Applications

Corollary 2.4.1. A complex submanifoldM of a bounded symmetric Banach space domain D is a symmetric metric Banach manifold with the Carat´eodory pseudodistance if it is the range of some holomorphic projection of D.

Proof. Assume P is a holomorphic projection of D onto a complex submanifold M ⊂ D.

Consider any point o ∈ M and let v∂/∂z be a tangent vector of M at o. By [102]

Corollary 17.13, to establish the symmetry of M, it suffices to see that there is a complete holomorphic vector field Y ∈ aut(M) with Y(o) = v∂/∂z. By assumption, the domain D is symmetric. Hence (also by [102] Corollary 17.13) for every w ∈ E there is a complete

holomorphic vector field X_w ∈aut(D) withX_w(o) = w∂/∂z. On the other hand, as being a bounded domain, its Carath´eodory pseudodistanced_D is a complete metric. Furthermore the derivative P^′(o) is a linear projection of T_o(D) = {

w∂/∂z : w ∈ E}

onto T_o(M) = {w∂/∂z : ∃ x:R →M w =x^′(0)}. Therefore. by Corollary 2.3.2, the choice Y :=P^′X suits our requirements because Y(o) = P^′X_v(o) =P^′(o)X_v(o) =P^′(o)v∂/∂z=v∂/∂z.

In view of the partial J*-triples associated with the unit balls of Banach spaces (see Section 1.10), with the notation

Sym(F) :=EB(F) ={

a∈F : ∃X ∈aut(B(F)) X(0) =a ∂/∂z} we get the following.

Corollary 2.4.2. If P : E →E is a contractive linear projection then PSym(E) ⊂ Sym(P E). In particular, if the unit ball B(E) is a symmetric manifold then so isB(P E), too.

Remark 2.4.3. As mentioned in Section 1.5, the above corollary of the Projection Princi-ple gave rise to Dineen’s bidualization of JB*-triPrinci-ples which enabled the a natural extension of the theory of von Neumann algebras to the category of all Banach spaces with symmetric unit ball and admitting a predual.

On the other extreme, by means of the Projection Principle one can conclude easily that the unit ball of most Banach spaces does not admit non-linear automorphisms at all.

Corollary 2.4.4. If there is a familyP of contractive linear projections E →E such that for everyP∈P, Aut(

B(P E))

consists only of linear transformations and ∩

P∈P

kerP ={0} then all the elements of Aut(

B(E))

are also linear.

Proof. If X ∈aut( B(E))

then P X(0) = 0 for all P ∈ P whence X(0) = 0 i.e. the vector fieldXis linear. It is well-known [11] since 1978 that Aut(

B(E))

=U(E)·exp aut(

B(E)) { =

Uexp(X) : U ∈ U(E), X ∈aut(

B(E))}

whereU(E) :={E−unitarities}.

Example 2.4.5. Let (Ω, µ) denote a measure space. In [11],[77] it was proved: The unit ball of E := L^p(Ω, µ) admits only linear automorphisms unless dimE = 1 or p = 1,∞. Corollary 2.4.4 reduces this result to Thullen’s classical 2-dimensional theorem [99].

Proof. Suppose p ∈ [1,∞]\{2} and dimE > 1. If g₁, g₂ are functions in E with norm 1 having disjoint supports then it is easily seen that the mapping

P_g1,g2 :E ∋f 7→

∑2 j=1

[ ∫

f g_j|g_j|^p−2dµ ]

g_j

(with the convention 0·0^p−2 := 0) is a contractive linear projection of E onto the sub-space E_g1,g2 :=

∑2 j=1

Cg_j. Notice that B(E_g1,g2) = {ζ₁g₁ +ζ₂g₂ : |ζ₁|^p +|ζ₂|^p < 1} is a

Reinhardt domain whose biholomorphic automorphisms are all linear by Thullen’s 2.4.4 establishes the linearity of Aut(

B(E)) .

Remark 2.4.6. The important special case Corollary 2.4.2 with contractive linear pro-jections can be derived immediately from a basic property of convex bodies ([77] Lemma) which, for later use, enables precise calculations concerning low-dimensional pieces of the triple product {..^∗.}_B(E). Given a holomorphic mapping f : E → E, the vector field f(z)∂/∂z is complete in B(E) if and only if it is tangent to the boundary ∂B(E) in the following sense: for any point x with ∥x∥ = 1, the line x+Rf(x) is contained in every supporting hyperplane of B(E) at the point x.

