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# Reflection positivity and Hankel operators -- the multiplicity free case

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## 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 representation of triples $(G,S,\tau)$, where $G$ is a group, $\tau$ an involutive automorphism of $G$ and $S \subseteq G$ a subsemigroup with $\tau(S) = S^{-1}$. For the triples $(\mathbb Z,\mathbb N,-id_{\mathbb Z})$, corresponding to reflection positive operators, and $(\mathbb R,\mathbb R_+,-id_{\mathbb R})$, 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 $\mu_H$ on $\mathbb R_+$ defined by a positive Hankel operator $H$ on $H^2(\mathbb C_+)$ to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for $H$.
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 SGa
subsemigroup with τ(S) = S1. 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() ............................ 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 SGa subsemigroup with τ(S) = S1. 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 SGis 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:GU(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 gG. (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 ,·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)tRall 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 HB(E+) satisfying
HU+(s) = U+(s)Hfor sS. (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 UU(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={zC:|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),
zT, 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:TT, h(z) = h(z) for zT.
The hardest part is to control the positivity of the form ,·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)H(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 δ1is 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+=
{zC: Im z > 0}is the upper half-plane, H2(C+) is the Hardy space on C+, and (U(t)f)(x) =
3
eitxf(x), xR, 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:RT, h(x) = h(x) for xR.
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()g()H(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
w1is 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 + λ21
z+λH(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 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={zC:|z|<1},C+=R+iR+(upper half plane), Cr=R++iR
(right half plane), Sβ={zC: 0 <Im z < β}(horizontal strip).
For a holomorphic function fon C+, we write ffor its non-tangential limit function on
R; likewise for functions on Dand Sβ.
We write ω:DC+, ω(z) := i1+z
1zfor the Cayley transform with ω1(w) = wi
w+i.
On the circle T={eC:θR}, we use the length measure of total volume 2π.
For wCwe write ew(z) := ezw for the corresponding exponential function on C.
For a function f:GC, we put f(g) := f(g1).
We write Efor 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 ,·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 SGis 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:GU(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 \
gG
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
θU (g)θ=U(τ(g)) for gG(6)
([N ´
O18, Def. 3.3.1].
Deﬁnition 1.3. If (S, ♯) is an involutive semigroup and U+:SB(F) a representation of S
by bounded operators on the Hilbert space F, then we call AB(F) a U+-Hankel operator if
AU+(s) = U+(s)Afor sS. (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 AB(E)satisﬁes
AU(g) = U(τ(g))A=U(g)Afor gG, (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)R1=U(τ(g)) for gG, then AB(H)
satisﬁes (8) if and only if A=BR for some BU(G).
Proof. For the ﬁrst assertion, we observe that, for sSand ξ, η ∈ E+, we have
hξ, HAU+(s)ηi=hξ , AU(s)ηi(8)
=hξ, U (τ(s))i=hU(s)ξ, Aη i
=hU+(s)ξ, HAηi=hξ , U+(s)HAηi.
The second assertion follows from the fact that B:= AR1commutes with U(G).
6
Lemma 1.5. Hankel operators have the following elementary properties:
(a) If HHanU+(F), then HHanU+(F).
(b) If HHanU+(F)and Bcommutes with U+(S), then HB and BHare Hankel operators.
Proof. (a) If HHanU+(F) and sS, then
HU+(s) = (U+(s)H)= (U
+(s)H)= (HU+(s))=U+(s)H.
(b) Let HHanU+(F) and suppose that Bcommutes with U+(S). Then
HBU+(s) = H U+(s)B=U
+(s)HB for sS
implies that HB HanU+(F). Taking adjoints, we obtain BH= (HB)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 HHanU+(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 gG(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 1B(F) is a U+-Hankel operator if and only if the two representations Uand
Ucoincide, 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 sS.
