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arXiv:2105.08522v1 [math.FA] 17 May 2021
Reflection positivity and Hankel operators—
the multiplicity free case
Maria Stella Adamo, Karl-Hermann Neeb, Jonas Schober
May 19, 2021
Abstract
We analyze reflection positive representations in terms of positive Hankel operators. This
is motivated by the fact that positive Hankel operators are described in terms of their Carleson
measures, whereas the compatibility condition between representations and reflection positive
Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representa-
tion of triples (G, S, τ ), where Gis a group, τan involutive automorphism of Gand S⊆Ga
subsemigroup with τ(S) = S−1. For the triples (Z,N,−idZ), corresponding to reflection pos-
itive operators, and (R,R+,−idR), corresponding to reflection positive one-parameter groups,
we show that every Hankel positive representation can be made reflection positive by a slight
change of the scalar product. A key method consists in using the measure µHon R+defined
by a positive Hankel operator Hon H2(C+) to define a Pick function whose imaginary part,
restricted to the imaginary axis, provides an operator symbol for H.
Keywords: Hankel operator, reflection positive representation, Hardy space, Widom Theorem,
Carleson measure,
MSC 2020: Primary 47B35; Secondary 47B32, 47B91.
Contents
1 Hankel operators for reflection positive representations 5
2 Reflection positivity and Hankel operators 8
3 Reflection positive one-parameter groups 11
3.1 Hankel operators on H2(C+) ............................ 12
3.2 Widom’s Theorem for the upper half-plane . . . . . . . . . . . . . . . . . . . . 14
3.3 The symbol kernel of a positive Hankel operator . . . . . . . . . . . . . . . . . 15
4 Schober’s representation theorem 16
4.1 An operator symbol for H.............................. 16
4.2 From Hankel positivity to reflection positivity . . . . . . . . . . . . . . . . . . . 21
A Widom’s Theorem on Hankel operators on the disc 23
B The Banach ∗-algebra (H∞(Ω), ♯)25
B.1 The weak topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
B.2 Subsemigroups spanning weakly dense subalgebras . . . . . . . . . . . . . . . . 27
B.3 Weakly continuous positive functionals . . . . . . . . . . . . . . . . . . . . . . . 29
B.4 Weakly continuous representations . . . . . . . . . . . . . . . . . . . . . . . . . 31
1
B.5 The unit group of H∞................................ 32
B.6 The representation on H2(Ω) ............................ 32
B.7 The Carleson measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
C Cauchy and Poisson kernels 34
Introduction
This paper contributes to the operator theoretic background of reflection positivity, a basic con-
cept in constructive quantum field theory ([GJ81, JOl98, JOl00, Ja08]) that recently required
some interest from the perspective of the representation theory of Lie groups (see [N ´
O14, N ´
O15]
and the survey booklet [N ´
O18] which contains further references).
The main novelty of this paper is that we analyze reflection positive representations in terms
of positive Hankel operators. This is motivated by the fact that positive Hankel operators can
be described nicely in terms of their Carleson measures, whereas the compatibility condition
between representations and reflection positive Hilbert spaces is quite intricate. This leads us
to the concept of a Hankel positive representation of a triple (G, S, τ ), where Gis a group, τ
an involutive automorphism of Gand S⊆Ga subsemigroup with τ(S) = S−1. For the triples
(Z,N,−idZ), corresponding to reflection positive operators, and (R,R+,−idR), corresponding
to reflection positive one-parameter groups, we show that every Hankel positive representation
can be made reflection positive by a slight change of the scalar product.
To introduce our abstract conceptual background, we define a symmetric semigroup as a
triple (G, S, τ ), where Gis a group and S⊆Gis a subsemigroup satisfying τ(S)−1=S, so
that s♯:= τ(s)−1defines an involution on S. A representation of the pair (G, S) is a triple
(E,E+, U ), where U:G→U(E) is a unitary representation and E+⊆ E is a closed subspace
satisfying U(S)E+⊆ E+. It is said to be regular if E+contains no non-zero U(G)-invariant
subspace and the smallest U(G)-invariant subspace containing E+is E.
Additional positivity is introduced by the concept of a reflection positive Hilbert space, which
is a triple (E,E+, θ), consisting of a Hilbert space Ewith a unitary involution θand a closed
subspace E+satisfying
hξ, ξ iθ:= hξ, θξ i ≥ 0 for ξ∈ E+.(1)
Areflection positive representation of (G, S, τ ) is a quadruple (E,E+, θ, U ), where (E,E+, θ) is
a reflection positive Hilbert space and (E,E+, U ) is a representation of the pair (G, S) where U
and θsatisfy the following compatibility condition
θU (g)θ=U(τ(g)) for g∈G. (2)
Any reflection positive representation specifies three representations:
(L1) the unitary representation Uof the group Gon E,
(L2) the representation U+of the semigroup Son E+by isometries,
(L3) a ∗-representation ( b
E,b
U) of the involutive semigroup (S, ♯), induced by U+on the Hilbert
space b
Eobtained from the positive semidefinite form h·,·iθon E+.
The difficulty in classifying reflection positive representations lies in the complicated com-
patibility conditions between E+,θand U. For the groups G=Zand Rthat we study in this
paper, it is rather easy, resp., classical, to understand the regular representation (E,E+, U ) of the
pair (G, S). For (G, S) = (Z,N), this amounts to describe for a unitary operator Uall invariant
subspaces E+, and for (G, S) = (R,R+), one has to describe for a unitary one-parameter group
(Ut)t∈Rall subspaces E+invariant under (Ut)t>0. Beuerling’s Theorems for the disc and the
2
upper half plane solve this problem in terms of inner functions (cf. [Pa88, Thm. 6.4], [Sh64]).
Adding to such triples (E,E+, U ) a unitary involution θsuch that (E,E+, θ, U ) is reflection
positive is tricky because the θ-positivity of E+is hard to control.
Similarly, the description of all triples (E, θ , U) satisfying (2) is the unitary representation
theory of the semidirect product G⋊{idG, τ }, which is well-known for Zand R. To fit in
subspaces E+becomes complicated by the two requirements of θ-positivity and U(S)-invariance
of E+.
The new strategy that we follow in this paper is to focus on the intermediate level (L2) of
the representation U+of the involutive semigroup (S, ♯) by isometries on E+. On this level, we
introduce the concept of a U+-Hankel operator. These are the operators H∈B(E+) satisfying
HU+(s) = U+(s♯)∗Hfor s∈S. (3)
Although it plays no role for the representations of the pair (G, S ), the involution ♯on Sis a
crucial ingredient of the concept of a Hankel operator.
To illustrate these structures, let us take a closer look at the triple (Z,N0,−idZ), i.e., we
study reflection positive unitary operators U∈U(E) on a reflection positive Hilbert space
(E,E+, θ), which means that
UE+⊆ E+and θU θ =U∗.(4)
Classical normal form results for the isometry S:= U+(1) on E+(assuming regularity) imply
that the triple (E,E+, U ) is equivalent to (L2(T,K), H2(D,K), U ), where Kis a multiplicity
space,
D={z∈C:|z|<1}
is the open unit disc, H2(D,K) is the K-valued Hardy space on D, and (U(1)f)(z) = zf(z),
z∈T, is the multiplication operator corresponding to the bilateral shift on L2(T,K). Our
assumption of multiplicity freeness means that K=C. In this case U+-Hankel operators are
precisely classical Hankel operators, realized as operators on H2(D). The difficult part in the
classification of reflection positive operators consists in a description of all unitary involutions
θturning (E,E+, U ) = (L2(T), H2(D), U ) into a reflection positive representation. The com-
patibility with Uis easy to accommodate. It means that θis of the form
(θhf)(z) = h(z)f(z) with h:T→T, h(z) = h(z) for z∈T.
The hardest part is to control the positivity of the form h·,·iθon E+=H2(D). Here the key
observation is that, if P+is the orthogonal projection L2(T)→H2(D), then Hh:= P+θhP∗
+is
a Hankel operator whose positivity is equivalent to (L2(T), H2(D), θh) being reflection positive.
Positive Hankel operators Hon H2(D) are most nicely classified in terms of their Carleson
measures µHon the interval (−1,1) via the relation
hξ, H ηiH2(D)=Z1
−1
ξ(x)η(x)dµH(x) for ξ, η ∈H2(D).
