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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