Since the supporting hyperplanes at x∈∂B(E) are exactly those in the form{

z ∈E : Re⟨ϕ, z−x⟩= 0}

with some continuous linear functionalϕ∈E^∗such that∥ϕ∥= 1 =⟨ϕ, x⟩, the above geometric statement can be formulated analytically as

f(z)∂/∂x∈aut(

Example 2.4.8. We outline how 2.4.7 was used in [79] to achieve precise structural infor-mation on{..^∗.}_B(E) in case of minimal atomic Banach lattices (min. B-latticesfor short).

In Chapter 7 we shall see technical details in the setting of continuous Reinhardt domains.

Throughout this subsection let E denote a min. B-lattice. According to a well-known representation lemma [76] p.143 Ex.7(b), we may assume that for a fixed set Ω, E is a sublattice of {Ω→Cfunctions} such that

Span{1_ω :ω ∈Ω}=E where 1_ω ∈E and ∥1_ω∥= 1 (ω ∈Ω) (2.4.9) holds with the indictor functions 1_ω := [

Ω ∋ y 7→ 1 if y = ω and 0 elsewhere]

For the sake of simplicity we shall write increasing positive-homogeneous convex function and C is a cone of Lebesgue measure 0.

Let us fix arbitrary vectors ϱ∈Rⁿ+\C, ϑ ∈Rⁿ and set

Since B is a bounded circular domain and a(z)∂/∂z ∈ aut(B), in view of Section 1.10 we have a(z) = a − {za^∗z}_B +ℓz with suitable a ∈ E and ℓ ∈ L(E). Since |a| = e^iϑ1ωa+ 1Ω\{ω}a for any ϑ ∈R and ω ∈Ω it follows that a∈ Sym(E)⇒ 1ωa ∈Sym(E) for any base space point ω ∈Ω. Hence, for some subset Ω₀ ⊂Ω, we have Sym(E) ={f ∈ E : support(f) ⊂ Ω₀}. Thus, without loss of generality, we need only to investigate the cases a(z) =ℓz and a(z) = {z[1y1]^∗z}_B, respectively. By setting

α_jk :=⟨ℓ(1_yj),1_yk⟩, β_jk^ℓ :=⟨1^∗_y

ℓ,{

[1_yj][1_y1]^∗[1_yk]}

⟩,

from 2.4.12 we obtain the equations

Theorem 2.4.12. Assume E is a Banach lattice of functions Ω → C with the min. B-lattice property 2.4.9. Then, the relation ω₁ ∼ ω₂ :⇐⇒ ⟨ℓ(1_ω1),1_ω2⟩ ̸= 0 for some linear vector field ℓ(z)∂/∂z ∈ aut(

B(E))

is an equivalence on Ω. By writing Ω = ∪

i∈IΩ_i for the partition by its equivalence classes, the bands H_i := Span

ω∈Ωi along with a subfamily I0 ⊂ I of indices such that

(i) Sym(E)( (iii) a mapping g : B(E) → E belongs to the identity component of Aut(

B(E))

if and only if, by denoting the band projection onto H_i by P_i, we have

P_ig(f) =U_i function [I0 ∋j 7→ϱj] assuming values in R+ and vanishing at infinity, respectively, with the M¨obius- and co-M¨obius transformations

M_τ(ζ) := ζ+ tanh(τ)

1 +ζtanh(τ), M_τ^⊥(ζ) := {1−(tanh(τ))²}^1/2

1 +ζtanh(τ) (τ∈R, |ζ|<1).

Example 2.4.13. Recently considerable interest is payed [8] by researchers of Banach space geometry to the structure of strongly continuous one-parameter groups of isometries of spaces of multilinear functionals E₁ × · · · ×E_n → C. In particular it is a question of fundamental importance in which case are they always of the form[

U₁^t⊗ · · · ⊗U_n^t : t∈R] with suitable strongly continuous one parameter groups [

U_k^t : t ∈ R]

on the respective components. The perhaps first complete result of this kind was achieved in my work [89] concerning the classical case of Hilbert spaces. As an application of the Projection

Principle, here we establish the linear tensor product form of the automorphisms of the unit ball for more than two components with dimension> 1 and then devote the next chapter to the details of [89].

Henceforth let H,H⁽¹⁾, . . . ,H^(N) with N > 2 denote arbitrarily fixed complex Hilbert spaces of dimensions≥ 2 and consider the holomorphic automorphisms of the unit ball B :=B(E) of the space customary tensor product notations ([74] Sec.1.3).²

Proposition 2.4.14. For N ≥3, all the elements of Aut(B) are linear.

