On C_{0}-semigroups of holomorphic Carathéodory isometries in Hilbert space^{}

L.L. Stachó^{a}

aBolyai Institute, University of Szeged, Aradi Vértanúk tere 1, H-6720 Szeged, Hungary

Abstract

We establish closed formulas for all strongly continuous one-parameter semigroups of holomorphic Carathéodory isometries of the unit ball of a Hilbert space in terms of spectral resolutions of skew self-adjoint dilations related to Vesentini’s non-linear infinitesimal generator.

Keywords: strongly continuous one-parameter group, holomorphic Carathéodory isometry, Hilbert space, dilation 2000 MSC: 47D03, 32H15, 46G20

1. Introduction

Throughout this work H denotes an arbitrarily fixed complex Hilbert space with scalar product ⟨x| y⟩ being antilinear in y and the canonical norm ∥x∥ := ⟨x|x⟩^{1/2}. We also keep fixed the standard notations B := {x ∈ H : ∥x∥ < 1}, a^{∗} := [x ↦ ⟨x|a⟩] for the open unit ball, and the adjoint representation of bounded linear functionals, respectively. We regard the elements h, h^{∗} (h ∈ H) as column resp. row matrices and, given a linear map A : S → H on some linear submanifold of H. We use the canonical matrix identifications x⊕ξ ≡ [{x \atop ξ}] resp. [ {A b \atop c^{∗} d}] ≡ [x ⊕ ξ ↦ (Ax + b) ⊕ (c^{∗}x + d)] with x, b, c ∈ H and ξ, d ∈ ℂ. This gives rise to the familiar linear representation of fractional linear maps on H:

𝔉([{A b \atop c^{∗} d}]) := [x ↦ (c^{∗}x + d)^{−1}(Ax + b)].

Our object of chief interest will be the semigroup Iso(d_{B}) of all holomorphic isometries of B with respect to the Carathéodory metric d_{B}. Recall [3] that all its elements are fractional linear maps (restricted to B), namely they are compositions of Möbius transformations^{1} with linear isometries of H.

The author’s research was supported by the project TÁMOP-4.1.1.C-12/1/KONV-2012-0005 Impulse Lasers for Use in Materials Science and Biophotonics of the European Union and co-financed by the European Social Fund.

1Fractional linear transformations mapping B injectively onto itself.

Preprint submitted to Elsevier December 13, 2015

In 1987, in his pioneering work [9] Vesentini studied subsemigroups of Iso(d_{B}) arising from strongly continuous one-parameter matrix-semigroups.^{2} He established that the correspondence 𝔉^{#} : [U^{t} : t ∈ ℝ_{+}] ↦ [𝔉(U^{t})|B : t ∈ ℝ_{+}] maps the family 𝔖 of all strongly continuous one-parameter semigroups of ℂ-linear isometries of the indefinite norm ∥x∥^{2} − |ξ|^{2} on H ⊕ ℂ into the family 𝔅 of all strongly continuous one-parameter semigroups [Ψ^{t} : t ∈ ℝ_{+}] ⊂ Iso(d_{B}). According to [9, Th.VII], given [U^{t} : t ∈ ℝ_{+}] ∈ 𝔖 with the infinitesimal generator A = \frac{d}{dt} |_{t=0+}U^{t}, for the corresponding non-linear objects Ψ^{t} := 𝔉(U^{t})|B we have {p ∈ B : t ↦ Ψ^{t}(p) is differentiable} = {x ∈ B : x⊕1 ∈ dom(A)} which is dense in the ball B. It is well-known [9,6] that here we can identify the in H⊕ℂ densely defined linear operator A with an H⊕ℂ-split matrix if and only if the orbit t ↦ Ψ^{t}(0) is differentiable. This happens if and only if the generator A has the form

A = [\begin{matrix}iA + ν & b \\ b^{∗} & ν \end{matrix}],  ν ∈ℂ, b∈H, A∈Her_{s}(H)  (1.1)

with dom(A) = dom(A) ⊕ ℂ where Her_{s}(H) stands for the family of all unbounded H-hermitian operators (maximal symmetric, in H densely defined closed linear operators). Even the cases with non-differentiable 0-orbit can be treated by passing to a semigroup [Φ^{t} : t ∈ ℝ_{+}] of the form Θ^{−1} ∘ Ψ^{t} ∘ Θ with any Möbius transformation Θ such that Θ(0) ∈ dom(Γ). Indeed, since the Möbius group is transitive on B, hence any strongly continuous one-parameter semigroup [Ψ^{t} : t ∈ ℝ_{+}] ∈ 𝔅 is equivalent up to a Möbius transformation (Möbius equivalent for short in the sequel) to a semigroup [Φ^{t} : t ∈ ℝ_{+}] ∈ 𝔅 whose infinitesimal generator has the form

Γ(x) = \frac{d}{dt}|_{t=0+}Φ^{t} = b − ⟨x|b⟩x + iAx, x ∈ dom(R) ∩ B  (1.2)

with some maximal symmetric operator A defined densely on H and some vector b ∈ H. Also conversely, if iA is the infinitesimal generator for some strongly continuous one-parameter subsemigroup of L(H) then, for any b ∈ H, the vector field (1.2) is the infinitesimal generator of a strongly continuous one-parameter subsemigroup of Iso(d_{B}). In [10,6] these considerations were extended to semigroups of fractional linear transformations arising from a strongly continuous one parameter semigroup of automorphisms of a Krein space, concluding with a description of the fractional-linear image of a strongly continuous one parameter group by means of the solutions of Ricatti type equations ẋ = Γ(x) with analogous vector fields to (1.2) in Pontriagin spaces. It should be noted that as far no argument appeared in the literature concerning the seemingly plausible surjectivity of the map 𝔉^{#}. The question is rather harmless in our setting: in the case of the unit ball of a Hilbert space an argument with joint fixed points (Proposition 3.5) furnishes positive answer. However, e.g. in the case of the unit ball of L(H) the surjectivity of the respective 𝔉^{#} seems to be open and highly not trivial.

2Given a topological space X, a family [T^{t} : t ∈ ℝ_{+}] [resp. [T^{t} : t ∈ ℝ]] of self-maps T^{t} : X → X is a strongly continuous one-parameter semigroup [resp. group] if T^{s+t} = T^{s} ∘ T^{t} (s, t ∈ ℝ_{+} [resp. ℝ]) and all the orbits t ↦ T^{t}(x) (x ∈ X) are continuous.

