A fixed point approach to unitary C_{0}-groups in Hilbert space

L.L. STACHÓ

1. Introduction, main results

Throughout this work H denotes an arbitrarily fixed infinite dimensional complex Hilbert space with the scalar product ⟨x|y⟩ which is linear in x and conjugate linear in y, giving rise to the norm ∥x∥ = ⟨x|x⟩^{1/2}. We denote the open unit ball {e ∈ H : ∥e∥ < 1} with B and for any vector a ∈ H we shall write a^{∗} := [x ↦ ⟨x|a⟩] for its dual functional. Recall that the group Aut(B) of all holomorphic automorphisms of B consists the biholomorphic maps B ↔ B, and the H-unitary operators restricted to B form the isotropy subgroup of the origin of Aut(B). Stone’s classical theorem on strongly continuous oneparameter groups of unitary operators can be reformulated in terms of Aut(B) as a statement that the infinitesimal generator of a strongly continuous one-parameter subgroup of Aut(B) leaving fixed the origin can be identified canonically with the restriction iA|B where A is a possibly unbounded self-adjoint linear operator with dense domain in H. The first attempt to reach an analogous description for the strongly continuous one-parameter subgroups of Aut(B) formed by possibly non-linear maps can be found in Vesentini’s celebrated paper [15] in 1987 based on a linear model generalizing naturally a well-known analogous concept for finite dimensional Möbius groups. Later on [16] he returned to the theme with the aim of extending the results to strongly continuous one-parameter semigroups holomorphic automorphisms of the unit ball of a Cartan factor of type 1 that is a space of the form L(H_{1}, H_{2}) with Hilbert spaces H_{k} where a linear model is still available. Katshkevich-Reich-Shoiket [9] extended these investigations to general strongly continuous one-parameter semigroups of holomorphic fractional linear transformations. Nevertheless a simple explicit algebraic description for these semigroups seems not yet being appeared in the literature. Our purpose in this paper will be to develop an alternative shorter approach to the description of vector fields arising as infinitesimal generators of strongly continuous one-parameter subsemigroups of Aut(B). Though several details presented here seem to be contained implicitly in [15,16,9], our treatment treatment based on the existence of joint fixed points uses essentially different ideas which may be of independent geometric interest concerning the structure of the Banach-Lie group of the surjective isometries of a hyperbolic space. We try give a self-contained presentation staring only from the familiar form (2.3) for the Möbius shifts establish first in [5,Ch.X] in infinite dimensions. At the beginning we also provide some simple general results concerning the existence of joint fixed points and continuity of boundary extensions in the setting of reflexive spaces. We pay particular attention (Section 4) to the characterize the cases where a Kaup type formula (x ↦ b − ⟨x|b⟩ + iAx given first in [10] for the uniformly continuous case) is available for the vector fields of the infinitesimal generators. We focus to one-parameter groups, establishing the following main results.

Theorem 1.1. Assume [Ψ^{t} : t ∈ ℝ] is a strongly continuous one-parameter group of holomorphic automorphisms of B.^{1} Then there exists a vector \overline{x} with ∥\overline{x}∥ ≤ 1 along with a constant λ ∈ ℝ and a densely defined possibly unbounded self-adjoint operator A : Z → H with dense domain such that

(1.2) B ∩ (\overline{x} + Z) = D where D := \{x ∈ D : t ↦ Ψ^{t}(x) is differentiable on ℝ\},

(1.3)  \frac{d}{dt}|_{t=0}Ψ^{t}(x) = −⟨(iA − λ)(x − \overline{x})|\overline{x}⟩x + (iA + λ)(x − \overline{x})  (x ∈ D).

Given any tuple (A, \overline{x}, λ) consisting of a densely defined self-adjoint operator A : Z → H, a vector \overline{x} with ∥\overline{x}∥ ≤ 1 and a real number λ, there exists (a necessarily unique) strongly continuous one-parameter group [Ψ^{t} : t ∈ ℝ] satisfying (1.2) and (1.3) if and only if one of the following alternatives holds: (1) ∥\overline{x}∥=1; (2) ∥\overline{x}∥<1, λ=0.

Corollary 1.4. If ∥\overline{x}∥ = 1 above and \overline{x} is an eigenvector of the operator A with eigenvalue ρ(∈ ℝ) then the following alternatives hold: either for some 0 ≠ λ ∈ ℝ we have

Ψ^{t}(x_{0} + ξ\overline{x}) = \frac{e^{λt} exp(itA)x_{0}}{2λ − φ(ρ, λ, t)(ξ − 1)} + (1 + \frac{2λe^{2λt}(ξ − 1)}{2λ − φ(ρ, λ, t)(ξ − 1)})\overline{x}

where λ ≠ 0 and φ(ρ, λ, t) := (iρ − λ)e^{2λt} − (λ + iρ), or

Ψ^{t}(x_{0} + ξ\overline{x}) = \frac{exp(itA)x_{0}}{1 + iρt(ξ − 1)} + (1 + \frac{ξ − 1}{1 + iρt(ξ − 1)})\overline{x}

for all t ∈ ℝ, and x_{0} + ξ\overline{x} ∈ B with x_{0} ⊥ \overline{x}.

Corollary 1.5. If \overline{x} ≠ \overline{x}′ ∈ ∂B are the only common fixed poits of [\overline{Ψ^{t}} : t ∈ ℝ], there exists Θ ∈ Aut(B) along with a constant 0 ≠ λ ∈ ℝ such that \overline{Θ}(\overline{x}) = \overline{x}, \overline{Θ}(\overline{x}′) = −\overline{x} and

Θ ∘ Ψ^{t} ∘ Θ^{−1}(x + ξ\overline{x}) = \frac{e^{λt} exp(itA)x_{0}}{2λ − φ(0, λ, t)(ξ − 1)} + (1 + \frac{2λe^{2λt}(ξ − 1)}{2λ − φ(0, λ, t)(ξ − 1)})\overline{x}

for all t ∈ ℝ, and x_{0} + ξ\overline{x} ∈ B with x_{0} ⊥ \overline{x}.

2. Preliminaries: linear model with joint fixed points

Lemma 2.1. Assume K is a compact topological space and let [f_{t} : t ∈ ℝ_{+}] be a oneparameter semigroup of continuous maps K → K admitting fixed points such that all the functions t ↦ f_{t}(x) are continuous. Then also ⋂_{t∈ℝ_{+}} Fix(f_{t}) ≠ ∅.

1 That is Ψ^{t+s} = Ψ^{t} ∘ Ψ^{s} ∈ Aut(B) for all couples t, s ∈ ℝ and the functions [t ↦ Ψ^{t}(x)] are continuous ℝ → H for any fixed vector x ∈ B.

Proof. Consider any parameter t > 0 and a point x ∈ Fix(f_{t}). For n = 1, 2, … recursively we have f_{nt}(x) = f_{t}(f_{(n−1)t}(x)) = x. Thus Fix(f_{nt}) ⊃ Fix(f_{t}) ≠ ∅ (t ∈ ℝ_{+}, n = 1, 2, …). From the continuity of the maps f_{t} it follows that Fix(f_{1/n!}) (n = 1, 2, …) is a decreasing sequence of non-empty compact sets with non-empty intersection X := ⋂_{n} Fix(f_{1/n!}). Since any rational number 0 ≠ q ∈ ℚ_{+} can be written in the form q = m/n! for suitable integers m, n > 0, it follows even that ⋂_{q∈ℚ_{+}} Fix(f_{q}) = X ≠ ∅. Consider any parameter t > 0 and any point x ∈ X. Given any sequence q_{1}, q_{2}, … ∈ ℚ_{+} converging to t, the continuity of the orbit t ↦ f_{t}(x) ensures that x = lim_{n} f_{q_{n}}(x) = f_{t}(x).

Lemma 2.2. Let K be a domain in a Banach space E and let f_{t} : D_{t} → E (t ∈ ℝ_{+}] be a family of holomorphic maps defined on open neighborhoods of \overline{K} such that the restrictions [f_{t}|K : t ∈ ℝ_{+}] form a strongly continuous one-parameter semigroup. Assume that for every boundary point x ∈ ∂K there exists a 1-dimensional complex disc Δ_{x} centered in x and intersecting K such that Δ_{x} ⊂ ⋂_{t∈[0,δ_{x}]} D_{t} and ⋃_{t∈[0,δ_{x}]} f_{t}(Δ_{x}) is a bounded set for some δ_{x} > 0. Then [f_{t}|\overline{K} : t ∈ ℝ] is also a strongly continuous one-parameter semigroup.