Widom’s Theorem (see [Wi66] and Theorem A.1 in the appendix) characterizes these mea-
sures in very explicit terms. Our main result on positive Hankel operators on the disc asserts
that all these measures actually arise from reflection positive operators on weighted L2-spaces
L2(T, δ dz), where δis a bounded positive weight for which δ−1is also bounded (Theorem 4.8).
As a consequence, the corresponding weighted Hardy space H2(D, δ) coincides with H2(D),
endowed with a slightly modified scalar product.
The results for reflection positive one-parameter groups concerning (R,R+,−idR) are sim-
ilar. Here the Lax–Phillips Representation Theorem shows that a regular multiplicity free
representation (E,E+, U) of the pair (R,R+) is equivalent to (L2(R), H2(C+), U ), where C+=
{z∈C: Im z > 0}is the upper half-plane, H2(C+) is the Hardy space on C+, and (U(t)f)(x) =
3
eitxf(x), x∈R, is the multiplication representation. Again, U+-Hankel operators are the clas-
sical Hankel operators on H2(C+) (cf. [Pa88, p. 44]). The unitary involutions compatible with
Uin the sense of (2) are of the form
(θhf)(x) = h(x)f(−x) with h:R→T, h(−x) = h(x) for x∈R.
Now Hh:= P+θhP∗
+is a Hankel operator on H2(C+) and (L2(R), H 2(C+), θh) is reflection
positive if and only if Hhis positive. Instead of trying to determine all functions hfor which
this is the case, we focus on positive Hankel operators Hon H2(C+) because they completely
determine the ∗-representation in (L3). We prove a suitable version of Widom’s Theorem for
the upper half plane (Theorem 3.7) that characterizes the Carleson measures µHon R+which
are determined by
hf, H giH2(C+)=ZR+
f(iλ)g(iλ)dµH(λ) for f, g ∈H2(C+).
For C+we show that all these measures actually arise from reflection positive one-parameter
groups on weighted L2-spaces L2(R, w dx), where wis a bounded positive weight for which
w−1is also bounded. As a consequence, the corresponding weighted Hardy space H2(C+, w dz )
coincides with H2(C+) but is endowed with modified scalar product.
Our key method of proof is to observe that the measure µHdefines a holomorphic function
κ:C\(−∞,0] →C, κ (z) := ZR+
λ
1 + λ2−1
z+λdµH(λ)
whose imaginary part defines a bounded function
hH(p) := i
π·Im(κ(ip))
which is an operator symbol of H(Theorem 4.1). As h(R)⊆iR, adding real constants, we
obtain operator symbols for Hwhich are invertible in L∞(R), and this is used in Subsection 4.2
to show that, for (Z,N,−idZ) and (R,R+,−idR) all multiplicity free regular Hankel positive
representations can be made reflection positive by modifying the scalar product with an inverti-
ble intertwining operator. On the level of representations, this means to pass from the Hardy
space H2(D), resp., H2(C+) to the Hardy space corresponding to a boundary measure with a
positive bounded density whose inverse is also bounded.
Since the Banach algebras H∞(D)∼
=H∞(C+) play a central role in our arguments, we
decided to discuss some of their key features in an appendix. In view of the Riemann Mapping
Theorem, this can be done for an arbitrary proper simply connected domain Ω ⊆C, endowed
with an antiholomorphic involution σthat is used to define on H∞(Ω) the structure of a Banach
∗-algebra by f♯(z) := f(σ(z)). By Ando’s Theorem, this algebra has a unique predual, so that
it carries a canonical weak topology, which for H∞(D) and H∞(C+) is defined by integrating
boundary values against L1-functions on Tand R, respectively. In the literature, what will
be called weak topology on H∞(Ω) with respect to the canonical pairing (H∞(Ω)∗, H ∞(Ω))
is also known as the weak*-topology. For this algebra we determine in particular all weakly
continuous positive functionals and all weakly continuous characters.
Structure of this paper: In the short Section 1 we introduce the concepts on an abstract level.
In particular, we define reflection and Hankel positive representations of symmetric semigroups
(G, S, τ ). In particular, we show that reflection positive representations are in particular Hankel
positive and that every Hankel positive representation defines a ∗-representation of (S, ♯) by
bounded operators on the Hilbert space b
Edefined by the H-twisted scalar product on E+. We
thus obtain the same three levels (L1-3) as for reflection positive representations.
In Section 2 we connect our abstract setup with classical Hankel operators on H2(D). We
study reflection positive representations of the symmetric group (Z,N,−idZ), i.e., reflection
4
positive operators, and relate the problem of their classification to positive Hankel operators
on the Hardy space H2(D). As these operators are classified by their Carleson measures on
the interval (−1,1), we recall in Appendix A Widom’s classical theorem characterizing the
Carleson measures on positive Hankel operators. In Section 3 we proceed to reflection positive
one-parameter groups. In this context, the upper half plane C+plays the same role as the unit
disc does for the discrete context and any regular multiplicity free representation of the pair
(R,R+) is equivalent to the multiplication representation on (L2(R), H 2(C+)). We show that
in this context the Hankel operators coincide with the classical Hankel operators on H2(C+)
and translate Widom’s Theorem to an analogous result on the upper half plane, where we
realize the Carleson measures on the positive half-line R+(Theorem 3.7). The key result of
Subsection 4.1 is Theorem 4.1 asserting that hHis an operator symbol of H. The applications
to reflection positivity are discussed in Subsection 4.2, where we prove that Hankel positive
representations (E,E+, U, H) of (Z,N,−idZ) and (R,R+,−idR) respectively, are reflection pos-
itive if we change the inner product on E+obtained through a symbol for H. Appendix B is
devoted to the Banach ∗-algebras (H∞(Ω), ♯) and in the short Appendix C we collect some
formulas concerning Poisson and Szeg¨o kernels.
Notation:
•R+= (0,∞), D={z∈C:|z|<1},C+=R+iR+(upper half plane), Cr=R++iR
(right half plane), Sβ={z∈C: 0 <Im z < β}(horizontal strip).
•For a holomorphic function fon C+, we write f∗for its non-tangential limit function on
R; likewise for functions on Dand Sβ.
•We write ω:D→C+, ω(z) := i1+z
1−zfor the Cayley transform with ω−1(w) = w−i
w+i.
•On the circle T={eiθ ∈C:θ∈R}, we use the length measure of total volume 2π.
•For w∈Cwe write ew(z) := ezw for the corresponding exponential function on C.
•For a function f:G→C, we put f∨(g) := f(g−1).
•We write E∗for the dual of a Banach space E.
Acknowledgment: We are most grateful to Daniel Beltit¸˘a for pointing out the references
[Pa88] and [Pe98] on Hankel operators and for suggesting the connection of our work on reflec-
tion positivity with Hankel operators. We thank Christian Berg for a nice short argument for
the implication (b) ⇒(c) in Widom’s Theorem (Theorem A.1).
MSA wishes to thank the Department of Mathematics, University of Erlangen, and the
Mathematisches Forschungsinstitut Oberwolfach (MFO) for their hospitality. This work is part
of two Oberwolfach Leibniz Fellowships with projects entitled “Beurling–Lax type theorems and
their connection with standard subspaces in Algebraic QFT” and “Reflection positive repre-
sentations and standard subspaces in algebraic QFT”. MSA is part of the Gruppo Nazionale
per l’Analisi Matematica, la Probabilit`a e le loro applicationi (GNAMPA) of INdAM. MSA
acknowledges the University of Rome “Tor Vergata” funding scheme “Beyond Borders” CUP
E84I19002200005 and the support by the Deutsche Forschungsgemeinschaft (DFG) within the
Emmy Noether grant CA1850/1-1. MSA wishes to thank Yoh Tanimoto for his insightful
comments and suggestions. KHN acknowledges support by DFG-grant NE 413/10-1.
1 Hankel operators for reflection positive represen-
tations
In this section we first recall the concept of a reflection positive representations of symmetric
semigroups in the sense of [N ´
O18]. In this abstract context we introduce the notion of a Hankel
operator (Definition 1.3). Below it will play a key role in our analysis of the concrete symmetric
5
semigroups (Z,N0,−idZ) and (R,R+,−idR), where it specializes to the classical concept of a
Hankel operator on H2(D) and H2(C+), respectively.