2. Results

Henceforth, for short, C_{0}S [resp. C_{0}G ] will abbreviate the terms strongly continuous one-parameter semigroup [-group]. We shall write gen[U^{t} : t ∈ ℝ_{+}] or gen[Ũ^{t} : t ∈ ℝ] for the infinitesimal generator of the C_{0}S [U^{t} : t ∈ ℝ_{+}] or C_{0}G [Ũ^{t} : t ∈ ℝ], respectively. Given a closed subspace K in the Hilbert spaces H or ℂe⊕H we shall write P_{K} the orthogonal projection onto K without danger of confusion.

In this paper we develop a triangularization method leading to explicit algebraic formulas for a C_{0}S generated by a vector field (1.2) in terms of fixed points of Γ and quadratures of a C_{0}S formed by complex linear isometries of a 1-codimensional subspace of H. As a consequence we conclude that any C_{0}S of holomorphic Carathéodory isometries of B admits a dilation to a C_{0}G of surjective holomorphic Carathéodory isometries of the unit ball of some covering Hilbert space. Our fixed point approach seems to be new even in finite dimensions (with uniformly continuous one-parameter groups).

Recall [3] that any Carathéodory isometry Ψ ∈ Iso(d_{B}) admits a continuous extension \overline{Ψ} to the closed unit ball \overline{B}. Given a C_{0}S Ψ = [Ψ^{t}: t∈ℝ_{+}]⊂Iso(d_{B}), the extensions \overline{Ψ}:=[\overline{Ψ^{t}}: t∈ℝ_{+}] form also a C_{0}S (see [7] in more general setting). According to [9, Section 7] \overline{Ψ} admits common fixed points whose family Fix(\overline{Ψ}) consists of one or two boundary points or it is the intersection of \overline{B} with some closed complex-affine submanifold containing points from B in which case Ψ is simply Möbius equivalent to a C_{0}S of linear isometries of H restricted to B.

Our main goal is the following classification of the remaining cases with explicit formulas up to Möbius equivalence.

Theorem 2.1. Suppose the vector field (1.2) is the infinitesimal generator of a C_{0}S Φ := [Φ^{t} : t ∈ ℝ_{+}] ⊂ Iso(d_{B}) having a common boundary fixed point e ∈ Fix(\overline{Φ}) ∩ ∂B. Then we have

\begin{matrix}P_{ℂe} Φ^{t}(ξe+x_{0})=[1 − (1 − ξ)e^{−2λt}/φ_{λ,μ}(t, x_{0}, ξ)]e \\ P_{H_{0}}Φ^{t}(ξe+x_{0})=[(1−ξ)e^{−2λt}(∫_{0}^{t}e^{λs}V_{0}^{s} ds)b_{0}+e^{−λt}V_{0}^{t}x_{0}]/φ_{λ,μ}(t, x_{0}, ξ) \end{matrix} (2.2)

for all points x_{0} + ξe ∈ B with x_{0} ⊥ e where H_{0} := H ⊖ ℂe, λ := Re⟨e|b⟩, μ := Im⟨e|b⟩, b_{0} := P_{H_{0}}b, [V_{0}^{t} : t ∈ ℝ_{+}] is the C_{0}S of linear H_{0}-isometries generated by the skew-H_{0}-hermitian operator iP_{H_{0}}(A − μ)|H_{0} and

\begin{aligned}φ_{λ,μ}(t, x_{0}, ξ) & := 1 + (1 − ξ)⟨(∫_{0}^{t}e^{−2λs}∫_{0}^{s}e^{λr}V_{0}^{r} dr ds)b_{0}| b_{0}⟩− \\ & − (1 − ξ)(λ + iμ)∫_{0}^{t}e^{−2λs}ds + ⟨(∫_{0}^{t}e^{−λs}V_{0}^{s} ds)x_{0}| b_{0}⟩. \end{aligned}

Remark 2.3. The following converse can be discovered from the proofs later on (see Corollary 3.12). Given any couple of vectors e, b_{0} ∈ H such that ∥e∥ = 1 and b_{0} ⊥ e along with any C_{0}S [V_{0}^{t} : t ∈ ℝ_{+}] of linear isometries of H_{0} = H ⊖ (ℂe) and two real constants λ, μ, the maps (2.2) form a C_{0}S in Iso(d_{B}).

Remark 2.4. In case of λ ≠ 0, one can express the integrated operators in (2.2) in terms of the resolvent R(±λ, iS_{0}) of the H_{0}-hermitian operator S_{0} := i^{−1}gen[V_{0}^{t} : t ∈ ℝ_{+}]. Namely we have ∫_{0}^{t} e^{−λτ} V_{0}^{τ} dτ = (1 − e^{−λt}V_{0}^{t})R(λ, iS_{0}), ∫_{0}^{t}e^{−2λτ}∫_{0}^{τ}e^{λσ}V_{0}^{σ}dσ dτ = \frac{1}{2λ}(1−e^{−2λt})R(−λ, iS_{0})−(1−e^{−λt}V_{0}^{t})R(λ, iS_{0})R(−λ, iS_{0}).

Theorem 2.5. Let Ψ := [Ψ^{t} : t ∈ ℝ_{+}] ⊂ Iso(d_{B}) be a C_{0}S with e ∈ Fix(\overline{Ψ}) ⊂ ∂B. Then, with the notations of Theorem 2.1, we have the alternatives

(i) Fix(\overline{Ψ}) consists of two points and Ψ is Möbius equivalent to some C_{0}S [Φ^{t} : t ∈ ℝ_{+}] ⊂ Iso(d_{B}) of the form

Φ^{t}(ξe + x_{0}) = \frac{ξ + tanh(λt)}{1 + ξ tanh(λt)} e + \frac{e^{−λt}}{cosh(λt) + ξ sinh(λt)}V_{0}^{t}x_{0}; (2.6)

(ii) \{e\} = Fix(\overline{Ψ}), there is a Ψ-invariant disc of the form ∅ ≠ (e + ℂv) ∩ B and Ψ is Möbius equivalent to a C_{0}S [Φ^{t} : t ∈ ℝ_{+}] of the form

Φ^{t}(ξe + x_{0}) = \frac{1 + iμt}{1 − iμt} \frac{ξ − iμt/(1 + iμt)}{1 + iμtξ/(1 − iμt)} e + \frac{1}{1 − iμt(1 − ξ)} V_{0}^{t}x_{0}; (2.7)

(iii) there is no Ψ-invariant disc of the form ∅ ≠ (e+ℂv)∩B and Ψ is Möbius equivalent to a C_{0}S [Φ^{t} : t ∈ ℝ_{+}] of the form (2.2) with λ = 0.