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+:SB(F)be a representation of the involutive semigroup (S, ♯)by
bounded operators on Fand H0be 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=qU+(s)for sS. (9)
Proof. For every sSand 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 nN0and η∈ F,
we have
kb
U(ss)nq(η)k2
b
F=hU+(ss)nη, H U+(ss)nηi ≤ kHkkU+(ss)k2nkηk2.
Now [Ne99, Lemma II.3.8(ii)] implies that
kb
U(s)k ≤ pkU+(ss)k ≤ max(kU+(s)k,kU+(s)k) for sS.
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 RU(E) a unitary
involution with R(E+) = E
+. Then
A:= R1AR
deﬁnes an antilinear involution on B(E) leaving the subalgebra
M:= {AB(E): AE+⊆ E+}
invariant. In fact, A∈ M implies that AE
+⊆ E
+, so that
AE+=R1ARE+=R1AE
+R1E
+=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 zC+.(12)
Let H=P+hRP
+,hL(R) be a Hankel operator on H2(C+) (cf. Theorem 3.5) and
gH(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 xR. 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 DC.
8
Deﬁnition 2.1. Areﬂection positive operator on a reﬂection positive Hilbert space (E,E+, θ)
is a unitary operator UU(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 \
nZ
UnE+=\
n>0
UnE+
!
={0}and [
nZ
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
(2(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 (2(Z), ℓ2(N0), S), where
Sis the right shift. For our purposes, it is most convenient to identify 2(Z) with L2(T) and
2(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 zT.
We now want to understand the possibilities for adding a unitary involution θfor which
H2(D) is θ-positive. For f:TCwe deﬁne
f(z) := f(z) for zT
(cf. (11) and (37) in Appendix B). Then any involution θsatisfying θSθ =S1has the form
θh(f)(z) = h(z)f(z) for zT,
where hL(T) satisﬁes h=hand h(T)T(cf. Lemma 1.4). As any hL(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 gH(D)on H2(D). Then the following are equivalent:
(a) The Rosenblum relation 1DS =SDholds 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
gDfor all gH(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 hL(T)with D=P+mhRP
+for R(F)(z) := zF (z),zD, 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 fL1(T), h H(D).(15)
For f1, f2H2(D), we observe that
hf1, Dgf2i=hDf1, g f2i=ZT
(Df1)(z)g(z)f
2(z)dz =η(Df1)f
2(g)
(cf. Example B.2), and
hgf1, Df2i=ZT
gf1
(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(z1), zT. 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 gH(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 hL(T) can even
be chosen in such a way that
kHhk=khk.
As Hh= 0 if and only if hH(D) for D={zC:|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)nN0deﬁned 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,mN0is positive deﬁnite, i.e.,
(an)nNdeﬁnes 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
xnH(x) = anfor nN0.
Widom’s Theorem (see Theorem A.1 in Appendix A) implies that
hf, H giH2(D)=Z1
1
f(x)g(x)H(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)tRis a unitary one-parameter group on Esuch that
UtE+⊆ E+for t > 0 and θUtθ=Utfor tR.(19)
As in Deﬁnition 1.2, we call a reﬂection positive one-parameter group regular if
\
tR
UtE+=\
t>0
UtE+
!
={0}and [
tR
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)tRare 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 xR
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=Utfor tRare 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 hL(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 hL(R), the following assertions hold:
(a) H
h=Hh. In particular Hhis hermitian if h=h.
(b) Hh= 0 if and only if hH(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 =hHhf, 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 hH(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+)/C1֒L(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,t0, and the multiplication operators mg,
gH(C+)on H2(C+). We also consider the unitary isomorphism
Γ2:L2(T)L2(R),2f)(x) := 2
x+ifxi
x+i(20)
from [Ni19, p. 200] which maps H2(D)to H2(C+)and the operator
D:= Γ1
2CΓ2:H2(D)H2(D).
Then the following are equivalent:
12
(a) There exists hL(R)with C=Hh, i.e., Cis a Hankel operator in the sense of
Deﬁnition 3.3.
(b) Cmg=m
gCfor all gH(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 t0.