Proof. By assumption f_{s}(f_{t}(x)) = f_{s+t}(x) (x ∈ K, s, t ∈ ℝ_{+}). Since the maps f_{t}|\overline{K} are all continuous, and it follows f_{s}(f_{t}(x)) = f_{s+t}(x) (x ∈ \overline{K}, s, t ∈ ℝ_{+}) that is [f_{t}|\overline{K} : t ∈ ℝ_{+}] is a one-parameter semigroup of continuous maps on the closure \overline{K}. Hence, to complete the proof, it suffices to see only that for any x ∈ ∂K, the function t ↦ f_{t}(x) is continuous on some neighborhood the origin, namely on [0, δ_{x}). Fix any x ∈ ∂K and consider a convergent sequence t_{n} → t within [0, δ_{x}]. We show the convergence f_{t_{n}}(x) → f_{t}(x) as follows. We can write Δ_{x} = {x+ζv : |ζ| < 1} with a suitable vector v ∈ E. By assumption, the functions g_{n}(ζ) := f_{t_{n}}(x + ζv) (n = 1, 2, …) are uniformly bounded and holomorphic on the unit disc. Furthermore they are assumed to converge pointwise to g(ζ) := f_{t}(x+ζv) on the non-empty open complex domain {ζ ∈ ℂ : x + ζv ∈ K}. In Banach space setting, pointwise convergence implies uniform convergence on compact sets for holomorphic maps [12]. In particular, we have uniform convergence for [g_{n} : n = 1, 2, …] on some compact disc with positive radius. By a theorem of Vigué [17, 8], for a uniformly bounded sequence of holomorphic maps, the uniform convergence on some subdomain entails locally uniform (and hence pointwise) convergence on the whole domain. In particular, g_{n} → g pointwise and hence f_{t_{n}}(x) = g_{n}(0) → g(0) = f_{t}(x).

Corollary 2.3. If E is a JB*-triple, K is its open unit ball and [f_{t} : t ∈ ℝ] is a strongly continuous one-parameter subgroup of Aut(K) then the maps \overline{f}_{t} obtained with graph closure from the respective f_{t}, form a strongly continuous one-parameter group of maps \overline{K} → \overline{K}.

Proof. It is well-known [11] that we can write f_{t} = M_{f_{t}(0)} ∘ U_{t} with some invertible linear operator U_{t} ∈ L(E) and a so-called Möbius transformation with the fractional linear form x ↦ f_{t}(0) + B_{t}[I + L(x, f_{t}(0)]^{−1}x where B_{t}, L(x, f_{t}(0)) ∈ L(E) and ∥L(x, f_{t}(0))∥ ≤ ∥f_{t}(0)∥∥x∥. In particular each f_{t} extends holomorphically to the ball of radius 1/∥f_{t}(0)∥. Thus the conditions required by the lemma are fulfilled since lim_{t→0} f_{(}0) = 0 by assumption.

Henceforth we focus to the case of the unit ball B of an infinite dimensional Hilbert space.

Recall [5, Ch. VI] that the group of Aut(B) all holomorphic automorphisms of B admits a matrix representation. Namely each element Ψ of Aut(B) has the fractional linear form

(2.4)  Ψ(x) = \frac{Ax + b}{⟨x|c⟩ + d} ,  A ∈ L(H), b, c ∈ H, d ∈ ℂ

and we have

Ψ_{1}∘Ψ_{2}(x) = Ψ_{1}(Ψ_{2}(x)) = \frac{Ax + b}{⟨x|c⟩ + d} whenever [ \begin{matrix}A & b \\ c^{∗} & d \end{matrix} ] = [ \begin{matrix}A_{1} & b_{1} \\ c_{1}^{∗} & d_{1} \end{matrix} ] [ \begin{matrix}A_{2} & b_{2} \\ c_{2}^{∗} & d_{2} \end{matrix} ] .

This representation is unique up to a constant, since in (2.4) we necessarily have

[ \begin{matrix}A & b \\ c^{∗} & d \end{matrix} ] = d[ \begin{matrix}Q_{a} & a \\ a^{∗} & 1 \end{matrix} ][ \begin{matrix}U & 0 \\ 0 & 1 \end{matrix} ] where a:=Ψ(0), U = (β_{a}^{2}P_{a}+ β_{a}\overline{P}_{a})^{−1}Ψ′(0)

in terms of the the Fréchet derivative Ψ′ and the standard notations

P_{a}:=[orthogonal projection H→ℂa], β_{a} =\sqrt{1−∥a∥^{2}}, Q_{a} := P_{a} + β_{a}(I − P_{a}).

We call the matrix

Ψ̃ := \left[{Q_{a} a \atop a^{∗} 1}\right]\left[{U 0 \atop 0 1}\right] = \left[{Q_{a}U a \atop (U^{∗}a)^{∗} 1}\right]

corresponding to the case with constant d = 1 the canonical representation of Ψ. In the sequel we shall write

H :=H⊕ℂ=\{[{x \atop ξ}]: x∈H, ξ ∈ℂ\}

and identify the matrix M := [m_{ij}]_{i,j=1}^{2} where m_{11} ∈ L(H), m_{12} ∈ H, m_{2,1} ∈ H^{∗} and m_{22} ∈ℂ with the linear operator [{x \atop ξ}] ↦ M[{x \atop ξ}] on H. Notice that, by (2.4) we have

(2.5)  Ψ(x) = [Ψ̃ \left[{x \atop 1}\right] ]_{ℂ}^{−1}[Ψ̃ \left[{x \atop 1}\right] ]_{H}  (x ∈ B)

where [ ·]_{ℂ} resp. [ ·]_{H} are the standard notations for the canonical projections H → ℂ resp. H → H. It is immediate that any Ψ ∈ Aut(B) extends holomorphically to the ball (1 − ∥Ψ(0)∥)^{−1}B. Hence we can define the group of all automorphisms of the closed unit ball \overline{B} := \{x ∈ H : ∥x∥ ≤ 1\} as

Aut(\overline{B}) := \{\overline{Ψ} : Ψ ∈ Aut (B)\} where \overline{Ψ} := [continuous extension of Ψ to \overline{B}].

It is also well-known [5, Ch.VI] that any mapping \overline{Ψ} ∈ Aut(\overline{B}) is weakly continuous and preserves the Grassmann family Aff(\overline{B}) of all complex affine closed subspaces intersected with \overline{B}.^{1} By Schauder’s fixed point theorem, Fix(\overline{Ψ}) ≠ ∅ since \overline{B} is weakly compact. Moreover we have the following alternatives:

1 If x = ∑_{k=1}^{2}λ_{k}x_{k} with λ_{1}, λ_{2} ∈ ℂ and ∑_{k=1}^{2}λ_{k} = 1 then \overline{Ψ}(x) = ∑_{k=1}^{2}α_{k}\overline{Ψ}(x_{k}) for some α_{1}, α_{2}∈ℂ with ∑_{k=1}^{2}α_{k} =1 (namely α_{k} =λ_{k}[1+⟨x_{k}|U^{∗}a⟩]/[1+⟨λ_{1}x_{1}+λ_{2}x_{2}|U^{∗}a⟩]).

(1) Fix(\overline{Ψ}) ∈ Aff(\overline{B}), (2) Fix(\overline{Ψ}) consists of two boundary points.

In case (2) from the proof of [5, Thm.VI.4.8] we see even that \overline{Ψ} = \overline{Φ} ∘ \overline{Θ}_{a} ∘ \overline{Φ}^{−1} with a suitable automorphism Φ ∈ Aut(\overline{B}) and a Möbius shift

(2.6)  Θ_{a} : x ↦ \frac{Q_{a}x + a}{1 + ⟨x|a⟩}

for some 0 ≠ a ∈ B such that Fix(\overline{Θ}_{a}) = \{−e, e\} where e := a/∥a∥.

Remark 2.7. In finite dimensions, it is customary to normalize (2.4) by requiring det \left[{A b \atop c^{∗}d}\right]=1. Thus, in case of dim(H) = N, in this manner one can establish a canonical identification of Aut(B) with a subgroup of the classical matrix group SL(N + 1). Though in infinite dimensions such a normalization is not available, for one-parameter groups with common fixed point there is an alternative way as follows.