Remark 2.8. In the setting of Theorem 2.1 a non-emty dics (e + ℂv) ∩ B is [Φ^{t} : t ∈ ℝ_{+}]-invariant if and only if e ⊥̸ v ∈ dom(A) and (iA + ⟨e| b⟩)v ∈ ℂe as established in Lemma 3.15. Hence, in finite dimensions only cases (i),(ii) may apper. Example 3.16 with possible independent interest for physics or stochastic processes shows that case (iii) is not void.

Recall that, as an implicit simple special case^{3} of Deddens [1,Main Thm.], every C_{0}S [U^{t}: t∈ℝ_{+}] of isometries of H admits a unitary group dilation in the following sense: there exists a Hilbert space Ĥ containing H as a subspace along with a C_{0}G [Û^{t}: t∈ℝ] of unitary operators of Ĥ such that U^{t} =Û^{t}|H (t∈ℝ_{+}). Applying a unitary dilation [V̂_{0}^{t}: r ∈ ℝ] of the isometry semigroup [V_{0}^{t}: t ∈ ℝ_{+}] in (2.2), we readily obtain the following result with non-linear dilations.

Corollary 2.9. Given any C_{0}S [Ψ^{t} : t ∈ ℝ_{+}] of holomorphic Caratéodory isometries of B, there is a strongly continuous one parameter group [Ψ̂^{t} : t ∈ ℝ_{+}] of surjective holomorphic Carathéodory isometries of the unit ball B̂ of some Hilbert space Ĥ containing H as a subspace such that Ψ^{t} = Ψ̂^{t}|B (t ∈ ℝ_{+}).

3We begin Section 4 with an elementary proof in Banach space setting.

By means of the functional calculus of the skew self-adjoint generator iŜ_{0} of the dilation group [V̂_{0}^{t} : t ∈ ℝ] of the C_{0}S [V_{0}^{t} : t ∈ ℝ_{+}] in the setting of Theorem 2.1, we get the following.

Corollary 2.10. In (2.2) we can write

φ_{λ,μ}(t,x_{0},ξ)=⟨x_{0}|f_{1}(t, λ, Ŝ_{0})b_{0}⟩+(1−ξ)[⟨f_{2}(t, λ, Ŝ_{0})b_{0}|b_{0}⟩−(λ+iμ) ∫_{0}^{t}e^{−2λs}ds]+1,

P_{H_{0}}Φ^{t}(x) = φ_{λ,μ}(t, x_{0}, ξ)^{−1}[e^{−λt} exp(itŜ_{0})x_{0} + (1 − ξ)e^{−2λt}f_{1}(t, λ, Ŝ_{0})b_{0}]

with the bounded analytic functions f_{j}(t, λ, ·) : ℝ → ℂ (j = 1, 2; λ, t ∈ ℝ)

\begin{matrix}f_{1}(t, λ, σ) := \frac{1 − e^{−t(λ+iσ)}}{λ + iσ} = ∑_{n=0}^{∞} \frac{(−1)^{n}(λ + iσ)^{n}}{(n + 1)!} t^{n+1}, f_{2}(t, λ, σ) := \\ := \frac{e^{−2λt}}{2λ(λ+iσ)} + \frac{1}{2λ(λ−iσ)} − \frac{e^{−t(λ−iσ)}}{λ^{2}+σ^{2}} = ∑_{n=2}^{∞} [\frac{(−2λ)^{n}}{2λ(λ + iσ)} − \frac{(−λ + iσ)^{n}}{λ^{2} + σ^{2}} ]\frac{t^{n}}{n!} . \end{matrix}

3. Triangularization with boundary fixed points

Lemma 3.1. Assume Ψ = [Ψ^{t} : t ∈ ℝ_{+}] ⊂ Iso(d_{B}) is a C_{0}S where Ψ^{t} = 𝔉U_{t}|B with U_{t} ∈ L(H ⊕ ℂ) (t ∈ ℝ_{+}). Then there is a family [μ_{t} : t ∈ ℝ_{+}] ⊂ ℂ \ {0} such that [μ_{t}U_{t} : t ∈ ℝ_{+}] is a C_{0}S in L(H ⊕ ℂ).

Proof. Let e be a common fixed point of the the transformations \overline{Ψ^{t}} = 𝔉U_{t}|\overline{B} (t ∈ ℝ_{+}). We are going to show that the choice μ_{t} := [U_{t}(e ⊕ 1)]_{ℂ}^{−1} entailing μ_{t}U_{t}(e ⊕ 1) = e ⊕ 1 suits our requirements. Consider the matrices V^{t} := μ_{t}U_{t}. Clearly 𝔉V^{t} = 𝔉U_{t} (t ∈ ℝ_{+}). Since the map U ↦ 𝔉U|B is a homomorphism with respect to compositions, and since its preimages are unique up to nonzero factors, we have 𝔉V^{t+s}|B = Ψ^{t+s} = Ψ^{t} ∘ Ψ^{s} = 𝔉(V^{t}V^{s})|B and hence V^{t+s} = d_{t,s}V^{t}V^{s} (t, s ∈ ℝ) with suitable constants d_{t,s} ≠ 0. The fixed point property

V^{t}(e ⊕ 1) = e ⊕ 1 (t ∈ ℝ_{+})  (3.2)

ensures that d_{t,s} ≡ 1 that is the family [V^{t} : t ∈ ℝ_{+}] is a one-parameter matrix semigroup. To see its strong continuity, recall [3, Ch. VI] that the Möbius shifts

Θ_{a} := 𝔉M_{a}, M_{a} := [\begin{matrix}Q_{a} & a \\ a^{∗} & 1 \end{matrix}], Q_{a} := P_{ℂa} + \sqrt{1−∥a∥^{2}}(1 − P_{ℂa}) (a∈B) (3.3)

act transitively on B. Thus, since every element of Iso(d_{B}) keeping the origin fixed is a restriction of a linear isometry of H, we can write Ψ^{t} = Θ_{a_{t}} ∘U_{t} where a_{t} := Ψ^{t}(0) and U_{t} is a suitable linear isometry of H. Since U_{t} = 𝔉[{U_{t} 0 \atop 0 1}], with suitable constants δ_{t} ≠ 0 we can write

V_{t} := δ_{t}M_{a_{t}} [\begin{matrix}U_{t} & 0 \\ 0 & 1 \end{matrix}] = δ_{t} [\begin{matrix}Q_{a_{t}}U_{t} & a_{t} \\ [U_{t}^{∗}a_{t}]^{∗} & 1 \end{matrix}] (t∈ℝ_{+}).