Proof. Observe that the familyP :={

P₁⊗. . .⊗P_N: allP_j-s are orthogonalH^(j)-projections 2.4.4 it suffices to see only that the elements of the group Aut(

B(B(C²,C²,C²))

are linear which we establish below by Lemma 2.4.15.

Lemma 2.4.15. Span{

For each index j, let P_j denote the orthogonal projection ofH^(j) ontoCe_j and and let V_j be an H^(j)-unitary operator with Vjej =fj (notice: the unitary group is transitive on the unit sphere of a Hilbert space). Define U_j^(ϑ) := exp(iϑP_j)V_j^∗ and let

Φ^ϑ1,...,ϑN := [U₁^(ϑ1)⊗. . .⊗U_N^(ϑN)]Φ (ϑ_j ∈R; j = 1, . . . , N).

Since the operators U_j^ϑj are H^(j)-unitary, they can be written in the form exp(iA_j) with suitableH^(j)-self-adjoint operators, we haveU₁^ϑ1⊗· · ·⊗U_N^ϑN = exp(ℓ^ϑ1,...,ϑN) withℓ^ϑ1,...,ϑN := the inner product and the norm, respectively. The products⟨.|.⟩are supposed to be linear in their first and conjugate-linear in their second variables. With this convention,h^∗will denote the linear functionalx7→

⟨x|h⟩. Given a familyh1∈H⁽¹⁾, . . . ,hN ∈H^(N)of vectors, we shall writeh^∗₁⊗· · ·⊗h^∗_N for theelementary

Returning to the proof of 2.4.14, we may assume N = 3 and H^(j) = C² (j = 1,2,3).

Thus let E := B(C²,C²,C²). Proceeding by contradiction, assume Sym(E) ̸= 0. Now Lemma 2.4.15 establishes Sym(E) =E, that is E is an 8-dimensional JB*-triple with the triple product{..^∗.}_B. It is well-known that finite-dimensional JB*-triples are isometrically isomorphic toℓ^∞-direct sums of Cartan factors where the factors are preserved by the maps in exp aut of the unit ball [59]. Thus, by Lemma 2.4.15, the spaceE =B(C²,C²,C²) would be isometrically isomorphic to an 8-dimensional Cartan factor. The list of Cartan factors is also well-known and hence we would have either E ≃ C⁸ (with is Hilbert norm) or E ≃ B(C²,C⁴) or E ≃Spin(C⁸). The mentioned factor are of rank≤2 that is they do not admit three elements Ψ₁,Ψ₂,Ψ₃ with

∥ζΨ_j +ηΨ_k∥= max{|ζ|,|η|} (j ̸=k; ζ, η∈C). (2.4.16) However, with the canonical unit vectors e₁ := (1,0), e₂ := (0,1) in C², the functionals

Ψ₁ :=e₁⊗e₂⊗e₂, Ψ₂ :=e₂⊗e₁⊗e₂, Ψ₃ :=e₂⊗e₂⊗e₁ inE satisfy 2.4.16. The contradiction obtained establishes the proposition.

Theorem 2.4.17. The linearE(

=B(H⁽¹⁾, . . . ,H^(N)))

-unitary operators are exactly those operators F for which there exists a permutation π of the index set {1, . . . , N} and there are surjective linear isometries U_k:H^(k)→H^(π(k)) (k = 1, . . . , n) such that

F(Φ) =[

(f₁, . . . ,f_N)7→Φ(U₁⁻¹f_π(1), . . . , U_N⁻¹f_π(N))]

. (2.4.18)

A linear vector field V is complete in B(E) if and only if it is of the form V =i·

∑N k=1

id_H(1) ⊗. . .⊗id_H(k−1)⊗A_k⊗id_H(k+1)⊗. . .⊗id_H(N) (2.4.19) where the terms A_k are arbitrary self-adjoint H_k-operators.

Proof. The present approach can be divided into four steps:

(1) based on elementary compactness arguments, in a separate appendix (A.1) we estab-lish independently the validity of 2.4.18 if the spaces H^(k) are all finite dimensional;

(2) next in Chapter 3.6 we establish a Stone-type theorem stating that any strongly continuous one-parameter group[

U₁^t⊗· · ·⊗U_N^t : t∈R]

withH^(k)-unitary operators is of the form [

exp(itA₁)⊗ · · ·exp(itA_N) : t∈R]

with suitable possibly unbounded H^(k)-self-adjoint operators A_k, whence 2.4.19 is immediate for finite dimensional E;

(3) by the aid of the Projection principle, we extend 2.4.19 to infinite dimensions;

(4) by 2.4.19 we seeE-Hermitian projections have the form ⊗

j<m

id_H(j) ⊗P_m⊗ ⊗

j>m

id_H(j)

whence we deduce 2.4.18 in full generality.