Definition 2.8. Let ([Ψ^{t} : t ∈ ℝ], \overline{x}) be a couple of a one-parameter subgroup of Aut(B) with common fixed point \overline{x} for the continuous extensions of its members to \overline{B}: \overline{x} ∈ \overline{B}, \overline{Ψ}^{t}(\overline{x}) = \overline{x} (t ∈ ℝ). In terms of the canonical representations define

Ψ̂t ; \overline{x} := [Ψ̃^{t}[\overline{x} ; 1]]_{ℂ}^{−1}Ψ̃^{t} = \frac{1}{1 + ⟨U_{t}\overline{x}|a_{t}⟩} [ \begin{matrix}Q_{t} & a_{t} \\ a_{t}^{∗} & 1 \end{matrix} ] [ \begin{matrix}U_{t} & 0 \\ 0 & 1 \end{matrix} ] (t ∈ ℝ).

where a_{t} = Ψ_{t}(0) ∈ B, U_{t} ∈ L(H) is a suitable unitary operator and Q_{t} := Q_{a_{t}} = P_{t} + β_{t}(I − P_{t}) with P_{t}x := P_{a_{t}}x= ∥a_{t}∥^{−2}⟨x|a_{t}⟩a_{t}, β_{t} =\sqrt{1−∥a_{t}∥^{2}}. Later on, conveniently we shall simply write Ψ̂^{t} instead of Ψ̂t ; \overline{x} without danger of confusion.

Remark 2.9. As we have seen Ψ^{t}(x) = (1 + ⟨U_{t}x|a_{t}⟩)^{−1}[Q_{t}U^{t}x + a_{t}] = [Θ_{a_{t}} ∘ U_{t}](x). Thus, by construction we have

\overline{Ψ^{t}}(x) = [Ψ̂^{t} \left[{x \atop 1}\right] ]_{ℂ}^{−1}[Ψ̂^{t} \left[{x \atop 1}\right] ]_{H} (x ∈ \overline{B}),  Ψ̂^{t}[\overline{x} ; 1] = [\overline{x} ; 1].

It is worth to notice that the term ⟨U_{t}\overline{x}|a_{t}⟩ is actually independent of U_{t} as

(2.10)  ⟨U_{t}\overline{x}|a_{t}⟩ = \frac{⟨\overline{x} − a_{t}|a_{t}⟩}{1 − ⟨\overline{x}|a_{t}⟩},  Ψ̂^{t} = \frac{1 − ⟨\overline{x}|a_{t}⟩}{1 − ⟨a_{t}|a_{t}⟩} [ \begin{matrix}Q_{t} & a_{t} \\ a_{t}^{∗} & 1 \end{matrix} ] [ \begin{matrix}U_{t} & 0 \\ 0 & 1 \end{matrix} ] .

Proof: In general we have P_{t}y = ⟨y|a_{t}⟩⟨a_{t}|a_{t}⟩^{−2}a_{t} (0 ≠ a_{t}, y ∈ H). It follows ⟨P_{t}U_{t}\overline{x}|a_{t}⟩ = ⟨U_{t}\overline{x}|a_{t}⟩ with ⟨\overline{P}_{t}U_{t}\overline{x}|a_{t}⟩ = 0 for any t ∈ ℝ. Thus multiplying the fixed point equation \overline{x} = \overline{Ψ}^{t}(\overline{x}) = (1+⟨U_{t}\overline{x}|a_{t}⟩)^{−1}(P_{t}+β_{t}\overline{P}_{t})U_{t}\overline{x} with |a_{t}⟩, we get (1+⟨U_{t}\overline{x}|a_{t}⟩)^{−1}⟨U_{t}\overline{x}+a_{t}|a_{t}⟩ = ⟨\overline{x}|a_{t}⟩ whence the relations (2.10) are immediate.

The power style indexing of Ψ̂^{t} in t is justified by the proposition below.

Proposition 2.11. Given a strongly continuous one-parameter group [Ψ^{t} : t ∈ ℝ] in Aut(B) with common fixed point \overline{x} ∈ \overline{B}, the family [Ψ̂ t ; \overline{x} : t ∈ ℝ] is a strongly continuous one-parameter group of operators in H.

Proof. Since Ψ^{t} ∘Ψ^{s} = Ψ^{t+s} (t, s ∈ ℝ), for the representation matrices we have Ψ̂^{t}Ψ̂^{s} = d_{t,s}Ψ̂^{t+s} with suitable constants d_{t,s} ∈ ℂ_{∗}. The fixed point property \overline{Ψ}^{t}(\overline{x}) = \overline{x} implies

Ψ̂^{t} \left[{\overline{x} \atop 1}\right] = \left[{\overline{x} \atop 1}\right] (t ∈ ℝ).

Hence necessarily d_{t,s} =1 (t, s∈ℝ), thus [Ψ̂^{t} : t∈ℝ] is a one-parameter matrix group. By assumption, the function t ↦ a_{t} = Ψ^{t}(0) is norm-continuous ℝ → B. Hence we can deduce the strong continuity of the H-unitary operator valued function t ↦ U_{t}. Namely consider any vector x ∈ H. To establish the norm-continuity of the function t ↦ U_{t}, we may assume without loss of generality that x ∈ B. Then, by the aid of the Möbius shifts (2.3) we can write

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

Observe that the norm continuity of t ↦ a_{t} implies the continuity of t ↦ ⟨x|a_{t}⟩ and t ↦ β_{t} ∈ [0, 1) entailing the norm-continuity of t ↦ P_{t} + β_{t}\overline{P}_{t} ∈ \overline{Ball}(L(H)). Hence the required norm-continuity of t ↦ U_{t}x = (1 − ⟨x|a_{t}⟩)^{−1}[(P_{t}x − a_{t} + β_{t}\overline{P}_{t}x] is immediate. In general, the product of two bounded strongly continuous linear operator valued functions ℝ → L(X) over a normed space X is strongly continuous. Hence we conclude that the entries (1, 1), (1, 2), (2, 1) resp. (2, 2) of the matrices Ψ̂^{t} are strongly continuous functions ℝ → L(H), ℝ → H, ℝ → H^{∗}≃H resp. ℝ → ℝ which completes the proof.

Corollary 2.12. Given a strongly continuous one-parameter group [T^{t} : t ∈ ℝ] in L(H), the following statements are equivalent

(i) for all t ∈ ℝ, the maps x ↦ [T^{t}[{x \atop 1}]]_{ℂ}^{−1}[T^{t}[{x \atop 1}]]_{H} belong to Aut(B); (ii) we have T^{t} = e^{μt}Ψ̂^{t} (t ∈ ℝ) for some strongly continuous one-parameter subgroup [Ψ^{t} : t ∈ ℝ] of Aut(B) and a constant μ ∈ ℂ;

(iii) each operator T^{t} maps the cone K := \{[{x \atop ξ}] : |ξ|^{2} > ∥x∥^{2}\} onto itself; (iv) each operator T^{t} maps ∂K := \{[{x \atop ξ}] : |ξ|^{2} = ∥x∥^{2}\} onto itself.

Proof. The implication (ii)⇒(i) is trivial by (2.5).

Proof of (i)⇒(ii): By assumption the maps Ψ^{t}(x) := [T^{t}[{x \atop 1}]]_{ℂ}^{−1}[T^{t}[{x \atop 1}]]_{H} (t ∈ ℝ, x ∈ B). are well-defined holomorphic automorphisms of the unit ball B. By (2.5) we have T^{t} = d_{t}Ψ̂^{t} (t ∈ ℝ) with suitable constants d_{t} ∈ ℂ_{∗}. Fixing any point x ∈ B, the strong continuity of the group [T^{t} : t ∈ ℝ] implies the continuity of the function T^{t}[{x \atop 1}] whence we deduce also the continuity of t ↦ Ψ^{t}(x) which entails the continuity of t ↦ Ψ̂^{t}[{x \atop 1}] = d_{t}^{−1}T^{t}[{x \atop 1}] and hence the continuity of t ↦ d_{t}. By the one-parameter group property, all the relations T^{t+s} = T^{t}T^{s}, Ψ̂^{t+s} = Ψ̂^{t}Ψ̂^{s} (t, s ∈ ℝ) hold. Therefore d_{t+s} = d_{t}d_{s} (t, s ∈ ℝ) and the continuity of t ↦ d_{t} establishes the existence of a constant μ ∈ ℂ with d_{t} = e^{μt} (t ∈ ℝ).

Proof of (i)⇔(iii)⇔(iv): Consider the projective Hilbert space H_{∗}/≈ associated with H regarded as the set of all nontrivial punctured complex rays ℂ_{∗}[{x \atop ξ}] with the factor topology.^{2} By homogeneity, any injective linear operator T ∈ L(H) acts holomorphically on H_{∗}/≈ by its factorization T_{≈} : ℂ_{∗}[{x \atop ξ}] ↦ ℂ_{∗}T [{x \atop ξ}]. In particular, as admitting a continuous inverse, each map T_{≈}^{t} is a holomorphic automorphism of H/ ≈. Hence the equivalences (i)⇔(iii)⇔(iv) are straightforward consequences of the facts that, with the embedding Π : x ↦ ℂ[{x \atop 1}] and its inverse π(ℂ[{x \atop ξ}]) := x/ξ (ξ ≠ 0), we have ΠB := K, πK = B and

[T^{t}[{x \atop 1}]]_{ℂ}^{−1}[T^{t}[{x \atop 1}]]_{H} = π ∘ T_{≈}^{t} ∘ Π[{x \atop 1}] whenever [T^{t}[{x \atop 1}]]_{ℂ} ≠ 0.