The value of δ_{t} is determined unambiguously by (3.2): δ_{t} = [1 + ⟨U_{t}e|a_{t}⟩]^{−1}. Thus to complete the proof, it suffices to see the continuity of the functions t ↦ a_{t}, t ↦ [U_{t}x, Q_{a_{t}}x] (x ∈ H). It is an immediate consequence of [2,App. A6] that the product t ↦ A_{t}B_{t} is strongly continuous for any couple of uniformly bounded strongly continuous operator valued functions t ↦ A_{t} ∈ L(X_{1}, X_{2}), t ↦ B_{t} ∈ L(X_{2}, X_{3} in case of normed spaces X_{k}. By assumption, the orbit t ↦ a_{t} = Ψ^{t}(0) is a norm-continuous map ℝ_{+} → B entailing the norm continuity of the function t ↦ Q_{a_{t}}. We deduce the strong continuity of the H-isometry valued function t ↦ U_{t} as follows. Consider any vector x ∈ H. We may assume x ∈ B without loss of generality. Then, by the aid of the Möbius shifts (3.3) we can write

U_{t}x = [Θ_{a_{t}}^{−1} ∘ Ψ_{t}](x) = Θ_{−a_{t}}(Ψ(x)) (t∈ℝ_{+})

whence the continuity of t ↦ U_{t}x = (1 − ⟨x|a_{t}⟩)^{−1}[Q_{a_{t}}x − a_{t}] is immediate.

3.4 Standard notations, assumptions. Henceforth, for the proofs for Section 2, we assume without loss of generality the following facts.

(i) Ψ := [Ψ^{t}: t∈ℝ_{+}] is an arbitrarily given C_{0}S of holomorphic Carathéodory isometries of B having no common fixed point within B.

(ii) Φ^{t} := Θ ∘ Ψ^{t} ∘ Θ^{−1} (t ∈ ℝ_{+}) with a suitable Möbius transformation Θ;

(iii) the orbit t ↦ Φ^{t}(0) is differentiable and Φ^{t} = 𝔉^{#}U^{t}|B with some C_{0}S [U^{t} : t∈ℝ_{+}] of linear H-isometries,

A := gen[U^{t} : t∈ℝ_{+}] = [\begin{matrix}iA & b \\ b^{∗} & 0 \end{matrix}], b∈H, iA = gen[U^{t} : t ∈ ℝ_{+}];

(iv) e ∈ ∂B is a joint boundary fixed point of the maps \overline{Φ^{t}}, we write

H_{0} := H⊖ℂe, P := P_{ℂe}, P_{0} := P_{H_{0}} = 1−P, T: x↦x+e, T := [\begin{matrix}id_{H} & e \\ 0 & 1 \end{matrix}].

Proposition 3.5. We have e∈dom(A) with A(e⊕1) = ν(e⊕1) and b = (ν−iA)e for some ν ∈ℂ. The possibly unbounded operator

A_{0} := P_{0}A|H_{0} ∩ dom(A) is H_{0}-hermitian and, in terms of (ℂe ⊕ H_{0} ⊕ ℂ)- matrices, we have

T^{−1}AT = [\begin{matrix}−\overline{ν} & 0 & 0 \\ −b_{0} & iA_{0} & 0 \\ ν & b_{0}^{∗} & ν \end{matrix}] where b_{0} := P_{0}b, ν = ⟨e| b⟩.  (3.6)

Proof. By assumption 3.4(ii), e⊕1 is a joint eigenvector of the linear operators U^{t}. Hence U^{t}(e⊕1) = ζ_{t}(e⊕1) (t ∈ ℝ_{+}) with a continuous solution [t ↦ ζ_{t}] of the Cauchy equation ζ_{s+t} = ζ_{s}ζ_{t}. Thus for some ν ∈ ℂ, ζ_{t} = e^{νt} and we have

e⊕1 ∈ dom(A) = {𝔷 : t ↦ U^{t}𝔷 is differentiable}, A(e⊕1) = ν(e⊕1).

As a consequence, e ∈ dom(A) = P_{H}dom(A) and the operator

Ã_{0} := A − P A − AP + P AP = (1 − P)A(1 − P) = P_{0}AP_{0}

is a bounded perturbation ranging in H_{0} of A ∈ Her_{s}(H) with a self-adjoint operator of finite rank. Hence its restriction A_{0} to H_{0} is a well-defined unbounded H_{0}-hermitian operator. Since A is a (ℂe ⊕ H_{0})-matrix operator, we can write

A=[\begin{matrix}iA & b \\ b^{∗} & 0 \end{matrix}]=[\begin{matrix}iα & ia_{0}^{∗} & β \\ ia_{0} & iA_{0} & b_{0} \\ \overline{β} & b_{0}^{∗} & 0 \end{matrix}], b_{0} :=P_{0}b, β:=⟨b|e⟩, a_{0} :=P_{0}Ae, αv:=⟨Ae|e⟩

in terms of (ℂe⊕H_{0} ⊕ℂ)-matrices. The eigenvector equation A(e⊕1) = ν(e⊕1) means iAe + b = νe with ⟨e| b⟩ = ν entailing iα + β = ν, ia_{0} + b_{0} = 0, \overline{β} = ν. Since

T = [\begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] , T^{−1} = [\begin{matrix}1 & 0 & −1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] ,

in (ℂe⊕H_{0}⊕ℂ)-matrix form, hence (3.6) is immediate.

Notation 3.7. Henceforth [U_{0}^{t} : t ∈ ℝ_{+}] denotes the C_{0}S of H_{0}-isometries generated by iA_{0}.

Lemma 3.8. Let E_{1}, E_{2} be Banach spaces, G := [{G_{1} 0 \atop H G_{2}} ] with H ∈ L(E_{1}, E_{2}) and G_{k} =gen[W_{k}^{t}: t∈ℝ_{+}] for some C_{0}S [W_{k}^{t}: t∈ℝ_{+}]⊂L(E_{k}). Then the family

S^{t} := [\begin{matrix}W_{1}^{t} & 0 \\ ∫_{0}^{t} W_{2}^{t−s}HW_{1}^{s}ds & W_{2}^{t} \end{matrix}] (t ∈ ℝ_{+})

is a C_{0}S in L(E_{1} ⊕ E_{2}) such that gen[S^{t} : t ∈ ℝ_{+}] = G.

