Henkin measures for the Drury-Arveson space

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by  Michael Hartz

HENKIN MEASURES FOR THE DRURY-ARVESON SPACE
7 MICHAEL HARTZ
1
0
Abstract. We exhibit Borel probability measures on the unit sphere
2
inCdford≥2whichareHenkinforthemultiplieralgebraoftheDrury-
n
Arveson space, but not Henkin in the classical sense. This provides a
a
negative answer to a conjecture of Clouâtre and Davidson.
J

6
2

] 1. Introduction
A
F Let Bd denote the open unit ball in Cd and let A(Bd) be the ball algebra,
. which is the algebra of all analytic functions on B which extend to be
h d
t continuous on Bd. A regular complex Borel measure µ on the unit sphere
a S = ∂B is said to be Henkin if the functional
m d d
[ A(Bd) C, f fdµ,
1 → 7→ ZSd
v extends to a weak- continuous functional on H∞(B ), the algebra of all
d
7 bounded analyticfu∗nctions on B . Equivalently, whenever (f )is asequence
7 d n
7 in A(Bd) which is uniformly bounded on Bd and satisfies limn→∞fn(z) = 0
7 for all z B , then
d
0 ∈
. lim f dµ = 0.
1 n
n→∞ZS
0 d
7 Henkinmeasuresplayaprominentroleinthedescriptionofthedualspace
1 of A(B ) and of peak interpolation sets for the ball algebra, see Chapter
d
:
v 9 and 10 of [17] for background material. Such measures are completely
i characterized by a theorem of Henkin [14] and Cole-Range [8]. To state the
X
theorem, recall that a Borel probability measure τ on S is said to be a
r d
a representing measure for the origin if
f dτ = f(0)
ZS
d
for all f A(B ).
d
∈
Theorem 1.1 (Henkin, Cole-Range). A regular complex Borel measure µ
on S is Henkin if and only if it is absolutely continuous with respect to some
d
representing measure for the origin.
2010 Mathematics Subject Classification. Primary 46E22; Secondary 47A13.
Key words and phrases. Drury-Arveson space, Henkin measure, totally singular mea-
sure, totally nullset.
The author was partially supported bya Feodor Lynen Fellowship.
1
2 MICHAELHARTZ
If d = 1, then the only representing measure for the origin is the normal-
ized Lebesgue measure on the unit circle, hence the Henkin measures on the
unitcircle are precisely thosemeasures which areabsolutely continuous with
respect to Lebesgue measure.
In addition to their importance in complex analysis, Henkin measures
alsoplayaroleinmultivariableoperatortheory[13]. However, ithasbecome
clearovertheyearsthatforthepurposesofmultivariableoperatortheory,the
“correct” generalization of H∞, the algebra of bounded analytic functions on
theunitdisc,tohigherdimensions isnotH∞(B ), butthemultiplier algebra
d
oftheDrury-ArvesonspaceH2. This isthereproducing kernelHilbertspace
d
on B with reproducing kernel
d
1
K(z,w) = .
1 z,w
−h i
A theorem of Drury [11] shows that H2 hosts a version of von Neumann’s
d
inequality for commuting row contractions, that is, tuples T = (T ,...,T )
1 d
of commuting operators on a Hilbert space such that the row operator
H
[T ,...,T ] : d is a contraction. The corresponding dilation theorem
1 d
H → H
is due to Müller-Vasilescu [15] and Arveson [5]. The Drury-Arveson space
is also known as symmetric Fock space [5, 10], it plays a distinguished role
in the theory of Nevanlinna-Pick spaces [1, 2] and is an object of interest in
harmonic analysis [9, 4]. An overview of the various features of this space
can be found in [18].