Corollary 2.13. Given any Θ ∈ Aut(\overline{B}), the Θ-shifted automorphisms Φ^{t} := Θ∘Ψ^{t} ∘Θ^{−1} form strongly continuous one-parameter group with common fixed point \overline{y} := Θ(\overline{x}) when extended continuously to \overline{B} and Φ̂t ; \overline{y} = e^{μt}Θ̃^{−1}Ψ̂t ; \overline{x}Θ̃ (t∈ℝ) for some μ∈ℂ.

3. Infinitesimal generators

Throughout this section, let ([Ψ^{t}: t ∈ ℝ], \overline{x}) be an arbitrarily fixed couple of a strongly continuous one-parameter group in Aut(B) with a common fixed point for the continuous extensions in \overline{B}. Recalling the Hille–Yosida theorem [13, Kap.10], Proposition 2.11ensures that the differential

(3.1) A : 𝔥 ↦ \frac{d}{dt}Ψ̂^{t}𝔥 with D := dom(A) = \{𝔥 ∈ H : t ↦ Ψ̂^{t}𝔥 is differentable on ℝ\}

(called the infinitesimal generator of the linear model [Ψ̂^{t}: t∈ℝ] where Ψ̂^{t} ≡Ψ̂t ; \overline{x} for short) is a not necessarily bounded linear map with closed graph and [Ψ̂^{t}: t∈ℝ]-invariant domain being dense in H. Instead of the differential A = \frac{d}{dt}|_{t=0}Ψ̂^{t} of the representations we are primarily interested in the differential

Ω := \frac{d}{dt}|_{t=0}Ψ^{t} : D → H where D = dom(Ω) = \{x ∈ B : \frac{d}{dt}|_{t=0}Ψ^{t}(x) exists\}.

In order that we could regard the vector field Ω as a non-linear infinitesimal generator for [Ψ^{t} : t ∈ ℝ], we should see the density of D in \overline{B}. In order to establish a non-linear Stone-type theorem, we should determine precise links to self-adjoint linear operators.

Lemma 3.2. D is [\overline{Ψ}^{t} : t∈ℝ]-invariant. We have [{x \atop 1}]∈D ⟺ x∈D whenever x∈B.

2 As usually, H_{∗} := H\\{0\} with the equivalence relation [{x \atop ξ}] ∼ [{y \atop η} ] : ⟺ ℂ_{∗}[{y \atop η} ] = ℂ_{∗}[{x \atop ξ}] where ℂ_{∗} := ℂ\{0}. A subset of H/ ≈ is open if the union of it members (rays in H_{∗}) is open in H.

Proof. The [Ψ^{t} : t∈ℝ]-invariance of D is clear from the group property Ψ^{t+s} = Ψ^{t} ∘ Ψ^{s} (t, s ∈ℝ). Moreover even dom(\frac{d}{dt} |_{t=0}Θ∘Ψ^{t} ∘Θ^{−1}) = Θ(D) whenever Θ is any holomorphic automorphism of B. Hence, given any point x ∈ B, we have x ∈ D if and only if 0 = Θ_{−x}(x) ∈ dom(\frac{d}{dt} |_{t=0}Φ^{t}) with the one-parameter group of the maps Φ^{t} := Θ_{−x} ∘ Ψ^{t} ∘ Θ_{x} in terms of the Möbius transformations (2.3). That is, without loss of generality, it suffices only to see the equivalence 0 ∈ dom(\frac{d}{dt} |_{t=0}Φ^{t}) ⟺ [{0 \atop 1}] ∈ D. According to (2.10), by setting a_{t} := Ψ^{t}(0) and 𝔞_{t} := Ψ̂^{t}[{0 \atop 1}] we have 𝔞_{t} = (1 − ⟨a_{t}|\overline{x}⟩(1 − ∥a_{t}∥^{2})^{−1}[{a_{t} \atop 1} ] (t ∈ ℝ). Hence the curves t ↦ a_{t} resp. t ↦ 𝔞_{t} are differentiable in the same time, which completes the proof.

For later use we also introduce the notations

Z := \{z ∈ H : \left[{z \atop 0}\right] ∈ D\}; Bz:=[A[{z \atop 0}]]_{H}, Λz:=[A[{z \atop 0}]]_{ℂ} (z ∈ Z).

Lemma 3.3. Z is a dense linear submanifold in H with D = (\overline{x} + Z) ∩ B and D = [{Z \atop 0}] + ℂ[\overline{x} ; 1]. The set D is dense in B and

\frac{d}{dt}|_{t=0}\overline{Ψ}^{t}(x) = [Λ(\overline{x} − x)]x + B(x − \overline{x}) (x ∈ D).

Proof. By definition, [\overline{x} ; 1] ∈ D with A[\overline{x} ; 1] = 0 since Ψ̂^{t}[\overline{x} ; 1] = [\overline{x} ; 1] (t ∈ ℝ). Since D is closed for linear combinations, it follows that [{Z \atop 0}] + ℂ[\overline{x} ; 1] = D and that Z is the image of D by the bounded linear operator Π[{x \atop ξ}] := x − ξ\overline{x}. Since ΠH = H and since D is dense in H, Z = ΠD is also dense in H = ΠH. From Lemma 3.2 we know that D = B ∩ \{x : [{x \atop 1}] ∈ D\}. Hence the relation D = (\overline{x} + Z) ∩ B along with the density of D in B is immediate. Given any x ∈ D, the relation [{x \atop 1}] ∈ D implies A[{x \atop 1}] = \frac{d}{dt} |_{t=0}Ψ̂^{t}[{x \atop 1}].

Since \overline{Ψ}^{t}(x) = \{Ψ̂^{t}[{x \atop 1}]\}_{ℂ}^{−1}\{Ψ̂^{t}[{x \atop 1}]\}_{H} along with Ψ̂^{0} = Id and A[\overline{x} ; 1] = 0, we get

\frac{d}{dt}|_{t=0} \overline{Ψ}^{t}(x) = −[Ψ̂^{0}[{x \atop 1}]]_{ℂ}^{−2}(\frac{d}{dt}|_{t=0}\{Ψ̂^{0}[{x \atop 1}]\}_{ℂ})[Ψ̂^{0}[{x \atop 1}]]_{H}+ [Ψ̂^{0}[{x \atop 1}]]_{ℂ}^{−1} \frac{d}{dt}|_{t=0}\{Ψ̂^{0}[{x \atop 1}]\}_{H}=

= −[A[{x \atop 1}]]_{ℂ}x + [A[{x \atop 1}]]_{H} = −[A[{x − \overline{x} \atop 0}]]_{ℂ}x + [A[{x − \overline{x} \atop 0}]]_{H}.

Lemma 3.4. Suppose a Hilbert space W is the orthogonal sum of the subspaces W_{1}, W_{2} and C is the infinitesimal generator of a strongly continuous one-parameter subgroup [T^{t} : t∈ℝ] of L(W). Then, for the cone K:={w_{1}⊕w_{2} : ∥w_{1}∥>∥w_{2}∥}, we have T^{t}K=K (t∈ℝ) if and only if C is tangent to the boundary of K that is if

(3.5) Re⟨C(w_{1}⊕w_{2})|w_{1}⟩ = Re⟨C(w_{1}⊕ w_{2})|w_{2}⟩ (w_{1}⊕w_{2} ∈dom(C), ∥w_{1}∥=∥w_{2}∥).