Proof. The operator G is a bounded perturbation by the operator H := [AT OP 0 0H 0] of the generator G_{1} ⊕G_{2}(≡ [{G_{1} 0 \atop 0 G_{2}} ]) of the C_{0}S [W^{t} : t ∈ ℝ_{+}] with W^{t} := W_{1}^{t} ⊕ W_{2}^{t} and dom(G) = dom(G_{1}) ⊕ dom(G_{2}). According to [2,Thm.III.1.10], for every fixed 𝔷=x⊕y ∈dom(G) we have

S^{t}𝔷 = ∑_{n=0}^{∞} S_{n}(t) where S_{0}(t) := W^{t}𝔷, S_{n+1}(t) = ∫_{0}^{t}W^{t−s}HS_{n}^{(k)}(s) ds.

Since H is an off-diagonal 2×2 triangular operator matrix, S_{n}(t) = 0 for n > 1.

3.9. Proof of Theorem 2.1 Since T a bounded invertible H⊕ℂ-operator and A = gen[U^{t} : t ∈ ℝ_{+}], we have

T^{−1}AT = gen[V^{t} : t ∈ ℝ_{+}] for V^{t} := T^{−1}U^{t}T .

Since Φ^{t} = 𝔉(U^{t})|B (t ∈ ℝ_{+}), in terms of the translation T x := x + e we can interpret the C_{0}S [V^{t} : t ∈ ℝ^{+}] as the linear representation by means of 𝔉 of the semigroup [T^{−1} ∘ Φ^{t} ∘ T : t ∈ ℝ_{+}] formed by holomorphic isometries of the shifted ball B − e whose continuous extensions leave the origin fixed. Due to the projective identities 𝔉(T^{−1}VT ) = T^{−1} ∘ 𝔉(V) ∘ T (V ∈L(H ⊕ ℂ)), for the points x ∈ T^{−1}B = B − e we have

𝔉V^{t}(x) = [𝔉(T^{−1}U^{t}T )](x) = [T^{−1} ∘ Φ ∘ T](x) = Φ(x + e) − e.

Therefore

Φ^{t}(x) = 𝔉V^{t}(x − e) + e (x ∈ B).

By the aid of Lemma 3.8 and (3.6) we calculate a quadrature form for V^{t} as follows. Regarding the top left 2×2-corner of the matrix T^{−1}AT we get

[\begin{matrix}−\overline{ν} & 0 \\ −b_{0} & iA_{0} \end{matrix}] = gen[V^{t} : t ∈ ℝ_{+}], V^{t} = [\begin{matrix}e^{−\overline{ν}t} & 0 \\ ∫_{0}^{t} U_{0}^{t−s}e^{−\overline{ν}s}(−b_{0})ds & U_{0}^{t} \end{matrix}] . (3.10)

Another application of Lemma 3.8 to T^{−1}AT yields

V^{t} = [\begin{matrix}V^{t} & 0 \\ ∫_{0}^{t} e^{ν(t−s)}b^{∗}V^{s}ds & e^{νt} \end{matrix}]  (t ∈ ℝ_{+}).  (3.11)

As a consequence of (3.11), since V^{t}(x⊕1)=[V^{t}x] ⊕ e^{νt}[ ∫_{0}^{t}⟨e^{−νs}V^{τ} x| b⟩dτ + 1], we get

Φ^{t}(x) = \frac{e^{−νt}V^{t}(x − e)}{⟨∫_{0}^{t}e^{−νs}V^{s}(x − e) ds| b⟩ + 1} + e  (x ∈ B, t ∈ ℝ_{+}).  (3.12)

We substitute (3.10) into (3.12) in terms of the new parametrization

λ = Re ν, μ := Im ν, V_{0}^{t} := e^{−iμt}U_{0}^{t}.

Given any vector z = z_{0} + ζe, z_{0} ∈ H_{0}, and recalling the commutativity of convolutions,

\begin{matrix}e^{−νt}V^{t}z = ζe^{−2λt}[e −∫_{0}^{t}e^{λs}V_{0}^{s}b_{0} ds]+ e^{−λt}V_{0}^{t}z_{0}, ∫_{0}^{t}⟨e^{−νs}V^{s}z|b⟩ ds = \\ = ζ(λ+iμ)\frac{1−e^{−2λt}}{2λ} − ζ∫_{0}^{t}e^{−2λs}∫_{0}^{s}e^{λr}⟨V_{0}^{r}b_{0}|b_{0}⟩ dr + ∫_{0}^{t}e^{−λs}⟨V_{0}^{s}z_{0}|b_{0}⟩ ds ds. \end{matrix}

The statement of Theorem 2.1 is immediate from (3.12) with z:= x−e = x_{0} + (ξ−1)e.

Remark 3.13. It is discovered from the above proof that any tuple

𝔞 := (H, e, [V_{0}^{t} : t∈ℝ_{+}], b_{0}, λ, μ)

with a Hilbert space H, a unit vector e ∈ H, a C_{0}S [V_{0}^{t} : t ∈ ℝ_{+}] of H_{0}(:= H ⊖ ℂe)-isometries, a vector b_{0} ∈H_{0} and two real constants gives rise to a C_{0}S [Φ𝔞^{t} : t ∈ ℝ_{+}] of holomorphic Carathéodory isometries of the open unit ball B

of H ≡ ℂe⊕H_{0} whose generator Γ(x) = \frac{d}{dt} |_{t=0+}Φ𝔞^{t}(x) = \frac{d}{dt} |_{t=0+}𝔉(T V^{t}T^{−1})x

has the form (1.2) with

b = \left[{λ − iμ \atop b_{0}}\right] , A = \left[{2μ −ib_{0}^{∗} \atop ib_{0} A_{0}}\right] , iA_{0} = gen[V_{0}^{t} : t ∈ ℝ_{+}].  (3.14)

In particular we can extend [Φ𝔞^{t} : t ∈ ℝ_{+}] to a C_{0}G [Φ𝔞^{t} : t ∈ ℝ] ⊂ Iso(d_{B}) if and only if [V_{0}^{t} : t ∈ ℝ_{+}] consists of H_{0}-unitary operators (cf. [9,Thm.II]). Furthermore, given any tuple 𝔟 := (H, A, e, λ) with a densely defined maximal symmetric linear H-operator A, there is a unique C_{0}S [Ψ_{b}^{t} : t ∈ ℝ_{+}] ⊂ Iso(d_{B}) whose infinitesimal generator is of the form (1.2) with b := (ν − iA)e where ν := λ + iμ and μ = ⟨Ae| e⟩.