In [7], Clouâtre and Davidson generalize much of the classical theory of
Henkinmeasures to the Drury-Arveson space. Let denote the multiplier
d
M
algebra of H2 and let be the norm closure of the polynomials in . In
particular, fudnctions inAd belong to A(B ). Clouâtre and Davidson dMefidne a
d d
regular Borel measure µAon S to be -Henkin if the associated integration
d d
A
functional
C, f fdµ
d
A → 7→ ZS
d
extends to a weak- continuous functional on (see Subsection 2.1 for the
d
∗ M
definition of weak- topology). Equivalently, whenever (f ) is a sequence in
n
such that f ∗ 1 forall n N and lim f (z) = 0 for all z B ,
Ad || n||Md ≤ ∈ n→∞ n ∈ d
then
f (z)dµ = 0,
n
ZS
d
see [7, Theorem 3.3]. This notion, along with the complementary notion of
-totally singular measures, is crucial in the study of the dual space of
d d
A A
and of peak interpolation sets for in [7].
d
A
Compelling evidence of the importance of -Henkin measures in multi-
d
A
variable operator theory can be found in [6], where Clouâtre and Davidson
extend the Sz.-Nagy-Foias H∞–functional calculus to commuting row con-
tractions. Recall that every contraction T on a Hilbert space can be written
HENKIN MEASURES FOR THE DRURY-ARVESON SPACE 3
as T = T U, where U is a unitary operator and T is completely non-
cnu cnu
⊕
unitary (i.e. has no unitary summand). Sz.-Nagy and Foias showed that in
the separable case, T admits a weak- continuous H∞-functional calculus if
∗
and only if the spectral measure of U is absolutely continuous with respect
to Lebesgue measure on the unit circle, see [19] for a classical treatment.
Clouâtre and Davidson obtain a complete generalization of this result. The
appropriate generalization of a unitary is a spherical unitary, which is a tu-
ple of commuting normal operators whose joint spectrum is contained in
the unit sphere. Every commuting row contraction admits a decomposition
T = T U, where U is a spherical unitary and T is completely non-
cnu cnu
⊕
unitary (i.e. has no spherical unitary summand), see [6, Theorem 4.1]. The
following result is then a combination of Lemma 3.1 and Theorem 4.3 of [6].
Theorem 1.2 (Clouâtre-Davidson). Let T be a commuting row contraction
actingon aseparable Hilbert space withdecomposition T = T U as above.
cnu
⊕
Then T admits a weak- continuous -functional calculus if and only if the
d
∗ M
spectral measure of U is -Henkin.
d
A
This result shows that for the theory of commuting row contractions, -
d
A
Henkin measures are a more suitable generalization of absolutely continuous
measures on the unit circle than classical Henkin measures. Thus, a charac-
terization of -Henkin measures would be desirable.
d
Since the uAnit ball of is contained in the unit ball of A(B ), it is trivial
d d
A
that every classical Henkin measure is also -Henkin. Clouâtre and David-
d
A
sonconjectured[7,Conjecture5.1]thatconversely, every -Henkinmeasure
d
A
is also a classical Henkin measure, so that these two notions agree. If true,
the classical theory would apply to -Henkin measures and in particular,
d
A
theHenkinand Cole-Rangetheorem would provide acharacterization of -
d
A
Henkin measures. They also formulate a conjecture for the complementary
notion of totally singular measure, which turns out to be equivalent to their
conjecture on Henkin measures [7, Theorem 5.2]. Note that the conjecture
is vacuously true if d= 1, as = H∞.
1
M
The purpose of this note is to provide a counterexample to the conjec-
ture of Clouâtre and Davidson for d 2. To state the main result more
precisely, we require one more definit≥ion. A compact set K S is said
d
⊂
to be totally null if it is null for every representing measure of the origin.
By the Henkin and Cole-Range theorem, a totally null set cannot support a
non-zero classical Henkin measure.
Theorem 1.3. Let d 2 be an integer. There exists a Borel probability
measure µ on S which≥is -Henkin and whose support is totally null.
d d
A
In fact, every measure which is supported on a totally null set is totally
singular (i.e. it is singular with respect to every representing measure of
the origin). The measure in Theorem 1.3 therefore also serves at the same
timeacounterexample totheconjectureofClouâtreandDavidsonontotally
singular measures, even without invoking [7, Theorem 5.2].
4 MICHAELHARTZ
It is not hard to see that if µ is a measure on S which satisfies the
d
conclusion of Theorem 1.3, then so does the trivial extension of µ to Sd′ for
any d′ d(seeLemma 2.3), henceit suffices toprove Theorem 1.3 ford = 2.