Proof. It is immediate that T^{t}K⊂ K (t ∈ ℝ) ⇒ T^{t}∂K ⊂ ∂K (t ∈ ℝ) ⇒ \frac{d}{dt}|_{t=0}T^{t}(w_{1} ⊕ w_{2}) ∈ Tan_{w_{1}⊕w_{2}}(K) for w_{1} ⊕ w_{2} ∈ dom(C) ⇒ (3.5). Assume (3.5) and let P denote the canonical projection of W onto W_{1} and define T^{t,s} := exp(tC + sP) (s, t ∈ ℝ). By the theorem of bounded perturbations [4, p.158] the operators T^{t,s} are all well-defined. Moreover, by [4, Corollary 1.7 p. 161] (applied with B := sP and A := C there) we have lim_{s→0} T^{t,s}w = T^{t}w (w ∈ dom(C), t ∈ ℝ). Therefore, to establish that T^{t}K ⊂ K (t ∈ ℝ), it suffices to see only that T^{t,s}w ∈ K whenever w ∈ dom(C)∩K and t, s > 0. To proceed to contradiction, let s, t > 0 and w := w_{1} ⊕ w_{2} ∈ dom(C) with ∥w_{1}∥ > ∥w_{2}∥ but ∥[T^{t,s}w]_{1}∥ ≤ ∥[T^{t,s}w]_{2}∥. The function δ(τ) := ∥[T^{τ,s}w]_{1}∥^{2} − ∥[T^{τ,s}w]_{2}∥^{2} is differentiable in τ on the whole ℝ and δ(0) > 0 ≥ δ(t). Thus there exist a point t_{∗} ∈ (0, t] such that δ(τ) > 0 = δ(t_{∗}) (0 ≤ τ < t_{∗}). Since δ(t_{∗}) = 0, the vector w_{∗} := T^{t_{∗},s}w belongs to ∂K and hence Re⟨Cw_{∗}|[w_{∗}]_{1}⟩ = Re⟨Cw_{∗}|[w_{∗}]_{2}⟩. We get the contradiction 0 ≥ δ′(t_{∗}) = 2Re⟨(C + sP)w_{∗}|[w_{∗}]_{1}⟩ − 2Re⟨(C + sP)w_{∗}|[w_{∗}]_{2}⟩ = 2s∥[w_{∗}]_{1}∥^{2} > 0.

Corollary 3.6. Re( − Λv + ⟨Bv|\overline{x}⟩ + ⟨Bv|v⟩) = 0 whenever v ∈ Z with ∥\overline{x} + v∥ = 1.

Proof. By Corollary 2.12, we have Ψ̂^{t}K = K (t ∈ ℝ) where K := \{[{x \atop ξ}] : |ξ| > ∥x∥\} ⊂ H. An application of Lemma 3.4 with W_{1} := ℂ, W_{2} := H, K := K, T^{t} := Ψ̂^{t}, C := A establishes that Re [(Λx)\overline{ξ}] = Re ⟨Bx|x⟩ whenever [{x \atop ξ}] ∈ D and ∥x∥ = |ξ|. We obtain the statement with the choice x := v + \overline{x} and ξ := 1 if v ∈ Z with ∥v + \overline{x}∥ = 1 because then, by Lemma 3.3, we have [{x \atop 1}] ∈ D.

Proposition 3.7. For some symmetric linear operator A : Z → H and a suitable constant λ ∈ ℝ which is necessarily = 0 if ∥\overline{x}∥ ≠ 1, we have (1.3) as

B = iA + λI, Λz = ⟨(iA − λI)z|\overline{x}⟩ (z ∈ Z).

Proof. Consider any vector 0 ≠ z ∈ Z. Let ζ ∈ ℂ be the (unique) constant such that \overline{x} + ζz ⊥ z and define ϱ := \sqrt{1 − ∥\overline{x} + ζz∥^{2}}. Actually we have ζ = −⟨\overline{x}|z⟩/⟨z|z⟩ and 1 ≥ ∥\overline{x}∥^{2} = ∥\overline{x} + ζz∥^{2} + ∥ − ζz∥^{2} showing that both ζ and ϱ are well-defined. Consider the unit vectors

v_{φ} := \overline{x} + ζz + e^{iφ}ϱz (φ∈ℝ).

According to Corollary 3.6,

Re((ζ+e^{iφ}ϱ)[ − Λz + ⟨Bz | \overline{x}⟩] + |ζ+e^{iφ}ϱ|^{2}⟨Bz | z⟩) = 0 (φ ∈ ℝ).

Thus the identity Re(α + βe^{iφ} + γe^{−iφ}) = 0 for all φ ∈ ℝ holds with the constants α := ζ[ − Λz + ⟨Bz | \overline{x}⟩] + (|ζ|^{2} + ϱ^{2})⟨Bz | z⟩, β := ϱ [ − Λz + ⟨Bz | \overline{x}⟩ + \overline{ζ}⟨Bz | z⟩] and γ := ϱζ⟨Bz | z⟩. Since 2 Re(α + βe^{iφ} + γe^{−iφ}) = 2 Re(α) + (β + \overline{γ})e^{iφ} + (\overline{β} + γ)e^{−iφ}, we have necessarily Re(α) = β + \overline{γ} = 0. From the relation β + \overline{γ} = 0, it follows

(3.8)  Λz − ⟨Bz | \overline{x}⟩ = 2 \overline{ζ} Re ⟨Bz | z⟩ = −2 ⟨z | \overline{x}⟩ \frac{Re ⟨Bz | z⟩}{⟨z | z⟩} ,

and substituting this into the relation 0 = Re(α), we get

(3.8′)  0 = (ϱ^{2} − |ζ|^{2}) Re ⟨Bz | z⟩ = (1 − ∥\overline{x}∥^{2}) Re ⟨Bz | z⟩ .

From (3.8) we see also that

z ↦ \frac{Re⟨Bz | z⟩}{⟨z | z⟩} = −\frac{1}{2} \frac{Λz − ⟨Bz | \overline{x}⟩}{⟨z | \overline{x}⟩}

is a real valued Gâteaux holomorphic function on the algebraically open and in Z algebraically dense domain \{z ∈ Z : z ⊥̸ \overline{x}\} which is possible only if being constant on Z. By writing λ for this constant value, from (3.8) and (3.8′) we conclude that

Λz = ⟨(B−2λI)z | \overline{x}⟩, Re⟨(B−λI)z | z⟩ = 0 (z ∈ Z),

and, in particular, if ∥\overline{x}∥ < 1 then necessarily λ = 0 above. By setting A:= −i(B +λI), hence the statement including the symmetry of A is immediate.

The following geometric converse of Proposition 3.7 is elementary:

Remark 3.9. Given a dense linear submanifold Z in H, a symmetric linear operator A : Z → H, a vector \overline{x} ∈ \overline{B}, the vector field Ω_{λ}(x) := −⟨(iA−λI)(x−\overline{x}|\overline{x}⟩x+(iA+λI)(x−\overline{x}) define for \overline{x} + Z is tangent to the unit sphere ∂B at the points x ∈ \overline{x} + Z with ∥x∥ = 1 whenever either ∥\overline{x}∥ = 1 and λ ∈ ℝ or ∥\overline{x}∥ < 1 and λ = 0.

In the sequel we proceed to the problem if the operator A in Proposition 3.7 arising from the differential \frac{d}{dt} |_{t=0}Ψ^{t} of a strongly continuous one-parameter subgroup of Aut(B) is necessarily self-adjoint and conversely if every self-adjoint operator may appear there.

4. The Jordan case

We continue the previous investigations with unchanged notations but under the additional hypothesis that

(4.1)  0 ∈ D = dom(\frac{d}{dt}|_{t=0}Ψ^{t}) .

As we know, D = \{x ∈ B : [{x \atop 1}] ∈ D\} = B∩(\overline{x}+Z) where D = dom(A) with A = \frac{d}{dt} |_{t=0}Ψ̂t ; \overline{x} and Z = \{x ∈ H : [{x \atop 0}] ∈ D\} is a dense complex linear submanifold in H. Thus, as a consequence of (4.1), for the distinguished common fixed point of the extended automorphisms \overline{Ψ}^{t} we have \overline{x} ∈ Z = dom(B) = dom(Λ). Therefore also

[{0 \atop 1}]∈D =[{Z \atop ℂ}], A[{x \atop ξ}]=[{B(x − ξ\overline{x}) \atop Λ(x − ξ\overline{x})}]=[{B −B\overline{x} \atop Λ −Λ\overline{x}}][{x \atop ξ}] (x∈Z, ξ ∈ℂ).

Remark 4.2. Recall [10] that the complete holomorphic vector fields on B are the infinitesimal generators of the uniformly continuous one-parameter subgroups of Aut(B) and, with suitable c ∈ H and a bounded self-adjoint operator C ∈ L(H) they can be written in the form x ↦ a − {xc^{∗}x} + iCx by means of the Jordan triple product.

(4.3)  \{xy^{∗}z\} := \frac{1}{2}⟨x|y⟩z + \frac{1}{2}⟨z|y⟩x

In terms of the factorization Ψ^{t} = Θ_{a_{t}} ∘ U_{t}|B we introduce the following vector resp. not necessarily bounded symmetric linear operator:

b := \frac{d}{dt}|_{t=0}Ψ^{t}(0) = lim_{t→0} \frac{1}{t}a_{t}, R := −i\frac{d}{dt}|_{t=0}U_{t} : x ↦ lim_{t→0} \frac{1}{it}(U_{t} − I)x.

Proposition 4.4. Under hypothesis (4.1), we have D = B ∩ Z along with dom(R) ⊃ Z and the vector field Ω := \frac{d}{dt} |_{t=0}Ψ^{t} admits the Jordan form

(4.5)  Ω(x) = b − {xb^{∗}x} + iRx  (x ∈ D).