Lemma 3.15. The C_{0}S [Φ^{t} : t ∈ ℝ_{+}] ⊂ Iso(d_{B}) with generator (1.2) and joint boundary fixed point e ∈ ∂B admits no invariant disc of the form B∩(e+ℂe) ≠ ∅ if and only if the operator iA + ⟨e|b⟩ is not injective or e ∈ range(iA − ⟨e|b⟩).

Proof. Consider any vector v ∈ H such that e + v ∈ B. The disc Δ_{e}^{v} := B ∩ (e + ℂv) is [Φ^{t} : t ∈ ℝ_{+}]-invariant if and only if the vector field (1.2) is tangent to it that is if b − ⟨e + τv|b⟩(e + τv) + iA(e + τv) ∈ ℂv whenever e + τv ∈ B. This happens if and only if −⟨v|b⟩e + iAv = ζv for some ζ ∈ ℂ because we have e ∈ dom(Γ) = dom(A) and Γ(e) = b − ⟨e|b⟩ + iAe = 0 (due to the fact that the point e is [\overline{Φ^{t}} : t ∈ ℝ_{+}]-invariant). According to Proposition 3.5, here we have b = (ν − iA)e where ν = ⟨e|b⟩. Therefore ζv = −⟨v|(ν − iA)e⟩e + iAe = ⟨(−\overline{ν} − iA)v|e⟩e + iAe. Notice that, in general, rmP_{ℂe}x = ⟨x|e⟩e = x − P_{H_{0}}x (x ∈ H). Thus the disc Δ_{e}^{v} is [Φ^{t} : t ∈ ℝ_{+}]- invariant if and only if −\overline{ν}P_{ℂe}v + P_{H_{0}}(iAv) − ζv = 0 i.e. P_{H_{0}}(iAv − ζv) = 0 and P_{ℂe}(−\overline{ν} − ζ)e = 0 for some ζ ∈ ℂ. By assumption Δ_{e}^{v} ≠ ∅ which is possible if and only if P_{ℂe}v ≠ 0 implying ζ = −\overline{ν}. Hence we conclude that the [Φ^{t} : t ∈ ℝ_{+}]-invariance of Δ_{e}^{v} is equivalent to the relation P_{H_{0}}(iAv + \overline{ν})v = 0 i.e. to (iA + \overline{ν})v ∈ ℂe which completes the proof.

Example 3.16. The C_{0}S of the type [Ψ𝔟^{t} : t ∈ ℝ_{+}] in Remark 3.13 with H := L^{2}(ℝ), Af := [x ↦ xf(x) (dom(A) := {f : ∫_{−∞}^{∞} |xf(x)|^{2} dx < ∞}), e:=(2π)^{−1/2}exp (−(x−1)^{2}/2) and λ:=0 admits no invariant 1-dimensional disc. Proof: We have ⟨Ae| e⟩ = (2π)^{−1} ∫_{−∞}^{∞} x exp(−(x − 1)^{2}) dx = (2π)^{1/2} ≠ 0. Thus, according to the construction of the C_{0}S Ψ𝔟, ν =⟨e| b⟩=iμ=i⟨Ae| e⟩/2∈iℝ\{0}. The relation (iA + \overline{ν})v = ζe would imply v = −iζ exp(−(x − 1)^{2}/2)/(x − μ) ∈ L^{2}(ℝ) which is possible only if v = 0.

3.17. Proof of Theorem 2.5 Recall [3] that the 1-dimensional complex affine discs the form Δ_{p,q} := (p+ℂ(q− p))∩B (q ≠ p, q ∈ ∂B) are the ranges of complex geodesics for the Carathéodory d_{B}, and d_{B}-isometries map complexmetric their family into itself. In particular, in the case when p ≠ q ∈ ∂B are joint fixed points of the continuous extensions \overline{Ψ^{t}} the disc Δ_{p,q} is automatically [Ψ^{t} : t ∈ ℝ_{+}]-invariant. Suppose Ψ^{t}(Δ_{p,q}) = Δ_{p,q} (t ∈ ℝ_{+}). Then the restricted maps ψ_{p,q}^{t} := Ψ^{t}|D_{p,q} form a C_{0}S of holomorphic automorphisms of a 1-dimensional Hilbert ball, thus their continuous extensions \overline{ψ_{p,q}^{t}} to \overline{D_{p,q}} admit at least one fixed point which is necessarily a joint fixed point for the maps \overline{Ψ^{t}}. A 1-dimensional application of Theorem 2.1 shows that all the orbits t ↦ ψ_{p,q}^{t}(x) = Ψ^{t}(x) (x ∈ Δ_{p,q}) are automatically real analytic. Hence, given any Möbius transformation Θ, the C_{0}S [Φ^{t} : t ∈ ℝ_{+}] with Φ^{t} := Θ ∘ Ψ^{t} ∘ Θ^{−1} leaves the d_{B}-geodesic D_{Θ(p),Θ(q)} invariant with differentiable 0-orbit t ↦ Φ^{t}(0). Also conversely: if [Φ^{t} : t ∈ ℝ_{+}] is a C_{0}S leaving the disc Δ_{e,−e}(= {ζe : |ζ| < 1}) invariant and Φ^{t}(e) = e, Ψ^{t} = Θ^{−1} ∘ Φ^{t} ∘ Θ (t ∈ ℝ_{+}) as in 3.4, then the image Θ(\overline{Δ_{e,−e}}) is a [\overline{Ψ^{t}} : t ∈ ℝ_{+}]-invariant 1dimensional affine section of \overline{B} containing a joint fixed point (the point Θ^{−1}(e)) of [\overline{Ψ^{t}} : t ∈ ℝ_{+}].

Proof of (i),(ii). It remains only to verify the possibility of the simplified representations (2.6),(2.7) by means of an appropriate choice for the coordinatizing Möbius transformation Θ in 3.4. By setting x_{0} := 0 in (2.2), it is straightforward to check that a C_{0}S [Φ^{t} : t ∈ ℝ_{+}] of the form 2.2 leaves the disc Δ_{e,−e} invariant if and only if b_{0} = 0 and Φ^{t}(ξe) = ω_{λ,μ}(t, ξ)e (|ξ| < 1) with the function

ω_{λ,μ}(t, ξ) := 1 − \frac{2λ(1 − ξ)e^{−2λt}}{2λ − (1 − ξ)(λ + iμ)(1 − e^{−λt})}.