Areflection positive Hilbert space is a triple (E,E+, θ), consisting of a Hilbert space Ewith
a unitary involution θand a closed subspace E+satisfying
hξ, ξ iθ:= hξ, θξ i ≥ 0 for ξ∈ E+.
This structure immediately leads to a new Hilbert space b
Ethat we obtain from the positive
semidefinite form h·,·iθon E+by completing the quotient of E+by the subspace of null vectors.
We write q:E+→b
E, ξ 7→ b
ξfor the natural map.
Definition 1.1. Asymmetric semigroup is a triple (G, S, τ ), where Gis a group, τis an
involutive automorphism of G, and S⊆Gis a subsemigroup invariant under s7→ s♯:= τ(s)−1.
In the present paper we shall only be concerned with the two examples (Z,N0,−idZ) and
(R,R+,−idR). As it creates no additional difficulties, we formulate the concepts in this short
section on the abstract level.
Definition 1.2. Let (G, S, τ ) be a symmetric semigroup.
(a) A representation of the pair (G, S) is a triple (E,E+, U ), where U:G→U(E) is a unitary
representation, E+⊆ E is a closed subspace and U(S)E+⊆ E+. We call (E,E+, U)regular if
span(U(G)E+) = Eand \
g∈G
U(g)E+={0}.(5)
This means that E+contains no non-zero U(G)-invariant subspace and that Eis the only closed
U(G)-invariant subspace containing E+.
(b) A reflection positive representation of (G, S, τ ) is a quadruple (E,E+, θ , U), where (E,E+, θ)
is a reflection positive Hilbert space, (E,E+, U ) is a representation of the pair (G, S) and, in
addition,
θU (g)θ=U(τ(g)) for g∈G(6)
([N ´
O18, Def. 3.3.1].
Definition 1.3. If (S, ♯) is an involutive semigroup and U+:S→B(F) a representation of S
by bounded operators on the Hilbert space F, then we call A∈B(F) a U+-Hankel operator if
AU+(s) = U+(s♯)∗Afor s∈S. (7)
We write HanU+(F)⊆B(F) for the subspace of U+-Hankel operators.
If U∨
+(s) := U+(s♯)∗denotes the dual representation of Son F, then (7) means that Hankel
operators are the intertwining operators (F, U+)→(F, U ∨
+).
Lemma 1.4. Let (G, S, τ)be a symmetric semigroup, (E,E+, U )be a representation of the pair
(G, S), and P+:E → E+be the orthogonal projection. If A∈B(E)satisfies
AU(g) = U(τ(g))A=U(g♯)∗Afor g∈G, (8)
then
HA:= P+AP ∗
+∈B(E+)
is a U+-Hankel operator for the representation of Sin E+by U+(s) := U(s)|E+.
If, in addition, Ris unitary in Esatisfying RU(g)R−1=U(τ(g)) for g∈G, then A∈B(H)
satisfies (8) if and only if A=BR for some B∈U(G)′.
Proof. For the first assertion, we observe that, for s∈Sand ξ, η ∈ E+, we have
hξ, HAU+(s)ηi=hξ , AU(s)ηi(8)
=hξ, U (τ(s))Aηi=hU(s♯)ξ, Aη i
=hU+(s♯)ξ, HAηi=hξ , U+(s♯)∗HAηi.
The second assertion follows from the fact that B:= AR−1commutes with U(G).
6
Lemma 1.5. Hankel operators have the following elementary properties:
(a) If H∈HanU+(F), then H∗∈HanU+(F).
(b) If H∈HanU+(F)and Bcommutes with U+(S), then HB and B∗Hare Hankel operators.
Proof. (a) If H∈HanU+(F) and s∈S, then
H∗U+(s) = (U+(s)∗H)∗= (U∨
+(s♯)H)∗= (HU+(s♯))∗=U+(s♯)∗H∗.
(b) Let H∈HanU+(F) and suppose that Bcommutes with U+(S). Then
HBU+(s) = H U+(s)B=U∨
+(s)HB for s∈S
implies that HB ∈HanU+(F). Taking adjoints, we obtain B∗H= (H∗B)∗∈HanU+(F)
with (a).
Definition 1.6. (Hankel positive representations) Let (G, S, τ ) be a symmetric semigroup.
Then a Hankel positive representation is a quadruple (U, E,E+, H ), where (E,E+, U) is a re-
presentation of the pair (G, S), and H∈HanU+(E+) is a positive Hankel operator for the
representation U+(s) := U(s)|E+of Sby isometries on E+.
Example 1.7. (a) Let (E,E+, U ) be a representation of the pair (G, S ) and θ:E → E a unitary
involution satisfying θU(g)θ=U(τ(g)) for g∈G(see (2)). Then Lemma 1.4 implies that
Hθ:= P+θP ∗
+∈B(E+)
is a U+-Hankel operator. It is positive if and only if (E,E+, θ ) is reflection positive.
(b) The identity 1∈B(F) is a U+-Hankel operator if and only if the two representations Uand
U∨coincide, i.e., if Uis a ∗-representation of the involutive semigroup (S, ♯). If U+(S) consists
of isometries, this is only possible if all operators U+(s) are unitary and U+(s♯) = U+(s)−1. In
the context of (a), this leads to the case where U+(s)E+=E+for s∈S.
The following proposition shows that a positive Hankel operator Himmediately leads to a
∗-representation of Son the Hilbert space defined by Hvia the scalar product hξ , ηiH:=
hξ, H ηi.
In the context of reflection positive representations (Example 1.7), the passage from the
representation (E+, U+) of Sby isometries to the ∗-representation on ( b
E,b
U) by contractions is
called the Osterwalder–Schrader transform, see [N ´
O18] for details. In this sense, the following
Proposition 1.8 generalizes the Osterwalder–Schrader transform.
Proposition 1.8. Let U+:S→B(F)be a representation of the involutive semigroup (S, ♯)by
bounded operators on Fand H≥0be a positive U+-Hankel operator on F. Then
hξ, ηiH:= hξ , HηiF
defines a positive semidefinite hermitian form on F. We write b
Ffor the associated Hilbert space
and q:F → b
Ffor the canonical map. Then there exists a uniquely determined ∗-representation
b
U: (S, ♯)→B(b
F)satisfying b
U(s)◦q=q◦U+(s)for s∈S. (9)
Proof. For every s∈Sand ξ, η ∈ F, we have
hξ, U+(s)ηiH=hξ , HU+(s)ηi=hξ, U+(s♯)∗H ηi=hU+(s♯)ξ, H ηi=hU+(s♯)ξ, ηiH.(10)
If q(η) = 0, i.e., hη, H ηi= 0, then this relation implies that q(U+(s)η) = 0. Therefore
b
U(s)q(η) := q(U+(s)η) defines a linear operator on the dense subspace D:= q(F)⊆b
F. It also
follows from (10) that ( b
U , D) is a ∗-representation of the involutive semigroup (S, ♯).
7
To see that the operators b
U(s) are bounded, we observe that, for every n∈N0and η∈ F,
we have
kb
U(s♯s)nq(η)k2
b
F=hU+(s♯s)nη, H U+(s♯s)nηi ≤ kHkkU+(s♯s)k2nkηk2.
Now [Ne99, Lemma II.3.8(ii)] implies that
kb
U(s)k ≤ pkU+(s♯s)k ≤ max(kU+(s)k,kU+(s♯)k) for s∈S.
We conclude that the operators b
U(s) are contractions, hence extend to operators on b
F. We
thus obtain a ∗-representation of (S, ♯). Clearly, this representation is uniquely determined by
the equivariance requirement (9).
The construction in Proposition 1.8 shows that every Hankel positive representation
(U, E,E+, H) of (G, S, τ ) defines a ∗-representation of Sby bounded operators on the Hilbert
space b
Edefined by the H-twisted scalar product on E+. So we obtain the same three levels
(L1-3) as for reflection positive representations.