≥
In fact, the construction of such a measure µ is easier in the case d = 4, so
will consider that case first.
The remainder of this note is organized as follows. In Section 2, we recall
some of the necessary background material. Section 3 contains the construc-
tion of a measure µ which satisfies the conclusion of Theorem 1.3 in the case
d= 4. In Section 4, we prove Theorem 1.3 in general.
2. Preliminaries
2.1. The Drury-Arveson space. As mentioned in the introduction, the
Drury-Arveson space H2 is the reproducing kernel Hilbert space on B with
d d
reproducing kernel
1
K(z,w) = .
1 z,w
−h i
For background material on reproducing kernel Hilbert spaces, see [16] and
[3]. We will require a more concrete description of H2. Recall that if α =
d
(α ,...,α ) Nd is a multi-index and if z = (z ,...,z ) Cd, one usually
1 d 1 d
∈ ∈
writes
zα = zα1...zαd, α! = α !...α !, α = α +...+α .
1 d 1 d | | 1 d
The monomials zα form an orthogonal basis of H2, and
d
α!
zα 2 =
|| ||Hd2 α!
| |
for every multi-index α, see [5, Lemma 3.8].
Let(x )and(y )betwosequencesofpositivenumbers. Wewritex y
n n n n
≃
to mean that there exist C ,C > 0 such that
1 2
C y x C y for all n N.
1 n n 2 n
≤ ≤ ∈
The following well-known result can be deduced from Stirling’s formula, see
[5, p.19].
Lemma 2.1. Let d N. Then
∈
(z z ...z )n 2 d−nd(n+1)(d−1)/2
|| 1 2 d ||Hd2 ≃
for all n N. (cid:3)
∈
The multiplier algebra of H2 is
d
= ϕ :B C : ϕf H2 for all f H2 .
Md { d → ∈ d ∈ d}
Every ϕ gives rise to a bounded multiplication operator M on H2,
∈ Md ϕ d
and we set ϕ = M . Moreover, we may identify with a unital
|| ||Md || ϕ|| Md
subalgebraofB(H2),thealgebraofboundedoperators onH2. Itisnothard
d d
to see that is WOT-closed, and hence weak- closed, inside of B(H2).
Md ∗ d
HENKIN MEASURES FOR THE DRURY-ARVESON SPACE 5
Thus, becomes a dual space in this way, and we endow it with the
d
M
resulting weak- topology. In particular, for every f,g H2, the functional
∗ ∈ d
C, ϕ M f,g ,
d ϕ
M → 7→ h i
is weak- continuous. Moreover, it is well known and not hard to see that
∗
on bounded subsets of , the weak- topology coincides with the topology
d
of pointwise convergencMe on B . ∗
d
2.2. Henkin measures and totally null sets. Let K S be a compact
d
set. A function f A(B ) is said to peak on K if f = 1 on⊂K and f(z) < 1
d
∈ | |
for all z B K. Recall that K is said to be totally null if it is null for
d
∈ \
every representing measure of the origin. In particular, if d = 1, then K is
totally null if and only if it is a Lebesgue null set. We will make repeated
use of the following characterization of totally null sets, see [17, Theorem
10.1.2].
Theorem 2.2. A compact set K S is totally null if and only if there
d
exists a function f A(B ) which p⊂eaks on K.
d
∈
If d′ d, then we may regard Sd Sd′ in an obvious way. Thus, every
regular≥Borel measure µ on Sd admits⊂a trivial extension µ to Sd′ defined by
µ(A) = µ(A S )
∩ d b
for Borel sets A ⊂ Sd′. The bfollowing easy lemma shows that it suffices to
prove Theorem 1.3 in the case d = 2.
Lemma 2.3. Let µ be a Borel probability measure on S , let d′ d and let
d
µ be the trivial extension of µ to Sd′. ≥
(a) If µ is d-Henkin, then µ is d′-Henkin.
b (b) If the sAupport of µ is a totallAy null subset of S , then the support of
d
µ is a totally null subsetbof Sd′.