Proof. The relation D = B ∩ Z is clear since \overline{x} ∈ Z. By the definition of the generator A,

A[{0 \atop 1}] = lim_{t→0} \frac{1}{t}(Ψ̂^{t} − Ψ̂^{0})[{0 \atop 1}] = lim_{t→0} \frac{1}{t}\{\frac{1}{1 + ⟨U_{t}\overline{x}|a_{t}⟩}[{a_{t} \atop 1} ] − [{0 \atop 1}]\}.

Since lim_{t→0} a_{t} = 0 and ∥U_{t}\overline{x}∥ ≤ 1 (t ∈ ℝ), taking (2.10) into account, we see that the limit

b := lim_{t→0} \frac{1}{t}a_{t} = \frac{d}{dt}|_{t=0}Ψ^{t}(0)

is well-defined and

B\overline{x} = −[A[{0 \atop 1}]]_{H} = −b, Λ\overline{x} = −[A[{0 \atop 1}]]_{ℂ} = ⟨\overline{x} | b⟩.

As a consequence we also have

β_{t} = \sqrt{1 − ∥a_{t}∥^{2}} = \sqrt{1 − ∥tb + o(t)∥^{2}} = 1 − \frac{1}{2}∥b∥^{2}t^{2} + o(t^{2}),
Q_{t} = P_{t} + β_{t}(I − P_{t}) = I + (1 − β_{t})P_{t} = I + o(t) (in operator norm).

Since also U_{t}z = z + o(1) in norm, hence we deduce that for any vector z ∈ Z, and ζ ∈ ℂ,

(4.6)  Ψ̂^{t}[{z \atop ζ} ] = \frac{1 − ⟨\overline{x}|a_{t}⟩}{1 − ∥a_{t}∥^{2}} \left[{Q_{t}U_{t} a_{t} \atop a_{t}^{∗}U_{t}z 1}\right] [{z \atop ζ} ] = \left[{U_{t}z − t⟨\overline{x}|b⟩z \atop t⟨U_{t}z|b⟩ + ζ}\right] + o(t) in norm.

By definition, [{Bz \atop Λz} ] = lim_{t→0} \frac{1}{t}(Ψ̂^{t} − I)[{z \atop 0}]. Hence with well-defined limits we conclude that

(4.7)  Bz = lim_{t→0} \frac{1}{it}(U_{t}−I)z − ⟨\overline{x} |b⟩z  (z ∈ Z).

The strong limit of t^{−1}(U_{t} − I)|Z is necessarily symmetric due to the fact that each U_{t} is unitary. Thus comparing (4.7) with Proposition 3.7 stating that the operator B has the form iA + λI with some λ ∈ ℝ and a symmetric operator A with dom(A) = Z, we get

(4.8)  λ = −Re⟨\overline{x} | b⟩ , A = lim_{t→0}^{strong} (U^{t} − I) − Im⟨\overline{x} | b⟩I|Z in 3.7.

We calculate Ω by substituting (4.7-8) into its form Ω(x) = [Λ(\overline{x}−x)]x+B(x−\overline{x}) applying also the relations B\overline{x} = −b, Λ\overline{x} = ⟨x|b⟩, B = iA+λI, Λx = ⟨(iA−λI)x|\overline{x}⟩ = ⟨(B−2λI)x|\overline{x}⟩. Namely, given any vector x ∈ D, taking into account the antisymmetry of the operator iA = B−λI, we can write

Ω(x) = [Λ\overline{x}]x − [Λx]x + Bx − B\overline{x} =
= ⟨\overline{x}|b⟩x − ⟨(B − 2λI)x|\overline{x}⟩x + Bx + b =
= b + [⟨\overline{x}|b⟩I + B]x − ⟨(B − λI)x|\overline{x}⟩x + λ⟨x|\overline{x}⟩x =
= b + iRx + ⟨x|(B − λI)\overline{x}⟩x + λ⟨x|\overline{x}⟩x =
= b + iRx + ⟨x|B\overline{x}⟩x = b + iRx − ⟨x|b⟩x.

Corollary 4.9. Z = dom(R) that is x ∈ Z if and only if the limit lim_{t→0} t^{−1}(U^{t}x − x) exists.

Proof. Recall that Z = \{x ∈ H : \frac{d}{dt} |_{t=0}Ψ̂^{t}[{x \atop 0}] is well-defined\}. From Proposition 4.4 we know that iRx = lim_{t→0} t^{−1}(U_{t} − I)x is well-defined for every vector x ∈ Z. Conversely,

suppose u = lim_{t→0} t^{−1}(U_{t}x − x) is well-defined. Then U_{t}x = x + tu + o^{norm}(t) and (4.6)

establishes that Ψ̂^{t}[{x \atop 0}] = [{x \atop 0}] + t[{u−⟨\overline{x}|a_{t}⟩x \atop ⟨x|b⟩}] + o^{norm}(t).

Lemma 4.10. We have U_{−t} = U_{t}^{−1} = U_{t}^{∗}, a_{−t} = −U_{t}^{∗}a_{t} (t ∈ ℝ).

Proof. Given any t ∈ ℝ, we have Ψ^{−t} = Ψ_{t}^{−1} that is Θ_{a_{−t}}U_{−t} = [Θ_{a_{t}}U_{t}]^{−1} = U_{t}^{−1}Θ_{a_{t}}^{−1} = U_{t}^{−1}Θ_{−a_{t}} = [U_{t}^{−1}Θ_{−a_{t}}U_{t}]U_{t}^{−1} = Θ_{U_{t}^{−1}(−a_{t})}U_{t}^{−1}. By the unambiguous decomposability of holomorphic automorphisms of circular domains into Möbius and unitary parts [2], hence we deduce that Θ_{a_{−t}} = Θ_{−U_{t}^{−1}a_{t}} and U_{−t} = U_{t}^{−1}.

Lemma 4.11. The operator R is self-adjoint with dom(R) = Z.

Proof. In view of (4.6), and since U_{t}x = x + itRx + o^{norm}(t) for any x ∈ Z = dom(R) = dom(\frac{d}{dt} |_{t=0}U_{t}), we conclude that

(4.12)  \frac{d}{dt}|_{t=0}Ψ̂^{t}[{x \atop ξ}] = [ {iR−⟨\overline{x}|b⟩I b \atop b^{∗} −⟨\overline{x}|b⟩}][{x \atop ξ}] for any [{x \atop ξ}] ∈ D = [{Z \atop ℂ}].

The linear operator in L(H) with matrix [ {−⟨\overline{x}|b⟩I b \atop b^{∗} −⟨\overline{x}|b⟩}] is bounded. Since A = \frac{d}{dt} |_{t=0}Ψ̂^{t} with domain D is the generator of a strongly continuous semigroup in L(H), by the theorem of bounded perturbations [4], also the operator with matrix [{iR 0 \atop 0 0}] with domain D is the generator of a strongly continuous one-parameter subgroup of L(H) entailing that iR is the generator of a strongly continuous group [V_{t} : t ∈ ℝ] in L(H). Since U_{−t} = U_{t}^{−1} = U_{t}^{∗}, the arguments on sun adjoint semigroups in [4, p. 69] show that lim_{t→0} t^{−1}(U_{t}^{∗} − I) = −iR is the

generator of the sun adjoint group [V_{t}^{∗} : t ∈ ℝ] = [V_{−t} : t ∈ ℝ] and we have −iR = (iR)^{∗} which completes the proof.

Theorem 4.13. Any vector field of the form (4.5) where R is a not necessarily bounded self-adjoint operator with dense domain Z ⊂ H, is the infinitesimal generator defined on D := Z ∩ B of a pointwise continuous one-parameter group [Φ^{t} : t ∈ ℝ] of holomorphic automorphisms of B.^{3}

Proof. It suffices to see that there is a strongly (i.e. pointwise) continuous one-parameter group [V^{t} : t ∈ ℝ] of bounded linear operators of the space H such that

\frac{d}{dt}|_{t=0}V^{t}[{x \atop ξ}] = [{iR b \atop b^{∗} 0}][{x \atop ξ}] (x ∈ Z, ξ ∈ ℂ), V^{t}K ⊂ K := \{[{x \atop ξ}] : ∥x∥^{2} ≥ |ξ|^{2}\}.