Remark 1.9. Let Ebe a Hilbert space, E+⊆ E a closed subspace, and R∈U(E) a unitary
involution with R(E+) = E⊥
+. Then
A♯:= R−1A∗R
defines an antilinear involution on B(E) leaving the subalgebra
M:= {A∈B(E): AE+⊆ E+}
invariant. In fact, A∈ M implies that A∗E⊥
+⊆ E⊥
+, so that
A♯E+=R−1A∗RE+=R−1A∗E⊥
+⊆R−1E⊥
+=E+.
Examples 1.10. (a) For E=L2(T)⊇ E+=H2(D) and (Rf)(z) = zf (z), we have M=
H∞(D) ([Ni19, §1.8.3]) and the corresponding involution is given by
f♯(z) := f(z) for z∈ D.(11)
For this example Hankel operators will be discussed in Theorem 2.2.
(b) For E=L2(R)⊇ E+=H2(C+), we have (Rf)(x) = f(−x) with M=H∞(C+), endowed
with the involution
f♯(z) := f(−z) for z∈C+.(12)
Let H=P+hRP ∗
+,h∈L∞(R) be a Hankel operator on H2(C+) (cf. Theorem 3.5) and
g∈H∞(C+). Then the corresponding multiplication operator mgon H2(C+) satisfies
Hmg=P+hRP ∗
+mg=P+hRmgP∗
+=P+h(g∗)∨RP ∗
+,
where (g∗)∨(x) = g∗(−x) for x∈R. This also is a Hankel operator, where hhas been
modified by (g∗)∨. We shall use this procedure in Theorem 4.5 to pass from Hankel positive
representations to reflection positive ones.
2 Reflection positivity and Hankel operators
In this section we connect the abstract context from the previous section with classical Hankel
operators on H2(D). We study reflection positive operators as reflection positive representations
of the symmetric group (Z,N,−idZ) and relate the problem of their classification to positive
Hankel operators on the Hardy space H2(D) on the open unit disc D⊆C.
8
Definition 2.1. Areflection positive operator on a reflection positive Hilbert space (E,E+, θ)
is a unitary operator U∈U(E) such that
UE+⊆ E+and θU θ =U∗.(13)
It is easy to see that reflection positive operators are in one-to-one correspondence with
reflection positive representations of (Z,N,−idZ): If (E,E+, θ, U ) is a reflection positive repre-
sentation of the symmetric semigroup (Z,N,−idZ), then U(1) is a reflection positive operator.
If, conversely, Uis a reflection positive operator, then U(n) := Undefines a reflection positive
representation of (Z,N,−idZ). Accordingly, we say that a reflection positive operator Uis
regular if \
n∈Z
UnE+=\
n>0
UnE+
!
={0}and [
n∈Z
UnE+=[
n<0
UnE+
!
=E
(cf. Definition 1.2). If this is the case, we obtain for K:= E+∩(UE+)⊥a unitary equivalence
from (E,E+, U ) to
(ℓ2(Z,K), ℓ2(N0,K), S),
where Sis the right shift (Wold decomposition, [SzNBK10, Thm. I.1.1]).
We would like to classify quadruples (E,E+, θ, U ), where Uis a regular reflection positive
operator, up to unitary equivalence. In the present paper we restrict ourselves to the multiplicity
free case, where K=C, so that the triple (E,E+, U ) is equivalent to (ℓ2(Z), ℓ2(N0), S), where
Sis the right shift. For our purposes, it is most convenient to identify ℓ2(Z) with L2(T) and
ℓ2(N0) with the Hardy space H2(D) of the unit disc D, so that the shift operator acts by
(Sf )(z) = z f (z) for z∈T.
We now want to understand the possibilities for adding a unitary involution θfor which
H2(D) is θ-positive. For f:T→Cwe define
f♯(z) := f(z) for z∈T
(cf. (11) and (37) in Appendix B). Then any involution θsatisfying θSθ =S−1has the form
θh(f)(z) = h(z)f(z) for z∈T,
where h∈L∞(T) satisfies h♯=hand h(T)⊆T(cf. Lemma 1.4). As any h∈L∞(T) defines a
Hankel operator
Hh:= P+θhP∗
+∈B(H2(D)),(14)
this leads us naturally to Hankel operators on H2(D). If his unimodular with h♯=h, so that
θhis a unitary involution, then Hhis positive if and only if E+is θh-positive.
The following theorem characterizes Hankel operators from several perspectives. Condition
(a) provides the consistency with the abstract concept of a U-Hankel operator from Defini-
tion 1.3. The equivalence of (a) and (c) is well known ([Ni02, p. 180]).
Theorem 2.2. (Characterization of Hankel Operators on the disc) Consider a bounded op-
erator Don H2(D), the shift operator (S F )(z) = zF (z), and the multiplication operators mg
defined by g∈H∞(D)on H2(D). Then the following are equivalent:
(a) The Rosenblum relation 1DS =S∗Dholds for the shift operator (Sf )(z) = z f (z), i.e.,
Dis a Hankel operator for the representation of (N,+) on H2(D)defined by U+(n) := Sn.
(b) Dmg=m∗
g♯Dfor all g∈H∞(D), i.e., Dis a Hankel operator for the representation of
the involutive algebra (H∞(D), ♯)on H2(D)by multiplication operators.
(c) There exists h∈L∞(T)with D=P+mhRP ∗
+for R(F)(z) := zF (z),z∈D, i.e., Dis a
bounded Hankel operator on H2(D)in the classical sense.
1See [Ro66], [Ni02, p. 205].
9
Proof. (b) ⇒(a) is trivial.
(a) ⇒(b): We recall from Example B.2(b) that the weak topology on the Banach algebra
H∞(D) is defined by the linear functionals
ηf(h) = ZT
f(z)h∗(z)dz for f∈L1(T), h ∈H∞(D).(15)
For f1, f2∈H2(D), we observe that
hf1, Dgf2i=hD∗f1, g f2i=ZT
(D∗f1)∗(z)g∗(z)f∗
2(z)dz =η(D∗f1)∗f∗
2(g)
(cf. Example B.2), and
hg♯f1, Df2i=ZT
g♯f1
∗(z)(Df2)∗(z)dz =ZT
f∗
1(z)g∗(z)(Df2)∗(z)dz
=ZT
(f♯
1)∗(z)g∗(z)(Df2)∗(z)dz =η(f♯
1)∗(Df2)∗,∨(g),
where we use the notation h∨(z) = h(z−1), z∈T. Both define weakly continuous linear
functionals on H∞(D) because L2(T)L2(T) = L1(T), which by (a) coincide on polynomials. As
these span a weakly dense subspace (Lemma B.6(a)), we obtain equality for every g∈H∞(D),
which is (b).
(a) ⇔(c): It is well known that (a) characterizes bounded Hankel operators on H2(D) (cf. [Pe98,
Thm. 2.6]). This relation immediately implies
hzj, Dzki=hzj, D Sk1i=hzj,(S∗)kD1i=hSkzj, D1i=hzj+k, D1i,(16)
so that the matrix of Dis a Hankel matrix [Ni02, Def. 6.1.1]. The converse requires Nehari’s
Theorem. We refer to [Ni02, Part B, 1.4.1] for a nice short functional analytic proof.
Remark 2.3. In the proof above we have used Nehari’s Theorem ([Pe98, Thm. 2.1], [Ni02,
Part B, 1.4.1], [Ni19, Thm. 4.7.1]) which actually contains the finer information that every
bounded Hankel operator on H2(D) is of the form Hh(see (14)), where h∈L∞(T) can even
be chosen in such a way that
kHhk=khk∞.
As Hh= 0 if and only if h∈H∞(D−) for D−={z∈C:|z|>1},bounded Hankel operators
on H2(D) are parametrized by the quotient space L∞(T)/H∞(D−) (cf. [Pa88, Cor. 3.4]). As
kHhk= distL∞(T)(h, H∞(D−)),(17)
the embedding L∞(T)/H∞(D−)֒→B(H2(D)) is isometric.
We now recall how positive Hankel operators can be classified by using Hamburger’s The-
orem on moment sequences.
Definition 2.4. (The Carleson measure µH) Suppose that His a positive Hankel operator.