(d) Dis a Hankel operator on H2(D).
Proof. (a) (b): Suppose that C=Hhfor some hL(R). For f1, f2H2(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 =hgf1, Hhf2i,
which is (b).
(b) (c) follows from e
it =eit for t > 0.
(c) (b): For f1, f2H2(C+) and gH(C+), we observe that
hf1, Cg f2i=hCf1, gf2i=ZR
Cf1
(x)g(x)f
2(x)dx =ηCf1f2(g)
(see Example B.2 for the functionals ηf) and
hgf1, Cf2i=ZR
(gf1)(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 xR. 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 gH(C+), which is
(b).
(b) (d): The Cayley transform ω:DC+, ω(z) := i1+z
1zdeﬁnes an isometric isomorphism
L(T)L(R), g 7→ gω1which restricts to an isomorphism H(D)H(C+) and
satisﬁes
Γ2mg=mgω1Γ2.
Therefore (b) is equivalent to
Dmgω=m
gωDfor gH(C+),
which is (d) by Theorem 2.2.
(d) (a): Suppose that D=Dkas in Theorem 2.2. For fH2(D), we then have for xR
(CΓ2(f))(x) = Γ2(Df )(x) = 2
x+i(Df )(ω1(x)) = 2k(ω1(x))
x+if(ω1(x))
=k(ω1(x)) ix
(i+x)
2
(x+i)f(ω1(x)) = k(ω1(x)) ix
i+xΓ2(f)(x).
The assertion now follows with
h(x) := k(ω1(x)) ix
i+x=k(ω1(x))ω1(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 t0, 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
eλt H(λ) for t > 0.(23)
Widom’s Theorem for C+(Theorem 3.7 below) now implies that
hf, H giH2(C+)=Z
0
f()g()H(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
(λ) := (λ)
1 + λ2.
Then the following are equivalent:
(a) There exists an αRwith
ZR+|f()|2(λ)αkfk2for fH2(C+),(24)
i.e., µis the Carleson measure of a positive Hankel operator on H2(C+).
(b) ρ((0, x)) = O(x)and ρ((x1,)) = O(x)for x0+.
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) = λ1
λ+ 1 =ω1().
14
For fH2(D), we then have
Z1
1|f(t)|2D(t) = hf, Df iH2(D)=hΓ2(f), CΓ2(f)iH2(C+)
=ZR+|Γ2(f)()|2C(λ) = 2 ZR+
|f(ω1())|2
(1 + λ)2C(λ)
= 2 ZR+
|f(γ(λ))|2
(1 + λ)2C(λ) = 2 Z1
1
|f(t)|2
(1 + γ1(t))2d(γµC)(t).
As γ1(t) = (t) = 1+t
1tand 1 + (1+t)
(1t)=2
(1t),it follows that
D(t) = (1 t)2
2d(γµC)(t).
We conclude that
µD((1 x, 1)) = Z1
1x
(1 t)2
2d(γµC)(t) = Z
γ1(1x)
(1 γ(λ))2
2C(λ)
=Z
2
x1
2
(λ+ 1)2C(λ) = 2 Z
2
x1
1 + λ2
(λ+ 1)2(λ).
Therefore µD((1 x, 1)) has for x0+the same asymptotics as ρ((x1,)). Likewise
µD((1,1 + x)) = Z1+x
1
(1 t)2
2d(γµC)(t) = Zγ1(x1)
0
(1 γ(λ))2
2C(λ)
=Zx
2x
0
2
(λ+ 1)2C(λ) = 2 Zx
2x
0
1 + λ2
(λ+ 1)2(λ).