Proof. (a) Let P : Cd′ Cd denote the orthogonal projection onto the first
b
→
d coordinates. It follows from the concrete description of the Drury-Arveson
space at the beginning of Subsection 2.1 that
V : Hd2 → Hd2′, f 7→ f ◦P,
is an isometry. Moreover, V∗MϕV = Mϕ|Bd for every ϕ ∈Md′, so that
d′ d, ϕ ϕ B ,
M 7→ M 7→ d
(cid:12)
is weak- -weak- continuous and maps d′ into(cid:12) d.
∗ ∗ A A
Suppose now that µ is -Henkin. Then there exists a weak- continuous
d
A ∗
functional Φ on which extends the integration functional given by µ,
d
M
thus
ϕdµ = ϕdµ = Φ(ϕ )
B
ZSd′ ZSd (cid:12) d
forϕ d′. Sincetheright-habnd sidedefines awe(cid:12)ak- continuous functional
∈ A ∗
on d′, we see that µ is d′-Henkin.
M A
b
6 MICHAELHARTZ
(b) We have to show that if K S is totally null, then K is also totally
d
null as a subset of Sd′. But this⊂is immediate from Theorem 2.2 and the
observation that if f A(Bd) peaks on K, then f P A(Bd′) peaks on K
as well, where P deno∈tes the orthogonal projection◦fro∈m (a). (cid:3)
3. The case d= 4
The goal of this section is to prove Theorem 1.3 in the case d = 4 (and
hence for all d 4 by Lemma 2.3). To prepare and motivate the construc-
≥
tionofthemeasureµ,webeginbyconsideringanalogues ofHenkinmeasures
for more general reproducing kernel Hilbert spaces on the unit disc. Sup-
pose that is a reproducing kernel Hilbert space on the unit disc D with
H
reproducing kernel of the form
∞
(1) K(z,w) = a (zw)n,
n
nX=0
where a = 1 and a > 0 for all n N. If ∞ a < , then the series
0 n ∈ n=0 n ∞
above converges uniformly on D D, and Pbecomes a reproducing kernel
Hilbert space of continuous funct×ions on DHin this way. In particular, eval-
uation at 1 is a continuous functional on and hence a weak- continuous
H ∗
functional on Mult( ). Indeed,
H
ϕ(1) = M 1,K(,1)
ϕ H
h · i
for ϕ Mult( ). Therefore, the Dirac measure δ induces a weak- con-
1
∈ H ∗
tinuous functional on Mult( ), but it is not absolutely continuous with re-
H
spect to Lebesgue measure, and hence not Henkin. (In fact, every regular
Borel measure on the unit circle induces a weak- continuous functional on
∗
Mult( ).)
H
ThemainideaoftheconstructionistoembedareproducingkernelHilbert
space as in the preceding paragraph into H2.
4
To find the desired space on the disc, recall that by the inequality of
H
arithmetic and geometric means,
sup z z ...z : z B = d−d/2,
1 2 d d
{| | ∈ }
andthe supremum is attained if andonly if z = ... = z = d−1/2. Hence,
1 d
| | | |
r :B D, z 16z z z z ,
4 1 2 3 4
→ 7→
indeed takes values in D, and it maps B onto D. For n N, let
4
∈
a = r(z)n −2,
n || ||H42
and let be the reproducing kernel Hilbert space on D with reproducing
H
kernel
∞
K(z,w) = a (zw)n.
n
nX=0
HENKIN MEASURES FOR THE DRURY-ARVESON SPACE 7
Lemma 3.1. The map
H2, f f r,
H → 4 7→ ◦
is an isometry, and ∞ a < .
n=0 n ∞
Proof. It is well knowPn that for any space on D with kernel as in Equation
(2),themonomials zn formanorthogonal basisand zn 2 = 1/a forn N.
n
|| || ∈
Thus, with our choice of (a ) above, we have
n
1
zn 2 = = r(z)n 2 .
|| || an || ||H42
Since the sequence r(z)n is an orthogonal sequence in H2, it follows that V
4
is an isometry. Moreover, an application of Lemma 2.1 shows that
r(z)n 2 = 44n (z ,...,z )n 2 (n+1)3/2,
1 4
|| || || || ≃
so that a (n+1)−3/2, and hence ∞ a < . (cid:3)
n ≃ n=0 n ∞
P
Let
h: T3 S , (ζ ,ζ ,ζ ) 1/2(ζ ,ζ ,ζ ,ζ ζ ζ )
4 1 2 3 1 2 3 1 2 3
→ 7→
and observe that the range of h is contained in r−1( 1 ). Let µ be the
pushforward of the normalized Lebesgue measure m on{T}3 by h, that is,
µ(A)= m(h−1(A))
for a Borel subset A of S . We will show that µ satisfies the conclusion of
4
Theorem 1.3.