Then (16) shows that the sequence (an)n∈N0defined by
an:= hSn1, H1i(18)
satisfies
an+m=hSn+m1, H1i=hSn1, H Sm1i,
so that the positivity of Himplies that the kernel (an+m)n,m∈N0is positive definite, i.e.,
(an)n∈Ndefines a bounded positive definite function on the involutive semigroup (N0,+,id)
10
whose bounded spectrum is [−1,1]. By Hamburger’s Theorem ([BCR84, Thm. 6.2.2], [Ni02,
Chap. 6]), there exists a unique positive Borel measure µHon [−1,1] with
Z1
−1
xndµH(x) = anfor n∈N0.
Widom’s Theorem (see Theorem A.1 in Appendix A) implies that
hf, H giH2(D)=Z1
−1
f(x)g(x)dµH(x) for f, g ∈H2(D)
and it characterizes the measures on [−1,1] which arise in this context. In particular, all these
measures are finite and satisfy µH({1,−1}) = 0. We call µHthe Carleson measure of H.
We shall return to positive Hankel operators on the disc Din Theorem 4.8.
3 Reflection positive one-parameter groups
In this section we proceed from the discrete to the continuous by studying reflection positive
one-parameter groups instead of single reflection positive operators. In this context, the upper
half plane C+plays the same role as the unit disc does for the discrete context.
Definition 3.1. Areflection positive one-parameter group is a quadruple (E,E+, θ, U ) defining a
reflection positive strongly continuous representation of the symmetric semigroup (R,R+,−idR).
This means that (Ut)t∈Ris a unitary one-parameter group on Esuch that
UtE+⊆ E+for t > 0 and θUtθ=U−tfor t∈R.(19)
As in Definition 1.2, we call a reflection positive one-parameter group regular if
\
t∈R
UtE+=\
t>0
UtE+
!
={0}and [
t∈R
UtE+=[
t<0
UtE+
!
=E.
If this is the case, then the representation theorem of Lax–Philipps provides a unitary equiva-
lence from (E,E+, U ) to
(L2(R,K), L2(R+,K), S),
where (St)t∈Rare the unitary shift operators on L2(R,K) and Kis a Hilbert space (the multi-
plicity space) ([N ´
O18, Thm. 4.4.1], [LP64, LP67, LP81]).
To classify reflection positive one-parameter groups, we consider in this paper the multiplic-
ity free case, where K=C. Again, it is more convenient to work in the spectral representation,
i.e., to use the Fourier transform and to consider on E=L2(R) the unitary multiplication
operators
(Stf)(x) = eitxf(x) for x∈R
and the Hardy space E+:= H2(C+) which is invariant under the semigroup (St)t>0.
Remark 3.2. (Representations of (R,R+)) The closed invariant subspaces E+⊆H2(C+)⊆
L2(R) under the semigroup (St)t >0are of the form hH2(C+) for an inner function h. This is
Beurling’s Theorem for the upper half plane. It follows from Beurling’s Theorem for the disc
([Pa88, Thm. 6.4]) and Lemma B.6 by translation with Γ2(see Theorem 3.5).
The involutions θsatisfying θUtθ=U−tfor t∈Rare of the form θh=hR, where (Rf)(x) =
f(−x) and his a measurable unimodular function on Rsatisfying h♯=h, where h♯(x) = h(−x)
as in (12).
11
3.1 Hankel operators on H2(C+)
Definition 3.3. For h∈L∞(R), we define on H2(C+) the Hankel operator
Hh:= P+hRP ∗
+,where (Rf)(x) := f(−x), x ∈R,
P+:L2(R)→H2(C+) is the orthogonal projection, and hR stands for the composition of R
with multiplication by h(cf. [Pa88, p. 44]).
Let
j±:H∞(C±)→L∞(R,C), f 7→ f∗
denote the isometric embedding defined by the non-tangential boundary values. Accordingly,
we identify H∞(C±) with its image under this map in L∞(R,C).
Lemma 3.4. For h∈L∞(R), the following assertions hold:
(a) H∗
h=Hh♯. In particular Hhis hermitian if h♯=h.
(b) Hh= 0 if and only if h∈H∞(C−).
(c) kHhk ≤ khk.
Proof. (a) follows from the following relation for f, g ∈H2(C+):
hf, Hhgi=ZR
f∗(x)h(x)g∗(−x)dx =ZR
h(−x)f∗(−x)g∗(x)dx
=ZR
h♯(x)f∗(−x)g∗(x)dx =hHh♯f, gi.
(b) (cf. [Pa88, Cor. 4.8]) The operator Hhvanishes if and only if
hH2(C−) = θhH2(C+)⊆H2(C+)⊥=H2(C−),
which is equivalent to h∈H∞(C−).
(c) follows from kP+k=kRk= 1.
The preceding lemma shows that we have a continuous linear map
L∞(R)/H∞(C−)→B(H2(C+)),[h]7→ Hh
which is compatible with the involution ♯on the left and ∗on the right. By Nehari’s Theorem
([Pa88, Cor. 4.7]), this map is isometric. As H∞(C+)∩H∞(C−) = C1, we obtain in particular
an embedding
H∞(C+)/C1֒→L∞(R)/H∞(C−)→B(H2(C+)),[h]7→ Hh.
In Proposition 1.8, we have used a positive Hankel operator Hto define a new scalar product
that led us to a ∗-representation of (S, ♯). Here the key ingredient was the Hankel relation, an
abstract form of the Rosenblum relation in Theorem 2.2(b). As the following theorem shows,
this relation actually characterizes Hankel operators on H2(C+), so that the classical definition
(Definition 3.3) and Definition 1.3 are consistent.
Theorem 3.5. (Characterization of Hankel Operators on the upper half plane) Consider a
bounded operator C, the isometries Stf=eitf,t≥0, and the multiplication operators mg,
g∈H∞(C+)on H2(C+). We also consider the unitary isomorphism
Γ2:L2(T)→L2(R),(Γ2f)(x) := √2
x+ifx−i
x+i(20)
from [Ni19, p. 200] which maps H2(D)to H2(C+)and the operator
D:= Γ−1
2CΓ2:H2(D)→H2(D).
Then the following are equivalent:
12
(a) There exists h∈L∞(R)with C=Hh, i.e., Cis a Hankel operator in the sense of
Definition 3.3.
(b) Cmg=m∗
g♯Cfor all g∈H∞(C+), where g♯(z) = g(−z), i.e., Cis a U+-Hankel operator
for the representation of the involutive algebra (H∞(C+), ♯)on H2(C+)by multiplication
operators U+(g) = mg.
(c) CSt=S∗
tCfor all t > 0, i.e., Cis a U+-Hankel operator for the representation of R+on
H2(C+)defined by U+(t)f:= eitffor t≥0.
(d) Dis a Hankel operator on H2(D).
Proof. (a) ⇒(b): Suppose that C=Hhfor some h∈L∞(R). For f1, f2∈H2(C+) we then
have
hf1, Hhgf2i=ZR
f∗
1(x)h(x)g∗(−x)f∗
2(−x)dx =ZR
g∗,♯(x)f∗
1(x)h(x)f∗
2(−x)dx =hg♯f1, Hhf2i,
which is (b).
(b) ⇒(c) follows from e♯
it =eit for t > 0.
(c) ⇒(b): For f1, f2∈H2(C+) and g∈H∞(C+), we observe that
hf1, Cg f2i=hC∗f1, gf2i=ZR
C∗f1
∗(x)g∗(x)f∗
2(x)dx =ηC∗f1f2(g)
(see Example B.2 for the functionals ηf) and
hg♯f1, Cf2i=ZR
(g♯f1)∗(x)(Cf2)∗(x)dx =ZR
f∗
1(x)g∗(−x)(Cf2)∗(x)dx
=ZR
(f♯
1)∗(x)g∗(x)(Cf2)∗(−x)dx =ηf♯
1(Cf2)∨(g),
where we use the notation h∨(x) := h(−x) for x∈R. Both define weakly continuous linear
functionals on H∞(C+), which by (c) coincide on the functions eit ,t > 0. As these span a
weakly dense subspace (Lemma B.6(b)), we obtain equality for every g∈H∞(C+), which is
(b).
(b) ⇔(d): The Cayley transform ω:D→C+, ω(z) := i1+z
1−zdefines an isometric isomorphism
L∞(T)→L∞(R), g 7→ g◦ω−1which restricts to an isomorphism H∞(D)→H∞(C+) and
satisfies
Γ2◦mg=mg◦ω−1◦Γ2.
