ is called the Cauchy kernel. This is justified by the following observation. For each holomorphic functionf onDextending continuously to the boundary, we obtain the simpler formula for the norm:

We see, in particular, that such a function is contained inH1and thus f(z) =hf, K_z¹i= 1

where the latter integral denotes a complex line integral. This means that the fact thatK¹ is the reproducing kernel forH1is equivalent toCauchy’s integral formula

We have already seen how the spaces H_m decompose under the unitary representation of the group T, but the spaces H_m carry form ∈ N a unitary representation of the larger group

G:= SU_1,1(C) :=n

We claim that

σ_g(z) :=g.z:= (az+b)(bz+a)⁻¹

defines a continuous action of G on D. Note that this expression is always defined because |z| <1 and |b|<|a|implies thatbz+a6= 0. Thatσg(z)∈ D forz∈ Dfollows from

|az+b|²=|a|²|z|²+ (abz+abz) +|b|²<|b|²|z|²+ (abz+abz) +|a|²=|bz+a|². The relationsσ1(z) =zandσgg⁰ =σgσg⁰ are easily verified (see Exercise 5.2.5).

To see that this action is transitive, we note that for|z|<1, g:= 1

p1− |z|² 1 z

z 1

∈SU1,1(C) satisfiesg.0 =z.

To obtain a unitary action ofGonH_m, we have to see how the corresponding kernelK^m transforms under the action ofG. For the kernelQ(z, w) = 1−zw an easy calculation shows that

Q(g.z, g.w) = 1−(az+b) (bz+a)

(aw+b)

(bw+a) = (bz+a)(a+bw)−(az+b)(aw+b) (bz+a)(a+bw)

= (|a|²− |b|²)(1−zw)

(bz+a)(a+bw) = Q(z, w) (bz+a)(a+bw). Finally, we note that,

J(g, z) :=a−bz

defines a 1-cocycle for the action of G on D, which can be verified by direct calculation), and

J(g, g.z) =a−baz+b

bz+a= abz+|a|²−baz− |b|²

bz+a = 1

bz+a, so that we obtain for

J^m(g, z) :=J(g, z)^−m the relation

K^m(g.z, g.w) =J^m(g, g.z)K(z, w)J^m(g, g.w), and Proposition 5.1.6 show that, form∈N,

(π_m(g)f)(z) =J^m(g, z)f(g⁻¹.z) = (a−bz)^−mf

az−b a−bz

defines a continuous unitary representation ofG= SU1,1(C) onH^m. Form= 2, the spaceH2 is the Bergman space of the disc and

J²(g, z) = 1

(a−bz)² =σ⁰_g−1(z)

for g ∈ SU_1,1(C), so that the existence of the representation in this case also follows from Exercise 5.2.4.

Remark 5.2.7. From Example 3.3.6 we recall the positive definite kernels K^s(z, w) := (1−zw)^−s, s >0,

on the open unit discD ⊆C. We have seen in Example 5.2.6 that fors∈N, we have a unitary representation of SU_1,1(C) on the corresponding Hilbert space.

The reason for restricting to integral values ofsis that otherwise we don’t have a corresponding cocycle. However, forQ(z, w) = 1−zw, we have

Q(g.z, g.w) = Q(z, w)

(bz+a)(a+bw).= Q(z, w)

|a|²(1 + (b/a)z)(1 + (b/a)w)Q(z, w), and therefore

K^s(g.z, g.w) =θg(z)θg(w)K^s(z, w) for

θ_g(z) :=|a|^s(1 + (b/a)z)^s,

where, in view of|b|<|a|, the right hand side can be defined by a power series converging inD.

As we shall see below, these considerations lead to a projective unitary rep-resentation of SU1,1(C) onHsby

(πs(g)f)(z) :=θg(g⁻¹.z)f(g⁻¹.z).

Im Dokument An Introduction to Unitary Representations of Lie Groups (Seite 111-116)