Namely, in this case the maps

Φ^{t}(x) := [V^{t}[{x \atop 1}]]_{ℂ}^{−1}[V^{t}[{x \atop 1}]]_{H} (t ∈ ℝ, x ∈ D)

suit the requirements of the theorem since x ∈ D ⇒ [{x \atop 1}] ∈ K ⇒ V^{t}[{x \atop 1}] ⇒ Φ^{t}(x) ∈ D and x ∈ D ⇒ \frac{d}{dt} |_{t=0}Φ^{t}(x) = −[V^{0}[{x \atop 1}]]_{ℂ}^{−2} \frac{d}{dt} |_{t=0}[V^{t}[{x \atop 1}]]_{ℂ}[V^{0}[{x \atop 1}]]_{H} + \{V^{0}[{x \atop 1}]\}_{ℂ}^{−1} \frac{d}{dt} |_{t=0}[V^{t}[{x \atop 1}]]_{H} = −[[{iR b \atop b^{∗} 0}][{x \atop 1}]]_{ℂ}x+[[{iR b \atop b^{∗} 0}][{x \atop 1}]]_{H} = −⟨x|b⟩x + iRx + b = Ω(x). Notice that, by Corollary 2.12, a strongly continuous one parameter group of linear operator leaves the cone K invariant if all its members map the boundary ∂K=\{[{x \atop ξ}]: ∥x∥=|ξ|\}=\{[{x \atop e^{iτ} ∥x∥}]: x∈H, τ∈ℝ\} into itself. Therefore it suffices to check that there is a (necessarily unique) strongly continuous one-parameter group in L(H) with domain Z ⊕ ℂ = [{Z \atop ℂ}] such that

\frac{d}{dt}V^{t}[{x \atop ξ}] = [{iR b \atop b^{∗} 0}]V^{t}[{x \atop ξ}], ∥[V^{t}[{x \atop ∥x∥}]]_{H}∥ = |[V^{t}[{x \atop ∥x∥}]]_{ℂ}| (x ∈ Z, t ∈ ℝ).

By Stone’s theorem, the H-unitary operators W^{t}[{x \atop ξ}] := [{exp(itR)x \atop ξ}] form a strongly continuous one-parameter group whose infinitesimal generator is defined on dom(R)⊕ℂ = Z⊕ℂ with the diagonal matrix [{iR 0 \atop 0 0}]. Since the matrix [{0 b \atop b^{∗} 0}] represents a bounded linear operator in H, by the theorem of bounded perturbations [4], there is a strongly continuous one-parameter group [V^{t} : t ∈ ℝ] whose generator is defined on Z ⊕ ℂ with the matrix [{iR b \atop b^{∗} 0}]. In particular \frac{d}{dt}V^{t}[{x \atop ξ}] = [{iR b \atop b^{∗} 0}]V^{t}[{x \atop ξ}] (t ∈ ℝ, x ∈ Z). To complete the proof, we show that necessarily

\frac{d}{dt} [∥[V^{t}[{x \atop ∥x∥}]]_{H}∥^{2} − |[V^{t}[{x \atop ∥x∥}]]_{ℂ}|^{2}] = 0 (t ∈ ℝ, x ∈ Z).

3 That is, for all x ∈ D := Z ∩ B, the functions t ↦ Φ^{t}(x) range in Z, they are differentiable and satisfy the identity \frac{d}{dt}Φ^{t}(x) = V(Φ^{t}(x)) (t ∈ ℝ).

Consider any vector x ∈ Z and write [{x_{t} \atop ξ_{t}} ] := V^{t}[{x \atop ∥x∥}] for all parameters t ∈ ℝ. Then

\frac{d}{dt}[∥x_{t}∥^{2} − |ξ_{t}|^{2}] = 2 Re[⟨dx_{t}/dt|x_{t}⟩ − (dξ_{t}/dt)\overline{ξ_{t}}] =

= 2 Re \{⟨[[{iR b \atop b^{∗} 0}][{x_{t} \atop ξ_{t}} ]]_{H}|x_{t}⟩ − [[{iR b \atop b^{∗} 0}][{x_{t} \atop ξ_{t}} ]]_{ℂ}(dξ_{t}/dt)\overline{ξ_{t}}\} =

= 2 Re [⟨iRx_{t} + ξ_{t}b|x_{t}⟩ − ⟨x_{t}|b⟩\overline{ξ_{t}}] =

= −2 Im⟨Rx_{t}|x_{t}⟩ + 2 Im(⟨ξb|x_{t}⟩ − ⟨x_{t}|ξb⟩) = 0.

5. Proof of the main results

Proof of Theorem 1.1. Consider any pointwise continuous one-parameter subgroup [Ψ^{t} : t ∈ ℝ] of Aut(B) and let us fix any common fixed point \overline{x} ∈ \overline{B} of the continuous extensions of the maps Ψ^{t} (t ∈ ℝ) (guaranteed by Lemma 2.4). From Proposition 3.7, we know already that (1.2-3) hold for some dense linear complex submanifold Z of the underlying Hilbert space with a symmetric linear operator A with dom(A) = Z. We have to see that A is even self-adjoint in any case and, conversely every self-adjoint operator with domain Z may appear in (1.3) with any constant λ ∈ ℝ if \overline{x} ∈ ∂B or λ = 0 if \overline{x} ∈ B. In order to establish a link to the Jordan case, fix any point c∈D=dom(\frac{dΨ^{t}}{dt} |_{t=0}) and let

Φ^{t} := Θ_{−c}Ψ^{t}Θ_{c} (t ∈ ℝ), \overline{y} := Θ_{c}(\overline{x})

by means of the Möbius transformations (2.3). Clearly [\overline{Φ}^{t} : t ∈ ℝ] is a strongly continuous one-parameter subgroup of Aut(B) such that

(5.1)  0 = Θ_{−c}(c) ∈ Θ_{−c} (dom(\frac{∂Ψ^{t}}{dt} |_{t=0})) = dom(\frac{∂Φ^{t}}{dt} |_{t=0}), \overline{y} ∈ ⋂_{t∈ℝ} Fix(\overline{Φ}^{t}).

Thus we can apply the results of Section 4 in particular Lemma 4.11 to [Φ^{t} : t ∈ ℝ] to conclude that there is a dense complex linear submanifold Y ⊂ H along with a self-adjoint operator R with dom(R) = Y and a vector b ∈ H such that

Φ̂t ; \overline{y} = e^{−⟨\overline{x}|b⟩t} exp (t[{iR b \atop b^{∗} 0}]) (t ∈ ℝ).

Hence Corollary 2.11 establishes the existence of a constant ν ∈ ℂ with

Ψ̂t ; \overline{x} = e^{νt}β^{−2}Θ̃_{c}Φ̂t ; \overline{y}Θ̃_{−c} (t ∈ ℝ)

due to the identity Θ̃_{c}Θ̃_{−c} = β^{2}I = (1 − ∥c∥^{2})[{I 0 \atop 0 1}] for the canonical representations

Θ̃_{±c} = [{Q ± c \atop ±c^{∗} 1}] where Q := βI + (1 − β)P, β := \sqrt{1 − ∥c∥^{2}}, P := P_{c}.

By passing to infinitesimal generators, with μ := ν − ⟨\overline{x}|b⟩, we get

(5.2)  \frac{dΨ̂t ; \overline{x}}{dt} |_{t=0}[{x \atop ξ}] = μ[{x \atop ξ}] + β^{−2}Θ̃_{c}[{iR b \atop b^{∗} 0}]Θ̃_{−c}[{x \atop ξ}] for [{x \atop ξ}] ∈ dom(\frac{dΨ̂t ; \overline{x}}{dt} |_{t=0}).

From Lemma 3.3 and (5.1) we see that

[{Z \atop 0}] + ℂ[\overline{x} ; ξ] = dom(\frac{dΨ̂t ; \overline{x}}{dt} |_{t=0}) = Θ̃_{c}dom(\frac{dΦ̂t ; \overline{y}}{dt} |_{t=0}) = Θ̃_{c}[{Y \atop ℂ} ].

Thus, given any vector z ∈ H, we have z ∈ Z if and only if [{z \atop 0}] = Θ̃_{c}[{y \atop η} ] for some y ∈ Y and η ∈ ℂ that is if z = Qy − ⟨y|c⟩c = [Q − ∥c∥^{2}P]y for some y ∈ Y. It follows

(5.3)  Z = [Q − ∥c∥^{2}P ]Y = Q^{−1}Y

because the operators P, Q commute, we have β^{2} = 1 − ∥c∥^{2} > 0 and Q[Q − ∥c∥^{2}P ] = [βI + (1 − β)P][βI + (β^{2} − β)P ] = β^{2}I. We are now ready to establish the self-adjointness of the operator A in (1.3). By (5.2) we have

(iA + λI)z = [\frac{dΨ̂t ; \overline{x}}{dt} |_{t=0}[{z \atop 0}]]_{H} = μz + β[iQRQz − Qbc^{∗}z + cb^{∗}Qz] (z ∈ Z).