Therefore (b) is equivalent to
Dmg◦ω=m∗
g♯◦ωDfor g∈H∞(C+),
which is (d) by Theorem 2.2.
(d) ⇒(a): Suppose that D=Dkas in Theorem 2.2. For f∈H2(D), we then have for x∈R
(CΓ2(f))(x) = Γ2(Df )(x) = √2
x+i(Df )∗(ω−1(x)) = √2k(ω−1(x))
x+if∗(ω−1(x))
=k(ω−1(x)) i−x
(i+x)
√2
(−x+i)f∗(ω−1(−x)) = k(ω−1(x)) i−x
i+xΓ2(f)∗(−x).
The assertion now follows with
h(x) := k(ω−1(x)) i−x
i+x=−k(ω−1(x))ω−1(x) (21)
(cf. [Pa88, Thm. 4.6]).
13
3.2 Widom’s Theorem for the upper half-plane
In this subsection we translate Widom’s Theorem (Theorem A.1) characterizing the Carleson
measures of positive Hankel operators on the disc to a corresponding result on the upper half
plane. This is easily achieved by using Theorem 3.5 for the translation process.
Let Hbe a positive Hankel operator on H2(C+). For t≥0, the exponential functions
eit(z) = eitz in H∞(C+) satisfy e♯
it =eit. Therefore the function
ϕH:R+→R, ϕH(t) := heit/2, Heit/2iH2(C+)(22)
satisfies
ϕH(t+s) = hei(t+s)/2, He i(t+s)/2iH2(C+)=heit, H eisiH2(C+)for s, t > 0,
so that the kernel (ϕH(t+s))t,s>0is positive definite. This means that ϕHis a positive defi-
nite function on the involutive semigroup (R+,+,id) bounded on [1,∞). By the Hausdorff–
Bernstein–Widder Theorem ([BCR84, Thm. 6.5.12], [Ne99, Thm. VI.2.10]), there exists a
unique positive Borel measure µHon [0,∞) with
ϕH(t) = Z∞
0
e−λt dµH(λ) for t > 0.(23)
Widom’s Theorem for C+(Theorem 3.7 below) now implies that
hf, H giH2(C+)=Z∞
0
f(iλ)g(iλ)dµH(λ) for f, g ∈H2(C+)
and it characterizes the measures µHon [0,∞) which correspond to positive bounded Hankel
operators. In particular, all these measures satisfy µH({0}) = 0.
Definition 3.6. The measure µHon R+is called the Carleson measure of H.
Theorem 3.7. (Widom’s Theorem for the upper half-plane) For a positive Borel measure µ
on R+, we consider the measure ρon R+defined by
dρ(λ) := dµ(λ)
1 + λ2.
Then the following are equivalent:
(a) There exists an α∈Rwith
ZR+|f(iλ)|2dµ(λ)≤αkfk2for f∈H2(C+),(24)
i.e., µis the Carleson measure of a positive Hankel operator on H2(C+).
(b) ρ((0, x)) = O(x)and ρ((x−1,∞)) = O(x)for x→0+.
If these conditions are satisfied, then ρ(R+)<∞and there exist β, γ > 0such that
ρ((0, ε]) ≤βε and ρ([t, ∞)) ≤γ
tfor every ε, t ∈R+.
Proof. Condition (a) is equivalent to the existence of a positive Hankel operator Con H2(C+)
with µ=µC. Let Dbe the corresponding Hankel operator on H2(D) (Theorem 3.5) and
consider the diffeomorphism
γ:R+→(−1,1), γ(λ) = λ−1
λ+ 1 =ω−1(iλ).
14
For f∈H2(D), we then have
Z1
−1|f(t)|2dµD(t) = hf, Df iH2(D)=hΓ2(f), CΓ2(f)iH2(C+)
=ZR+|Γ2(f)(iλ)|2dµC(λ) = 2 ZR+
|f(ω−1(iλ))|2
(1 + λ)2dµC(λ)
= 2 ZR+
|f(γ(λ))|2
(1 + λ)2dµC(λ) = 2 Z1
−1
|f(t)|2
(1 + γ−1(t))2d(γ∗µC)(t).
As γ−1(t) = −iω(t) = 1+t
1−tand 1 + (1+t)
(1−t)=2
(1−t),it follows that
dµD(t) = (1 −t)2
2d(γ∗µC)(t).
We conclude that
µD((1 −x, 1)) = Z1
1−x
(1 −t)2
2d(γ∗µC)(t) = Z∞
γ−1(1−x)
(1 −γ(λ))2
2dµC(λ)
=Z∞
2
x−1
2
(λ+ 1)2dµC(λ) = 2 Z∞
2
x−1
1 + λ2
(λ+ 1)2dρ(λ).
Therefore µD((1 −x, 1)) has for x→0+the same asymptotics as ρ((x−1,∞)). Likewise
µD((−1,−1 + x)) = Z−1+x
−1
(1 −t)2
2d(γ∗µC)(t) = Zγ−1(x−1)
0
(1 −γ(λ))2
2dµC(λ)
=Zx
2−x
0
2
(λ+ 1)2dµC(λ) = 2 Zx
2−x
0
1 + λ2
(λ+ 1)2dρ(λ).
This shows that µD((−1,−1 + x)) has for x→0+ the same asymptotics as ρ((0, x)). Therefore
the assertion follows from Widom’s Theorem for the disc (Theorem A.1).
Now we assume that ρsatisfies (b). Then there exist β′, γ′>0 and ε0, t0∈R+such that
ρ((0, ε])
ε≤β′and ρ([t, ∞)) t≤γ′for every ε≤ε0, t ≥t0.
Then
ρ(R+) = ρ((0, ε0)) + ρ([ε0, t0]) + ρ((t0,∞)) ≤β′+ρ([ε0, t0]) + γ′<∞.
For ε > ε0and t < t0, we now find
ρ((0, ε])
ε≤ρ(R+)
ε0
and ρ([t, ∞)) t≤ρ(R+)t0.
This completes the proof.
3.3 The symbol kernel of a positive Hankel operator
Definition 3.8. Let Hbe a Hankel operator on H2(C+) and
Q(z, w) = Qw(z) = 1
2π
i
z−w
be the Szeg¨o kernel of C+(cf. Appendix C). Then we associate to Hits symbol kernel, i.e., the
kernel
QH(z, w) := hQz, H Qwi= (HQw)(z) = (H∗Qz)(w).(25)
Clearly, QHis holomorphic in the first argument and antiholomorphic in the second argument.
15
By [Ne99, Lemma I.2.4], the Hankel operator His positive if and only if its symbol kernel
QHis positive definite. Suppose that this is the case and let µHbe the corresponding Carleson
measure on R+. Then
QH(z, w) = Z∞
0
Qz(iλ)Qw(iλ)dµH(λ) = 1
4π2Z∞
0
dµH(λ)
(−iλ −z)(iλ −w)
=1
4π2Z∞
0
dµH(λ)
(λ−iz)(λ+iw).(26)
Definition 3.9. From Widom’s Theorem for the upper half plane (Theorem 3.7), we know
that the measure dµ(λ)
1+λ2is finite, so that,
κ(z) := ZR+
λ
1 + λ2−1
z+λdµH(λ) (27)
defines a holomorphic function on C\(−∞,0] ([Do74, Ch. II, Thm. 1]).
For z, w ∈Cr, we then have
κ(z)−κ(w) = ZR+
1
w+λ−1
z+λdµH(λ) = ZR+
z−w
(w+λ)(z+λ)dµH(λ),
so that κ(z)−κ(w)
z−w=ZR+
dµH(λ)
(w+λ)(z+λ)= 4π2QH(iz, iw).(28)
4 Schober’s representation theorem
In this section we explain how to find for every positive Hankel operator Hon H2(C+) an
explicit bounded function hH∈L∞(R) with values in iRsuch that h♯
H=hHand His the
corresponding Hankel operator, i.e., HhH=H. This supplements Nehari’s classical theorem
by a constructive component. Adding non-zero real constants then leads to functions fin the
unit group of L∞(R) with Hf=H, and we shall use this to shows that all Hankel positive
one-parameter groups are actually reflection positive for a slightly modified scalar product.