It is also easy to see that the constant 1 is a joint fixed point of all functions ω_{λ,μ}(t, ·) and, for fixed λ, μ ∈ ℝ, the family ω_{λ,μ}(t, ·) t ∈ ℝ_{+}) admits another fixed point namely the constant ξ_{λ,μ} := \frac{iμ−λ}{iμ+λ} with modulus 1 if and only if we have μ = 0. Due to folklore 2-transitivity properties of the Möbius group (for direct proof see [7]), given any two couples (e_{1}, e_{2}), (f_{1}, f_{2}) ∈ [∂B]^{2} of distinct boundary points there exists a Möbius transformation Θ^{(e_{1},f_{1},e_{2},f_{2})} with the effect e_{k} ↦ f_{k} (k = 1, 2). Thus in case if [\overline{Ψ^{t}} : t ∈ ℝ_{+}] has only a unique fixed point p ∈ ∂B but the disc Δ_{p,q} is [Ψ^{t} : t ∈ ℝ_{+}]-invariant, with any choice Θ := Θ^{(p,e,q,κe)} where |κ| = 1 we get a formula for Φ^{t} by substituting b_{0} = 0 and μ = 0 in (2.2) which is (2.7).

If [\overline{Ψ^{t}} : t ∈ ℝ_{+}] admits two distinct fixed points p, q ∈ ∂B then, as we have shown, the disc Δ_{p,q} is automatically [Ψ^{t} : t ∈ ℝ_{+}]-invariant, and with the choice Θ := Θ^{(p,e,q,−e)} we get a formula for Φ^{t} by substituting b_{0} = 0 and μ = 0 in (2.2) establishing (2.6).

Proof of (iii). Suppose indirectly that 0 ≠ λ = Re⟨e|b⟩. Then the skew symmetry of iA entails range(iA − ⟨e|b⟩) = H. By the previous lemma, we have a non-trivial Γ-invariant disc and we are in the settings of (i) or (iii). By assumption, (i) is not the case. However, in the case of (iii) we have ⟨e|b⟩ = iμ ∈ iℝ automatically.

3.18. Proof for Remark 2.4. The operator S_{0} is closed with dense domain in H_{0}. Since S_{0} is also symmetric, both ±iS_{0} are dissipative (namely Re⟨±iSx_{0}|x_{0}⟩ = 0 for x_{0} ∈ dom(S)) with the properties that both range(±iS_{0}+ δ) are dense in H for any δ > 0 and that the operators (iS + δ)^{−1} : range(S) → H_{0} (0 ≠ δ ∈ ℝ) are all bounded and densely defined.^{4} Given δ ∈ℝ \ {0}, by [2, II. Lemma 1.3] for any x_{0} ∈ range(iS_{0}−δ) and t > 0 we have ∫_{0}^{t} e^{−δτ} V_{0}^{τ} x_{0} dτ = ∫_{0}^{t} e^{−δτ} V_{0}^{τ} (iS_{0}−δ)[(iS_{0}−δ)^{−1}x_{0}] dτ = (e^{−δt}V_{0}^{t}−1)(iS_{0}−δ)^{−1}x_{0}. The boundedness of both the operators V^{t} and the resolvents R(δ, iS) = closure((δ −iS_{0})^{−1}) establishes 2.4 for t ∈ ℝ_{+} and 0 ≠ λ ∈ ℝ with integrals of strongly continuous bounded operator valued functions.

4. Dilation

Lemma 4.1. Let [U^{t} : t ∈ ℝ_{+}] be a C_{0}S of linear isometries of a Banach space E. Suppose E is a subspace of another Banach space F and there is a surjective isometry V ∈ L(F) such that U^{1} = V |E. Then there is a subspace E ⊂ Ê ⊂ F along with a C_{0}G [Û^{t} : t ∈ ℝ] of surjective linear isometries of Ê such that U^{t} = Û^{t}|E (t ∈ ℝ_{+}) with dom(gen[Û^{t} : t ∈ ℝ]) ⊃ dom(gen[U^{t} : t ∈ ℝ])

Proof. Let Ê := closure(E_{∞}) in F where E_{∞} := ⋃_{n=0}^{∞} with E_{n} := V^{−n}E. By assumption V E = U^{1}E ⊂ E whence, by induction we conclude that the subspaces E_{n} (n ∈ ℤ_{+}) form an increasing sequence. Therefore all the operators U_{n}^{t}:=V^{−2n}U^{t+n}V^{n}|E_{n} (t≥−n, n∈ℤ_{+}) are well-defined isometries E_{n} →E_{⌈n−t⌉}. We have U_{n}^{t} = U_{n+1}^{t}|E_{n} for all indices n∈ℤ_{+}. Indeed if x̂∈E_{n} and t≥−n then

\begin{aligned}U_{n+1}^{t}x̂ & = V^{−2n−2}U^{t+n+1}V^{n+1}x̂ = V^{−2n−2}U^{t+n+1}U^{1}V^{n}x̂ = \\ & = V^{−2n−2}U^{t+n+2}V^{n}x̂ = V^{−2n−2}V^{2}U^{t+n}V^{n}x̂ = V^{−2n}U^{t+n}V^{n}x̂ = U_{n}^{t}x̂ \end{aligned}

since V extends U^{1} and we have V^{n+1}x̂ ∈ E implying V^{n+1}x̂ = U^{1}V^{n}x̂. Hence

U_{∞}^{t}x̂ := lim_{n→∞} U_{n}^{t}x̂ = [U_{n}^{t}x̂ : n ∈ ℤ_{+}, n ≥ t] (x̂ ∈ E_{∞})

is a well-defined linear isometry of the linear manifold E_{∞} for any t ∈ ℝ. Since range(U_{n}^{t}) ⊃ V^{−2n}U^{⌈t⌉+n}E = V^{⌈t⌉−n}E for t ≥ −n, we have range(U_{∞}^{t}) = E_{∞} (t ∈ ℝ). Thus the operators Û^{t} := closure(U_{∞}^{t}) (t ∈ ℝ) are well-defined surjective linear Ê-isometries, each of which extending the respective U^{t}. We check they form a C_{0}G as follows. Since [Û^{t} : t ∈ ℝ] is an equilipschitzian family, it suffices to see that its restriction [Û^{t} : t ∈ ℝ] to the dense submanifold E_{∞} of Ê is a C_{0}G . Given s, t ∈ ℝ and x̂ ∈ Ê_{ℓ}, we have Û^{t}x̂ = V^{−2n}U^{t+n}V^{n}x̂ ∈ Ê_{2n} whenever n ≥ tℓ and Û^{s}(Û^{t}x̂) = V^{−2m}U^{t+m}V^{m}Û^{t}x̂ whenever m ≥ max{−s, 2n}. It follows Û^{s}Û^{t}x̂ = Û^{s+t}x̂ because hence, with k ≥ 2(|s| + |t| + ℓ) we have

\begin{aligned}Û^{s}(Û^{t}x̂) & = V^{−4k}U^{s+2k}V^{2k}V^{−2k}U^{t+k}V^{k}x̂ = \\ & = V^{−4k}U^{s+t+3k}V^{k}x̂ = V^{−4k}U^{s+t+2k}V^{2k}x̂V^{k} = Û^{s+t}x̂. \end{aligned}