That is, with the bounded self-adjoint operator S := iβ[Qbc^{∗} −cb^{∗}Qz] = [iQbc^{∗}]+[iQbc^{∗}]^{∗} we have A = βQRQ + S + i(λ − μ)I. We know the symmetry of A already entailing the relations μ = λ with A = βQRQ + S. Here the operator QRQ self-adjoint with dom(QRQ) = Q^{−1}dom(R) = Q^{−1}Y = Z = dom(A) since R is a self-adjoint operator with dom(R) = Y while Q is an invertible bounded self-adjoint operator, Therefore, as being the bounded self-adjoint perturbation, the operator A is necessarily self-adjoint. To see the converse, we need only to check the reversibility of some of our previous arguments. Assuming A to be self-adjoint in (1.2-3), it is the theorem of bounded perturbations [4] ensures that the operator A[{z+ξ\overline{x} \atop ξ}] := [{(iA+λI)z \atop ⟨(iA−λI)z|\overline{x}⟩(z+ξ\overline{x})}] (z ∈ Z, ξ ∈ ℂ) is the infinitesimal generator of a strongly continuous one-parameter subgroup [U^{t} : t ∈ ℝ] of L(H) with graph being tangent to the boundary of cone K in Corollary 2.12(iii) and we have K = U^{t}K (t ∈ ℝ). Hence the holomorphic maps Ψ^{t}(x) := [U^{t}[{x \atop 1}]]_{ℂ}^{−1}[U^{t}[{x \atop 1}]]_{H} are well-defined on the unit ball B leaving it invariant, and form a strongly continuous one-parameter subgroup in Aut(B).

Proof of Corollary 1.4. By assumption A\overline{x} = ρ\overline{x} in particular \overline{x} ∈ dom(A). Thus we are in the Jordan case 0 ∈ dom(Ω) and hence

\begin{aligned}Ω(x) & = ⟨x − \overline{x}|(iA + λI)\overline{x}⟩x + (iA + λI)(x − \overline{x} = \\ & = b − ⟨x|b⟩x + iR with b = Ω(0) = (−λ − iρ)\overline{x}, iR = Ω′(0) = iA + iρI . \end{aligned}

Recall that we have Ψ^{t}(x) = [Ψ̂^{t}({x \atop 1})]_{ℂ}^{−1}[Ψ̂^{t}({x \atop 1})]_{H} x ∈ H) for the strongly continuous oneparameter group [Ψ̂^{t} : t ∈ ℝ] which, according to (4.12), has the infinitesimal generator

\frac{d}{dt}|_{t=0}Ψ̂^{t} = [ \begin{matrix}iR − ⟨\overline{x}|b⟩ & b \\ b^{∗} & (−⟨\overline{x}|b⟩ \end{matrix} ] = [ \begin{matrix}iA + λI & (−λ − iρ)\overline{x} \\ (−λ + iρ)\overline{x}^{∗} & λ − iρ \end{matrix} ] .

Since the operator A is self-adjoint with eigenvector \overline{x}, it is reduced by the subspaces ℂ\overline{x} and H_{0} := \overline{x}^{⊥}(= \{x ∈ H : x ⊥ \overline{x}\}). Thus in terms of H_{0} ⊕ ℂ\overline{x} ⊕ ℂ-matrices,

\frac{d}{dt}|_{t=0}Ψ̂^{t} = [ \begin{matrix}(iA + λI)|H_{0} & 0 & 0 \\ 0 & λ + iρ & −λ − iρ \\ 0 & −λ + iρ & λ − iρ \end{matrix} ]

entailing that (a) in case 0 ≠ λ ∈ ℝ we have

Ψ̂^{t} = [ \begin{matrix}e^{iλt} exp(itA)|H_{0} & 0 \\ 0 & exp t \left[{λ+iρ −λ−iρ \atop −λ+iρ λ−iρ}\right] \end{matrix} ] =

= [ \begin{matrix}e^{iλt} exp(itA)|H_{0} & 0 & 0 \\ 0 & (λ + iρ)e^{2λt} + (λ + iρ) & (−λ − iρ)e^{2λt} + (λ − iρ) \\ 0 & (−λ + iρ)e^{2λt} + (λ + iρ) & (λ − iρ)e^{2λt} + (λ − iρ) \end{matrix} ] ,

(b) in case of λ = 0 we have

Ψ̂^{t} = [ \begin{matrix}e^{iλt} exp(itA)|H_{0} & 0 \\ 0 & exp itρ \left[{1 −1 \atop 1 −1}\right] \end{matrix} ] = [ \begin{matrix}e^{iλt} exp(itA)|H_{0} & 0 & 0 \\ 0 & 1 + itρ & −itρ \\ 0 & itρ & 1 − itρ \end{matrix} ] .

Hence the statement is straightforward from the relation

Ψ^{t}(x_{0} + ξ\overline{x}) = [Ψ̂^{t}[x_{0}, ξ, 1]^{T}]_{3}^{−1}([Ψ̂^{t}[x_{0}, ξ, 1]^{T}]_{2}\overline{x} + [Ψ̂^{t}(x_{0}, ξ, 1]^{T}]_{1}).

Lemma 5.4. The group Aut(\overline{B}) is 2-transitive on ∂B i.e. given any tuple (e_{1}, f_{1}, e_{2}, f_{2}) of unit vectors with e_{1} ≠ e_{2}, f_{1} ≠ f_{2}, there is a transformation Θ ∈ Aut(B)such that \overline{Θ}(e_{k}) = f_{k} (k = 1, 2).

Proof. Let e_{1}, f_{1}, e_{2}, f_{2} ∈ ∂B with e_{1} ≠ e_{2}, f_{1} ≠ f_{2}. Let c be the center of the circlar arc C in the 1-dimensional complex disc D := B ∩ {(1 − ζ)e_{1} + ζe_{2} : ζ ∈ ℂ} connecting the points e_{1}, e_{2} and being orthogonal in real sense to the boundary circle of D. Since holomorphic fractional linear transformations preserve affine lines, the Möbius shift Θ_{c}^{−1} = Θ_{−c}(x) = (1−⟨x|c⟩)^{−1}(Q_{c}x−c) maps D onto an affine disc passing through 0 = Θ_{−c}(c) and containg the points g_{k} := \overline{Θ_{−c}}(e_{k}) (k = 1, 2) as boundary points. Since the continuous extension \overline{Θ_{−c}} preserves the unit sphere ∂B, The image Θ_{−c}(D) is a disc of the form Θ_{−c}(D) = {ζg_{k} : |ζ| < 1} (k = 1, 2). Since holomorphic maps are conformal, the image Θ^{−1}(C) is an arc passing through the origin and being orhogonal to the boundary circle which is possible only if it is a diameter of the form {τg : −1 < τ < 1} with ∥g∥ = 1 that is we have \overline{Θ_{−c}}(e_{k}) = g_{k} = (−1)^{k}g (k = 1, 2) for some unit vector g. Similarly, \overline{Θ_{−d}}(f_{k}) = (−1)^{k}h (k = 1, 2) for some unit vector h and a Möbius shift Θ_{−d}. It is a well-known elementary fact that any the unitary group is transitive on ∂B, actually e.g. we have Ug = h with a twisted reflection U := κ[I − 2P_{h−κg}] where the constant κ ∈ ℂ is of modulus 1 such that ζg + h ⊥ ζg − h i.e. κ⟨g|h⟩ ∈ ℝ. Therefore the transformation Θ := Θ_{d} ∘ U ∘ Θ_{−c} suits the requirements of the lemma.

Proof of Corollary 1.5. Due to the 2-transitivity on the boundary, there exists Θ ∈ Aut(B) such that Θ(\overline{x}) = \overline{x}, Θ(\overline{x}′) = −\overline{x}. Thus, by passing to the group [Θ ∘ Ψ^{t} ∘ Θ^{−1} : t ∈ ℝ] instead of [Ψ^{t} : t ∈ ℝ], we may assume without loss of generality that \{±\overline{x}\} = Fix(\overline{Φ}^{t_{0}}) with the effect

−⟨(iA − λI)(x − \overline{x})|\overline{x}⟩x + (iA + λI)(x − \overline{x}) = 0 for x = ±\overline{x}.

Regarding the case x = −\overline{x}, since ⟨\overline{x}|\overline{x}⟩ = 1, it follows ⟨A\overline{x}|\overline{x}⟩\overline{x} + A\overline{x} = 0. However, since the operator A is self-adjoint, necessarily also A\overline{x} = ⟨A\overline{x}|\overline{x}⟩\overline{x} implying A\overline{x} = 0. Hence the statement is immediate from Corollary 1.4.

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L.L. STACHÓ Bolyai Institute University of Szeged Aradi Vértanúk tere 1 H-6720 SZEGED, HUNGARY

E-mail: stacho@math.u-szged.hu