4.1 An operator symbol for H
Theorem 4.1. Let Hbe a positive Hankel operator on H2(C+)with Carleson measure µH
and define
hH:R→iR, hH(p) := i
π·ZR+
p
λ2+p2dµH(λ).
Then hH∈L∞(R,C)and the associated Hankel operator HhHequals H.
Proof. Part 1: We first show that hHis bounded. Let dρ (λ) = dµH(λ)
1+λ2be the finite measure
on R+from Theorem 3.7. Then we have
ZR+
p
λ2+p2dµH(λ) = ZR+
p1 + λ2
λ2+p2dρ (λ).
For the integrand
fp(λ) := p1 + λ2
λ2+p2we have f′
p(λ) = 2pp2−1λ
(λ2+p2)2.
16
Hence the function fpis increasing for p≥1, and therefore
Z(0,1]
fp(λ)dρ (λ)≤fp(1) Z(0,1]
dρ (λ) = 2p
1 + p2·ρ((0,1]) ≤ρ((0,1]) .
Now, let γbe the constant from Theorem 3.7. Then integration by parts (cf. Lemma A.5) leads
for p≥1 to
Z(1,∞)
fp(λ)dρ (λ) = ρ((1,∞)) fp(1) + Z(1,∞)
ρ((t, ∞)) f′
p(t)dt
≤ρ((1,∞)) 2p
1 + p2+Z(1,∞)
γ
t·2pp2−1t
(t2+p2)2dt
≤ρ((1,∞)) ·1 + γp2−1
tp
t2+p2+ arctan t
p
p2
∞
1
=ρ((1,∞)) + γp2−1
p2π
2−p
1 + p2−arctan 1
p≤ρ((1,∞)) + γπ
2.
So, for every p≥1, we have
ZR+
p
λ2+p2dµH(λ) = ZR+
fp(λ)dρ (λ) = Z(0,1]
fp(λ)dρ (λ) + Z(1,∞)
fp(λ)dρ (λ)
≤ρ((0,1]) + ρ((1,∞)) + γπ
2=ρ(R+) + γπ
2.
For p∈(0,1), the function fpis decreasing and therefore
Z(1,∞)
fp(λ)dρ (λ)≤fp(1) Z(1,∞)
dρ (λ) = 2p
1 + p2·ρ((1,∞)) ≤ρ((1,∞)) .
Now, let βbe the constant from Theorem 3.7. Then, for p < 1, we have
Z(0,1]
fp(λ)dρ (λ) = ρ((0,1]) fp(1) −Z(0,1]
ρ((0, t]) f′
p(t)dt
≤ρ((0,1]) 2p
1 + p2−Z(0,1]
βt ·2pp2−1t
(t2+p2)2dt
≤ρ((0,1]) ·1 + β1−p2arctan t
p−tp
t2+p21
0
=ρ((0,1]) + β1−p2arctan 1
p−p
1 + p2≤ρ((0,1]) + βπ
2.
So, for every p∈(0,1), we have
ZR+
p
λ2+p2dµH(λ) = ZR+
fp(λ)dρ (λ) = Z(0,1]
fp(λ)dρ (λ) + Z(1,∞)
fp(λ)dρ (λ)
≤ρ((0,1]) + βπ
2+ρ((1,∞)) = ρ(R+) + βπ
2.
Therefore, for every p∈R+, we have
|hH(p)|=1
πZR+
p
λ2+p2dµH(λ)≤1
πρ(R+) + 1
2max{β, γ }.
17
Since hH(−p) = −hH(p), this yields
khHk∞≤1
πρ(R+) + 1
2max{β, γ}
and therefore hH∈L∞(R,C)♯, where h♯
H=hHfollows by hH(−p) = −hH(p) = hH(p).
Part 2: For the second statement, we recall the function
κ:C\(−∞,0] →C, κ(z) = ZR+
λ
1 + λ2−1
λ+zdµH(λ)
from (27). Then, for p∈R×, we have
Im (κ(ip)) = Im ZR+
λ
1 + λ2−1
λ+ip dµH(λ)!
= Im ZR+
λ
1 + λ2−λ−ip
λ2+p2dµH(λ)!=ZR+
p
λ2+p2dµH(λ),
so
hH(p) = i
π·Im (κ(ip)) .(29)
For the real part, we get
Re (κ(ip)) = Re ZR+
λ
1 + λ2−1
λ+ip dµH(λ)!= Re ZR+
λ
1 + λ2−λ−ip
λ2+p2dµH(λ)!
=ZR+
λ
1 + λ2−λ
λ2+p2dµH(λ) = p2−1ZR+
λ
(1 + λ2) (λ2+p2)dµH(λ)
and therefore
|Re (κ(ip))|=p2−1ZR+
λ
(1 + λ2) (λ2+p2)dµH(λ)
≤p2−1ZR+
4λ
(1 + λ)2(|p|+λ)2dµH(λ).
For p∈R×, we now define the function
np:C+→C, np(z) = 2√z
(1 −iz) (|p| − iz),
where by √·we denote the inverse of the function Cr∩C+→C+, z 7→ z2.Then npis holo-
morphic on C+and for y > 0, we have
|np(x+iy)|2=4px2+y2
((1 + y)2+x2) ((|p|+y)2+x2)≤4px2+y2
(1 + y2+x2) (p2+x2)≤2
p2+x2,
so
sup
y>0ZR|np(x+iy)|2dx ≤ZR
2
p2+x2dx =2π
|p|<∞
and therefore np∈H2(C+). Since µHis a Carleson measure, by Theorem 3.7(a), there is a
constant α≥0 such that
ZR+
f(iλ)g(iλ)dµH(λ)≤αkfk2kgk2for every f, g ∈H2(C+).
18
Then
|Re (κ(ip))| ≤ p2−1ZR+
4λ
(1 + λ)2(|p|+λ)2dµH(λ) = p2−1ZR+|np(iλ)|2dµH(λ)
≤p2−1αknpk2
2=αp2−1ZR
4|x|
(1 + x2) (p2+x2)dx
= 4αp2−1Z∞
0
2x
(1 + x2) (p2+x2)dx = 4αZ∞
0
2x
1 + x2−2x
p2+x2dx
= 4αlog 1 + x2−log p2+x2∞
0= 4αlog 1 + x2
p2+x2∞
0= 8α|log (|p|)|
for every p∈R×. This estimate together with khHk∞<∞shows that, for z, w ∈C+, the
integrals ZR
κ(ip)
(p−z) (p−w)dp and ZR
κ(ip)
(p−z) (p−w)dp
exist. We have ZR
κ(ip)
(p−z) (p−w)dp =ZR
κ(−ip)
(p+z) (p+w)dp = 0 (30)
because the function p→κ(−ip)
(p+z)(p+w)is holomorphic on C+.
By the Residue Theorem, for z , w ∈C+with z6=wand κ(−iz)6= 0 6=κ(−iw), we get
ZR
κ(ip)
(p−z) (p−w)dp =ZR
κ(−ip)
(p−z) (p−w)dp = 2πi κ(−iz)
z−w+κ(−iw)
w−z
= 2πi κ(−iz)−κ(−iw)
z−w
(28)
= (2π)3QH(z, −w).
By continuity of both sides in zand w, we get
ZR
κ(ip)
(p−z) (p−w)dp = (2π)3QH(z, −w) for every z , w ∈C+.(31)
For z, w ∈C+, we finally obtain
4π2QHhH(z, w) = 4π2hQz, hHRQwi=ZR
hH(p)
(p−z) (−p−w)dp =ZR
−hH(p)
(p−z) (p+w)dp
=ZR
−i
π·Im (κ(ip))
(p−z) (p+w)dp =1
2πZR
κ(ip)−κ(ip)
(p−z) (p+w)dp
=1
2π ZR
κ(ip)
(p−z) (p+w)dp −ZR
κ(ip)
(p−z) (p+w)dp!
(30)
=1
2πZR
κ(ip)
(p−z) (p+w)dp (31)
= 4π2QH(z, w)
This means that the operators Hand HhHhave the same symbol kernel, hence are equal by
[Ne99, Lemma I.2.4].
Lemma 4.2. Let H6= 0 be a positive Hankel operator on H2(C+). Then there exist c, a ∈R+
such that
|hH(p)| ≥ c·|p|
a2+p2for every p∈R×.