4Indeed y_{0} ⊥ range(±iS_{0} + δ) means 0 = ⟨±iS_{0}x_{0} − δx_{0}|y_{0}⟩ that is 0 = ⟨x_{0}| ∓ iS_{0}y_{0} − δy_{0}⟩ for (x_{0} ∈ dom(S)) entailing ∓iS_{0}y_{0} + δy_{0} = 0 with δ∥y_{0}∥^{2} = ±i⟨S_{0}y_{0}|y_{0}⟩ ∈ iℝ which is possible only if y = 0. Thus by the Lumer-Phillips theorem [2, II. Thm. 3.15], also the ope rator −iS_{0} generates a strongly continuous contraction (actually isometry) semigroup and all the values 0 ≠ δ ∈ ℝ belong to the resolvent set of iS.

To see strong continuity, consider any vector x̂ ∈ Ê_{ℓ}. Then for any integer n ≥ ℓ the orbit (−n, ∞) ∋ t ↦ Û^{t}x̂ = V^{−2n}U^{t+n}(V^{n}x̂) is continuous since V^{−2n} is an isometry and (V^{n}x̂) ∈ E. Hence we can see also the required generator domain inclusion property: with x̂ := x ∈ dom(gen[U^{t} : t ∈ ℝ_{+}]) we have V^{n}x = U^{n}x ∈ dom(gen[U^{t} : t ∈ ℝ_{+}]) entailing even the differentiability of the orbits (−n, ∞) ∋ t ↦ Û^{t}x.

In particular, since every linear isometry of a Hilbert space admits a unitary dilation [8], in our setting of interest we conclude the following.

Corollary 4.2. If [U^{t} : t ∈ ℝ_{+}] is a C_{0}S of linear H-isometries, there exists a Hilbert space Ĥ containing H as a subspace along with a C_{0}G [Û^{t} : t ∈ ℝ] of Ĥ-unitary operators such that U^{t} = Û^{t}|H (t ∈ ℝ_{+}) whose generator is an extension of gen[U^{t} : t ∈ ℝ_{+}].

4.3. Proof of Corollaries 2.9-10 Given any Hilbert space Ĥ containing H as a subspace, every Möbius transformation of H extends to a Möbius transformation of Ĥ. Hence it suffices to see only that any C_{0}S of the form of Theorem 2.1 admits a group a dilation of the same algebraic form in a larger Hilbert space. Let [Φ^{t} : t ∈ ℝ_{+}] be given as in Theorem 2.1. According to Corollary 3.13, for some tuple

𝔞 := (H, e, [V_{0}^{t} : t∈ℝ_{+}], b_{0}, λ, μ) we have Φ^{t} = Φ𝔞^{t} = 𝔉(U^{t}) (t ∈ ℝ_{+}) with

gen[U^{t} : t ∈ ℝ_{+}] = [\begin{matrix}iR & b \\ b^{∗} & 0 \end{matrix}] = [\begin{matrix}i(S_{0} + μ) & −b_{0} & b_{0} \\ b_{0}^{∗} & 2iμ & λ − iμ \\ b_{0}^{∗} & λ + iμ & 0 \end{matrix}] .  (4.4)

in terms of [H_{0} ⊕ (ℂe) ⊕ ℂ]-matrices. Let [V̂_{0}^{t} : t ∈ ℝ] be the dilation C_{0}G of [V_{0}^{t} : t ∈ ℝ_{+}] consisting of unitary operators of a covering Hilbert space Ĥ_{0} of H_{0} with the skew self-adjoint extension iŜ_{0} = gen[V̂_{0}^{t} : t ∈ ℝ] of iS_{0} guaranteed by Corollary 4.2. Also by Corollary 3.13, the tuple 𝔞̂ := (Ĥ, e, [V̂_{0}^{t} : t∈ℝ_{+}], b_{0}, λ, μ) where Ĥ := Ĥ_{0} ⊕ (ℂe) gives rise to a C_{0}G Φ𝔞̂^{t} : t ∈ ℝ] such that Φt ; â = 𝔉(Û^{t}) (t ∈ ℝ) whose infinitesimal generator can be written in the form of the right hand side of (4.4) when the entry S_{0} is replaced with Ŝ_{0}. Hence, by Theorem 2.1, the transformations Φ_{a}^{t} can be written in the form (2.3) with Ŝ_{0} in place of S_{0} and V̂_{0}^{t} in place of V_{0}^{t}. Since V̂_{0}^{t}|H_{0} = V_{0}^{t} (t ∈ ℝ_{+}), it readily follows Φ𝔞̂^{t}|H = Φ𝔞^{t} (t ∈ ℝ_{+}) which completes the proof of Corollary 2.9.

To prove Corollary 2.10, consider any C_{0}S [Φ_{a}^{t} : t ∈ ℝ_{+}] with its dilation group [Φ̂𝔞^{t} : t ∈ ℝ_{+}] as above. By construction, the dilation C_{0}G [V̂_{0}^{t} : t ∈ ℝ] consists of Ĥ_{0}-unitary operators. Thus, in view of Stone’s classical theorem, we can apply the functional calculus [8] with its skew self-adjoint generator iŜ_{0} when evaluating the transformations Φ𝔞^{t} by means of (3.12). Actually, for any t ∈ ℝ we have

∫_{0}^{t} e^{−λτ} V̂_{0}^{τ} dτ = g_{1,t}(Ŝ_{0}), ∫_{0}^{t} e^{−2λτ} ∫_{0}^{τ} e^{λσ}V̂_{0}^{σ} dσ dτ = g_{2,t}(Ŝ_{0})

with the functions s↦∫_{0}^{t} e^{−λτ} e^{iτs}dτ resp. s↦∫_{0}^{t} e^{−2λτ} ∫_{0}^{τ} e^{λσ}e^{iσs}dσ dτ which are real analytic ℝ → ℂ. Straightforward calculation establishes their algebraic form and the Taylor series appearing in Corollary 2.10.

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