This shows that µD((1,1 + x)) has for x0+ 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, t0R+such that
ρ((0, ε])
εβand ρ([t, )) tγfor every εε0, 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])
ερ(R+)
ε0
and ρ([t, )) tρ(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
zw
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) = (HQz)(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()Qw()H(λ) = 1
4π2Z
0
H(λ)
(z)(w)
=1
4π2Z
0
H(λ)
(λiz)(λ+iw).(26)
Deﬁnition 3.9. From Widom’s Theorem for the upper half plane (Theorem 3.7), we know
that the measure (λ)
1+λ2is ﬁnite, so that,
κ(z) := ZR+
λ
1 + λ21
z+λH(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+λ1
z+λH(λ) = ZR+
zw
(w+λ)(z+λ)H(λ),
so that κ(z)κ(w)
zw=ZR+
H(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 hHL(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:RiR, hH(p) := i
π·ZR+
p
λ2+p2H(λ).
Then hHL(R,C)and the associated Hankel operator HhHequals H.
Proof. Part 1: We ﬁrst show that hHis bounded. Let (λ) = H(λ)
1+λ2be the ﬁnite measure
on R+from Theorem 3.7. Then we have
ZR+
p
λ2+p2H(λ) = ZR+
p1 + λ2
λ2+p2(λ).
For the integrand
fp(λ) := p1 + λ2
λ2+p2we have f
p(λ) = 2pp21λ
(λ2+p2)2.
16
Hence the function fpis increasing for p1, and therefore
Z(0,1]
fp(λ)(λ)fp(1) Z(0,1]
(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 p1 to
Z(1,)
fp(λ)(λ) = ρ((1,)) fp(1) + Z(1,)
ρ((t, )) f
p(t)dt
ρ((1,)) 2p
1 + p2+Z(1,)
γ
t·2pp21t
(t2+p2)2dt
ρ((1,)) ·1 + γp21
tp
t2+p2+ arctan t
p
p2
1
=ρ((1,)) + γp21
p2π
2p
1 + p2arctan 1
pρ((1,)) + γπ
2.
So, for every p1, we have
ZR+
p
λ2+p2H(λ) = ZR+
fp(λ)(λ) = Z(0,1]
fp(λ)(λ) + Z(1,)
fp(λ)(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)(λ)fp(1) Z(1,)
(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)(λ) = ρ((0,1]) fp(1) Z(0,1]
ρ((0, t]) f
p(t)dt
ρ((0,1]) 2p
1 + p2Z(0,1]
βt ·2pp21t
(t2+p2)2dt
ρ((0,1]) ·1 + β1p2arctan t
ptp
t2+p21
0
=ρ((0,1]) + β1p2arctan 1
pp
1 + p2ρ((0,1]) + βπ
2.
So, for every p(0,1), we have
ZR+
p
λ2+p2H(λ) = ZR+
fp(λ)(λ) = Z(0,1]
fp(λ)(λ) + Z(1,)
fp(λ)(λ)
ρ((0,1]) + βπ
2+ρ((1,)) = ρ(R+) + βπ
2.
Therefore, for every pR+, we have
|hH(p)|=1
πZR+
p
λ2+p2H(λ)1
πρ(R+) + 1
2max{β, γ }.
17
Since hH(p) = hH(p), this yields
khHk1
πρ(R+) + 1
2max{β, γ}
and therefore hHL(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 + λ21
λ+zH(λ)
from (27). Then, for pR×, we have
Im (κ(ip)) = Im ZR+
λ
1 + λ21
λ+ip H(λ)!
= Im ZR+
λ
1 + λ2λip
λ2+p2H(λ)!=ZR+
p
λ2+p2H(λ),
so
hH(p) = i
π·Im (κ(ip)) .(29)
For the real part, we get
Re (κ(ip)) = Re ZR+
λ
1 + λ21
λ+ip H(λ)!= Re ZR+
λ
1 + λ2λip
λ2+p2H(λ)!
=ZR+
λ
1 + λ2λ
λ2+p2H(λ) = p21ZR+
λ
(1 + λ2) (λ2+p2)H(λ)
and therefore
|Re (κ(ip))|=p21ZR+
λ
(1 + λ2) (λ2+p2)H(λ)
p21ZR+
4λ
(1 + λ)2(|p|+λ)2H(λ).