19
Proof. Since H6= 0, we have µH6= 0, hence µH((0, a]) >0 for some a > 0. Then setting
c:= µH((0,a])
π, for p∈R×, we have
|hH(p)|=1
πZR+
|p|
λ2+p2dµH(λ)≥1
πZ(0,a]
|p|
λ2+p2dµH(λ)
≥1
πZ(0,a]
|p|
a2+p2dµH(λ) = c·|p|
a2+p2.
Choosing the measure µ=δafor an a∈R+shows that the estimate in this lemma is
optimal.
Definition 4.3. (cf. [RR94, Thm. 5.13]) A holomorphic function on C+is called an outer
function if it is of the form
Out(k, C )(z) = Cexp 1
πi ZR1
p−z−p
1 + p2log (k(p)) dp,
where C∈Tand k:R→R+satisfies RR
|log(k(p))|
1+p2dp < ∞.Then k=|Out(k, C )∗|. We write
Out(k) := Out(k, 1). If k1and k2are two such functions, then so is their product, and
Out(k1k2) = Out(k1) Out(k2).(32)
We also note that the function k∨(p) = k(−p) satisfies
Out(k∨) = Out(k)♯.(33)
Theorem 4.4. Let Hbe a positive Hankel operator on H2(C+). Then, for every c∈R×, we
have
δ:= hH+c1∈L∞(R,C)and 1
δ∈L∞(R,C).
Further Hδ=Hand there exists an outer function g∈H∞(C+)×(the unit group of this
Banach algebra) such that |g∗|2=|δ|.
Proof. Since hH(R)⊆iRwe have
kδk∞=qkhHk2
∞+c2<∞and
1
δ
∞≤1
|c|,
which shows the first statement. For the second statement, we notice that c1∈H∞(C−)
implies Hc1= 0 by Lemma 3.4, so that Hδ=HhH+Hc1=H+ 0 = Hby Lemma 3.4 and
Theorem 4.1.
Finally, we have
ZR
|log |δ(p)||
1 + p2dp ≤ZR
max log kδk∞,log
1
δ
∞
1 + p2dp < ∞
and ZRlog 1
δ(p)
1 + p2dp ≤ZR
max log kδk∞,log
1
δ
∞
1 + p2dp < ∞,
so we obtain bounded outer functions Out(|δ|1/2) and Out(|δ|−1/2) whose product is Out(1) = 1
([RR94, §5.12]). In particular, g:= Out(|δ|1/2) is invertible in H∞(C+) and |g∗|2=|δ|.
20
4.2 From Hankel positivity to reflection positivity
For a positive Hankel operator Hon H2(C+) and the corresponding function δfrom Theo-
rem 4.4, let νbe the measure on Rwith
dν (x) = |δ(x)|dx.
As δ(−x) = c+hH(−x) = c−hH(x) = δ(x), we have δ♯=δ, and in particular the function
|δ|is symmetric. We consider the weighted L2-space L2(R,C, ν) with the corresponding scalar
product h·,·iν. For the function
g:= Out(|δ|1/2)∈H∞(C+)×
we then have
|g∗|2=|δ|and g♯=g. (34)
Furthermore, gH2(C+) = H2(C+),and
mg∗:L2(R, ν)→L2(R), f 7→ g∗·f
is an isometric isomorphism of Hilbert spaces. We write
H2(C+, ν) := (H2(C+),k · kν)
for H2(C+), endowed with the scalar product from L2(R,C, ν ), so that we obtain a unitary
operator
mg:H2(C+, ν)→H2(C+).
For the unimodular function u:= δ
|δ|, we get with Theorem 4.4 for a, b ∈H2(C+):
ha, H biH2(C+)=ha∗, δRb∗iL2(R)=hp|δ|a∗,p|δ|uRb∗iL2(R)
=ha∗, uRb∗iL2(R,ν)=ha, HubiH2(C+,ν ).(35)
As νis symmetric and u♯=δ♯
|δ|♯=δ
|δ|=u,
θu(f)(x) := u(x)f(−x)
defines a unitary involution on L2(R, ν ) (and on L2(R)) for which the subspace H2(C+, ν) is
θu-positive by (35) (cf. Example 1.7). Therefore
(L2(R, ν), H 2(C+, ν), θu, U ) with (Utf)(x) = eitxf(x)
defines a reflection positive one-parameter group.
These are the essential ingredients in the proof of the following theorem:
Theorem 4.5. (Hankel positive representations are reflection positive) Let (E,E+, U, H )be a
regular multiplicity free Hankel positive representation of (R,R+,−idR). Then there exists an
invertible bounded operator g∈GL(E)with gE+=E+commuting with (Ut)t∈Rand a unitary
involution θ∈GL(E)such that:
(a) θUtθ=U−tfor t∈R.
(b) θis unitary for the scalar product hξ, η ig:= hgξ, gη i.
(c) With respect to h·,·ig, the quadruple (E,E+, θ, U )is a reflection positive representation.
(d) hξ, H ηi=hξ, θη ig=hgξ, gθ ηifor ξ, η ∈ E+.
21
Proof. As we have seen in the introduction to Section 3, the Lax–Phillips Representation
Theorem implies that, up to unitary equivalence, E=L2(R) and E+=H2(C+) with (Utf)(x) =
eitxf(x), so that Hcorresponds to a positive Hankel operator on H2(C+). We use the notation
from the preceding discussion and Theorem 4.4. Then mg∗defines an invertible operator on
L2(R) commuting with U, and θ:= uR satisfies (a) and (b). Further, (c) and (d) follow from
(35).
Remark 4.6. For H=Hδ=P+δRP ∗
+, we see with Example 1.10(b) that Hm−2
galso is a
Hankel operator Hhwith the operator symbol
h(x) = δ(x)
g∗(−x)2=δ(x)
g∗(x)2.
As |g∗|2=|δ|, the function his unimodular. Further g♯=gand δ♯=δimply h♯=h, so that
θh=hR is a unitary involution. We think of the factorization
H=Hhm2
g
as a “polar decomposition” of H.
Remark 4.7. The weighted Hardy space H2(C+, ν) has the reproducing kernel
Qν(z, w) = Q(z , w)
g(z)g(w).
In fact, for f∈H2(C+, ν) we have
hQν
w, f iH2(C+,ν)=f(w) = g(w)−1(fg)(w) = g(w)−1hQw, f giH2(C+)
=g(w)−1hg−1Qw, f iH2(C+,ν).
We have a similar result for the symmetric semigroup (Z,N,−idZ), which corresponds to
single unitary operators.
Theorem 4.8. (Hankel positive operators are reflection positive) Let (E,E+, U, H)be a regular
multiplicity free Hankel positive operator. Then there exists an invertible bounded operator
g∈GL(E)with gE+=E+commuting with Uand a unitary involution θ∈GL(E)such that:
(a) θU θ =U∗.
(b) θis unitary for the scalar product hξ, η ig:= hgξ, gη i.
(c) With respect to h·,·ig, the quadruple (E,E+, θ, U )is a reflection positive operator.
(d) hξ, H ηi=hξ, θη igfor ξ, η ∈ E+.
Proof. Up to unitary equivalence, we may assume that
E=L2(T),E+=H2(D) with (Uf)(z) = zf(z),
the shift operator (Wold decomposition), so that Hcorresponds to a positive Hankel operator
on H2(D).
Let C:= Γ2HΓ−1
2be the corresponding positive Hankel operator on H2(C+) (Theorem 3.5)
which we write as C=Hδas above in Theorem 4.4. Then (21) in the proof of Theorem 3.5
shows that H=Hkfor the function k:T→Cdefined by
k:T→C, k(z) := −δ(ω(z))zfor z∈T.
Then |k(z)|=|δ(ω(z))|is bounded with a bounded inverse.
22
We thus find an outer function g∈H∞(D)×with |g∗|2=|k|and consider the measure
dν(z) = |k(z)|dz on T([Ru86, Thm. 17.16]; see also Lemma B.13). Then
mg:H2(D, ν)→H2(D)
is unitary and the unimodular function u:= k
|k|on Tsatisfies, for a, b ∈H2(D):
ha, H biH2(D)=ha∗, kRb∗iL2(T)=hp