Step (3). LetV linear∈aut(B) and fixe₁ ∈∂B(H⁽¹⁾), . . . ,e_N ∈∂B(H^(N)) arbitrarily.

Consider the operator³ Ve :=V −⟨

δ_e1,...,eN, V(e^∗₁ ⊗ · · · ⊗e^∗_N)⟩

id_E. Since i·id_E ∈aut(B), we have also Ve ∈aut(B) with Ve(e^∗₁⊗. . .⊗e^∗_n) = 0. Introduce the family of mappings

P :={

P₁⊗ · · · ⊗P_N : P_k=P_k^∗=P_k²∈L(H^(k)), dimP_kH^(k)<∞, e_k∈P_kH^(k)} . Any element P := P₁ ⊗ · · · ⊗ P_N of P is a contractive linear projection of the space E onto its subspace B(P₁H⁽¹⁾, . . . , P_nH^(N))(≡ {Φ ∈ E : P₁ ⊗ · · ·P_NΦ = Φ}). By the Projection Principle, PVe

P E ∈aut(

B(P E))

for allP ∈ P. Hence (applying 2.4.19 to the finite dimensional B(P₁H⁽¹⁾, . . . , P_NH^(N))), for each P ∈ P, there exists a unique choice of self-adjoint operators A^P₁ ∈ L(H⁽¹⁾), . . . , A^P_N ∈ L(H^(N)) such that

A^P_kH^(k)⊂PkH^(k) (i.e. PkA^P_k =A^P_k) and ⟨A^P_kek|ek⟩= 0 (k = 1, . . . , N), PV Pe =

∑N k=1

i·id_H(1) ⊗ · · · ⊗id_H(k−1)⊗A_k⊗id_H(k+1) ⊗. . .⊗id_H(N).

Introduce the following partial ordering≤inP: ifP =P₁⊗. . .⊗P_N andQ=Q₁⊗. . .⊗Q_N then let P ≤ Q ⇐⇒^def P_kH_k ⊂ Q_kH_k (i.e. P_k ≤ Q_k) (k = 1, . . . , N). From the relation P ≤Q⇒PV Pe =P QV QPe we see immediately

A^P_k =P_kA^Q_kP_k (k= 1, . . . , N) whenever P ≤Q. (2.4.19) Observe that for any fixed P ∈ P and index k,

|⟨A^P_ke|f⟩| = ⟨(PVe)(e^∗₁ ⊗ · · · ⊗e^∗_k−1⊗e^∗⊗e^∗_k+1⊗ · · · ⊗e^∗_N), δ_{e1,...,ek−1,f,ek+1,...,eN}⟩ ≤

≤ ∥PVe∥ · ∥e^∗₁⊗ · · · ⊗e^∗ ⊗. . .⊗e^∗_N∥ · ∥δ_e1,...,f,...,eN∥=∥PVe∥ ≤ ∥Ve∥ for all unit vectors e,f ∈H^(k), that is

∥A^P_k∥ ≤ ∥Ve∥ (k = 1, . . . , N; P ∈ P). (2.4.20) Since obviously for all couples P, Q∈ P there exists R ∈ P with P, Q≤ R and since, by 2.4.19 and 2.4.20, the relation P ≤Q entails

|⟨A^Q_ke|f⟩ − ⟨A^P_ke|f⟩|=|⟨A^Q_k(e−P_ke)|f⟩+⟨A^Q_kP_ke|f −P_kf⟩| ≤ ∥Ve∥(∥e−P_ke∥+∥f−P_kf∥) for all e,f ∈∂B(H_k), the definitions

a_k(e,f) := lim

P∈P⟨A^p_ke|f⟩ (e,f ∈H^(k); k= 1, . . . , N)

make sense and determine bounded sesquilinear functionals. Therefore there exist self-adjoint operators A₁ ∈ L(H^(k)) (k = 1, . . . , N) such that a_k(e,f) =⟨A_ke|f⟩ and hence

⟨A^P_ke|f⟩=⟨A^P_k(Pke)|Pkf⟩=⟨AkPke|Pkf⟩=⟨AkPke|Pkf⟩=⟨PkAkPke|f⟩

3As usually, the symbolsδxstand for the evaluation functionals (Dirac deltas)δx:φ7→ ⟨δx, φ⟩=φ(x).