For pR×, we now deﬁne the function
np:C+C, np(z) = 2z
(1 iz) (|p| − iz),
where by ·we denote the inverse of the function CrC+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 npH2(C+). Since µHis a Carleson measure, by Theorem 3.7(a), there is a
constant α0 such that
ZR+
f()g()H(λ)αkfk2kgk2for every f, g H2(C+).
18
Then
|Re (κ(ip))| ≤ p21ZR+
4λ
(1 + λ)2(|p|+λ)2H(λ) = p21ZR+|np()|2H(λ)
p21αknpk2
2=αp21ZR
4|x|
(1 + x2) (p2+x2)dx
= 4αp21Z
0
2x
(1 + x2) (p2+x2)dx = 4αZ
0
2x
1 + x22x
p2+x2dx
= 4αlog 1 + x2log p2+x2
0= 4αlog 1 + x2
p2+x2
0= 8α|log (|p|)|
for every pR×. This estimate together with khHk<shows that, for z, w C+, the
integrals ZR
κ(ip)
(pz) (pw)dp and ZR
κ(ip)
(pz) (pw)dp
exist. We have ZR
κ(ip)
(pz) (pw)dp =ZR
κ(ip)
(p+z) (p+w)dp = 0 (30)
because the function pκ(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)
(pz) (pw)dp =ZR
κ(ip)
(pz) (pw)dp = 2πi κ(iz)
zw+κ(iw)
wz
= 2πi κ(iz)κ(iw)
zw
(28)
= (2π)3QH(z, w).
By continuity of both sides in zand w, we get
ZR
κ(ip)
(pz) (pw)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)
(pz) (pw)dp =ZR
hH(p)
(pz) (p+w)dp
=ZR
i
π·Im (κ(ip))
(pz) (p+w)dp =1
2πZR
κ(ip)κ(ip)
(pz) (p+w)dp
=1
2π ZR
κ(ip)
(pz) (p+w)dp ZR
κ(ip)
(pz) (p+w)dp!
(30)
=1
2πZR
κ(ip)
(pz) (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 pR×.
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 pR×, we have
|hH(p)|=1
πZR+
|p|
λ2+p2H(λ)1
πZ(0,a]
|p|
λ2+p2H(λ)
1
πZ(0,a]
|p|
a2+p2H(λ) = c·|p|
a2+p2.
Choosing the measure µ=δafor an aR+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
pzp
1 + p2log (k(p)) dp,
where CTand k:RR+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 cR×, we
have
δ:= hH+c1L(R,C)and 1
δL(R,C).
Further Hδ=Hand there exists an outer function gH(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 c1H(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
(x) = |δ(x)|dx.
As δ(x) = c+hH(x) = chH(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 ,·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, δRbiL2(R)=hp|δ|a,p|δ|uRbiL2(R)
=ha, uRbiL2(R)=ha, HubiH2(C+ ).(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 gGL(E)with gE+=E+commuting with (Ut)tRand a unitary
involution θGL(E)such that:
(a) θUtθ=Utfor tR.
(b) θis unitary for the scalar product hξ, η ig:= hgξ, gη i.
(c) With respect to ,·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 mgdeﬁnes 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 Hm2
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 fH2(C+, ν) we have
hQν
w, f iH2(C+)=f(w) = g(w)1(fg)(w) = g(w)1hQw, f giH2(C+)
=g(w)1hg1Qw, f iH2(C+).
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
gGL(E)with gE+=E+commuting with Uand a unitary involution θGL(E)such that:
(a) θU θ =U.
(b) θis unitary for the scalar product hξ, η ig:= hgξ, gη i.
(c) With respect to ,·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:= Γ2HΓ1
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:TCdeﬁned by
k:TC, k(z) := δ(ω(z))zfor zT.
Then |k(z)|=|δ(ω(z))|is bounded with a bounded inverse.
22
We thus ﬁnd an outer function gH(D)×with |g|2=|k|and consider the measure
(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, kRbiL2(T)=hp