Lemma 3.2. The support of µ is totally null.
Proof. Let X = r−1( 1 ), which is compact, and define f = 1+r. Then f
belongs to the unit ba{ll}of A(B ) and peaks on X, hence X is tot2ally null by
4
Theorem 2.2. Since h(T3) X, the support of µ is contained in X, so the
support of µ is totally null⊂as well. (cid:3)
The following lemma finishes the proof of Theorem 1.3 in the case d = 4.
Lemma 3.3. The measure µ is -Henkin.
4
A
Proof. Let α N4 be a multi-index. Then
∈
zαdµ = zα hdm = 2−|α| ζα1−α4ζα2−α4ζα3−α4dm.
ZS ZT3 ◦ ZT3 1 2 3
4
This integral is zero unless α = α = α = α =: k, in which case it equals
4 1 2 3
2−4k.
Let g = K(,1) r, where K denotes the reproducing kernel of . Then
· ◦ H
g H2 by Lemma 3.1, and it is a power series in z z z z . Thus, zα is
∈ 4 1 2 3 4
orthogonal to g unless α = ... = α =: k, in which case
1 4
hzα,giH42 = 2−4khr(z)k,giH42 = 2−4khzk,K(·,1)iH = 2−4k,
where we have used Lemma 3.1 again.
8 MICHAELHARTZ
Hence,
ZS ϕdµ = hMϕ1,giH42
4
for all polynomials ϕ, and hence for all ϕ . Since the right-hand side
4
∈ A
obviously extendstoaweak- continuous functional inϕon , weseethat
4
µ is -Henkin. ∗ M (cid:3)
4
A
4. The case d= 2
In this section, we will prove Theorem 1.3 in the case d = 2 and hence in
full generality by Lemma 2.3. To this end, we will also embed a reproducing
kernel Hilbert space on D into H2. Let
2
r : B D, z 2z z ,
2 1 2
→ 7→
and observe that r maps B onto D. For n N, let
2
∈
a = r(z)n −2,
n || ||H22
and consider the reproducing kernel Hilbert space on D with reproducing
kernel
∞
K(z,w) = a (zw)n.
n
nX=0
This space turns out to be the well-known weighted Dirichlet space ,
1/2
which is the reproducing kernel Hilbert space on D with reproducing kDernel
(1 zw)−1/2. This explicit description is not strictly necessary for what
−
follows, but it provides some context for the arguments involving capacity
below.
Lemma 4.1. The kernel K satisfies K(z,w) = (1 zw)−1/2.
−
Proof. The formula for the norm of monomials in Section 2 shows that
(2n)! 1/2
a = r(z)n −2 = 4−n = ( 1)n − ,
n || || (n!)2 − (cid:18) n (cid:19)
so that
∞
1/2
K(z,w) = ( 1)n − (zw)n = (1 zw)−1/2
− (cid:18) n (cid:19) −
nX=0
by the binomial series. (cid:3)
The analogue of Lemma 3.1 in the case d = 2 is the following result.
Lemma 4.2. The map
H2, f f r,
D1/2 → 2 7→ ◦
is an isometry. Moreover, a (n+1)−1/2 and zn 2 (n+1)1/2.
n ≃ || ||D1/2 ≃
HENKIN MEASURES FOR THE DRURY-ARVESON SPACE 9
Proof. AsintheproofofLemma3.1,weseethatV isanisometry. Moreover,
Lemma 2.1 shows that
zn 2 = r(z)n 2 = 22n (z z )n 2 (n+1)1/2
|| ||D1/2 || ||H22 || 1 2 ||H22 ≃
for n N. (cid:3)
∈
The crucial difference to the case d = 4 is that the functions in do
1/2
D
not all extend to continuous functions on D. This makes the construction of
the measure µ of Theorem 1.3 more complicated.
The following lemma provides a measure σ on the unit circle which will
serve as a replacement for the Dirac measure δ , which was used in the case
1
d = 4. It is very likely that this result is well known. Since the measure σ
is crucial for the construction of the measure µ, we explicitly indicate how
such a measure on the unit circle can arise.
Lemma 4.3. There exists a Borel probability measure σ on T such that
(a) the support of σ has Lebesgue measure 0, and
(b) the functional
C[z] C, p pdσ,
→ 7→ ZT
extends to a bounded functional on the space .
1/2
D
To prove Lemma 4.3, we require the notion of capacity. Background ma-
terial on capacity can be found in [12, Section 2]. Let k(t) = t−1/2. The
1/2-energy of a Borel probability measure ν on T is defined to be
I (ν)= k(x y )dν(x)dν(y).
k
ZTZT | − |
WesaythatacompactsubsetE Thaspositive Rieszcapacity of degree 1/2
⊂
ifthereexistsaBorelprobability measureν supportedonE withI (ν)< .
k
∞
Proof of Lemma 4.3. Let E T be a compact set with positive Riesz ca-
⊂
pacity of degree 1/2, but Lebesgue measure 0. For instance, since 1/2 <
log2/log3, the circular middle-third Cantor set has this property by [12,
Exercise 2.4.3 (ii)]. Thus, there exists a measure σ on T whose support is
contained in E with I (σ) < . Then (a) holds.
k
To prove (b), for n Z, let∞
∈
σ(n)= z−ndσ(z)
ZT
b
denote the n-th Fourier coefficient of σ. Since I (σ) < , an application of
k
∞
[12, Exercise 2.4.4] shows that
∞ σ(n)2
(2) | | < .
(n+1)1/2 ∞
nX=0 b
10 MICHAELHARTZ
Let now p be a polynomial, say
N
p(z) = α zn.
n
nX=0
Then using the Cauchy-Schwarz inequality, we see that
N
pdσ α σ( n)
n
(cid:12)(cid:12)ZT (cid:12)(cid:12) ≤ nX=0| || − |
(cid:12) (cid:12) b
N 1/2 N σ(n)2 1/2
(n+1)1/2 α 2 | | .
n
≤ (cid:16)nX=0 | | (cid:17) (cid:16)nX=0(nb+1)1/2(cid:17)
Lemma 4.2 shows that the first factor is dominated by C p for some
|| ||D1/2
constant C, and the second factor is bounded uniformly in N by (2). Thus,
(b) holds. (cid:3)
Remark 4.4. The last paragraph of the proof of [12, Theorem 2.3.5] in fact
shows that the Cantor measure on the circular middle-thirds Cantor set has
finite 1/2-energy, thus we can take σ to be this measure.
Let now σ be a measure provided by Lemma 4.3 and let E be the support
of σ. Let
1
h: T E S , (ζ ,ζ ) (ζ ,ζ ζ ),
2 1 2 1 1 2
× → 7→ √2
and observe that the range of h is contained in r−1(E). Define µ to be the
pushforward of m σ by h. We will show that µ satisfies the conclusion of
×
Theorem 1.3.
Lemma 4.5. The support of µ is totally null.
Proof. Let X = r−1(E). Since E has Lebesgue measure 0 by Lemma 4.3,
there exists by the Rudin-Carleson theorem (i.e. the d = 1 case of Theorem
2.2) a function f A(D) which peaks on E. Let f = f r. Then f belongs
0 0
to A(B ) and peak∈s on X, so that X is totally null by Th◦eorem 2.2. Finally,
d
the support of µ is contained in X, hence it is totally null as well. (cid:3)
The following lemma finishes the proof of Theorem 1.3.
Lemma 4.6. The measure µ is -Henkin.
2
A
Proof. For all m,n N, we have
∈
zmzndµ = 2−(m+n)/2 ζm−nζndm(ζ )dσ(ζ ).
ZS2 1 2 ZTZE 1 2 1 2
This quantity is zero unless m = n, in which case it equals
2−n ζndσ(ζ).
Z
E