for all e,f ∈ H^(k) i.e. A^P_k = P_kA_kP_k (P ∈ P; k = 1, . . . , N). Now for arbitrary Φ∈E, x₁ ∈H⁽¹⁾, . . . ,x_N ∈H⁽⁾N, the projections P_k:= Proj_Span{x

k,Akxk,ek} satisfy [VeΦ](x₁, . . . ,x_n) = [VeΦ](P₁x₁, . . . , P_Nx_N) = [PVeΦ](x₁, . . . ,x_N) =

=

∑N k=1

Φ(x₁, . . . , P_kA_kx_k, . . . ,x_N) =

∑N k=1

Φ(x₁, . . . , A_kx_k, . . . ,x_N).

Thus we can write VΦ(x₁, . . . ,x_N) =

∑N k=1

Φ(x₁, . . . , B_kx_k, . . . ,x_N) where B_j := A_j for j = 1, . . . , N−1 and B_N :=A_N+⟨V(e₁, . . . ,e_N),δ_e1,...,eN⟩id_E, proving (2.4.19) in general.

Step (4). To prove (2.4.18), let F be an arbitrarily given linear E-unitary operator and, for k = 1, . . . , N, introduce the families

Pk:={

P₁⊗ · · · ⊗P_n: P_k =P_k^∗ =P_k² ∈ L(H^(k)), P_j = id_H^(j) forj ̸=k} .

From 2.4.19 we seeiPk ⊂aut(B) and hence for every P ∈ Pk, the mappingQ:=F P F⁻¹ also has the properties iQ ∈ aut(B) and Q² =Q (since P² = P). By (2.4.19), this is possible only if Q∈ Pℓ_k(P) for some index ℓ_k(P) (k= 1, . . . , N).

Let k ∈ {1, . . . , N} be fixed. We show that ℓ_k(P₁) = ℓ_k(P₂) for P₁, P₂ ∈ Pk\ {id_E}. Indeed, if ℓ_k(R₁) ̸= ℓ_k(R₂) then the operators Q_j := F R_jF⁻¹ (j = 1,2) commute (i.e.

[Q₁, Q₂] := Q₁Q₂ − Q₂Q₁ = 0) whence we would have [R₁, R₂] = 0. Observe that

∀P₁,P₂∈Pk\{id_E} ∃P₃∈Pk [P₁,P₃],[P₂,P₃]̸= 0, thus (by taking R₁ :=P_j and R₂ :=P₃), we have ℓ_k(P_j) = ℓ_k(P₃) forj = 1,2. Therefore there exists a permutation π with

FPkF⁻¹ =Pπ(k) (k= 1, . . . , N). (2.4.21) Since the finite linear combinations of orthogonal projections form a dense submanifold of the algebra of linear operators in any Hilbert space, it directly follows the existence of surjective linear isometries S_k :L(H^(k))→ L(H^(π(k))) such that

F(id_H1 ⊗. . .⊗id_Hk−1 ⊗A_k⊗id_Hk+1 ⊗. . .⊗id_Hn)F⁻¹ =

= id_H1 ⊗. . .⊗id_Hπ(k)−1 ⊗S_k(A_k)⊗id_Hπ(k)+1⊗. . .⊗id_Hn

whenever Ak ∈ L(H^(k)) k = 1, . . . , N). As a consequence of the relations 2.4.21, the mappings S_k send orthogonal projections into orthogonal projections and therefore they constitute ∗-isomorphisms between the C^∗-algebras L(H^(k)) and L(H^(π(k))). It is well-known that now we can write Sk : Ak 7→ UkAkU_k⁻¹ (k = 1, . . . , N) for some surjective linear isometries U_k :H_k 7→H_π(k). Thus if we denote by σ the inverse of the permutation π, for any linearE-operator Aof the form A:=A₁⊗. . .⊗A_n with A_k∈ L(H^(k)) we have

(F AF⁻¹)Φ =[

(f₁, . . . ,f_n)7→Φ(U_σ(1)A_σ(1)U_σ(1)⁻¹ f, . . . , U_σ(n)A_σ(n)U_σ(n)⁻¹ f_N)]

(Φ∈E).

This means that F AF⁻¹ = U AU⁻¹ (A ∈ L(E)) holds for the E-unitary operator U defined by

U(Φ) := [(f₁, . . . ,f_N)7→Φ(U₁⁻¹f_π(1), . . . , U_N⁻¹f_π(N))] (Φ∈E).

Easily seen this is possible only ifF =e^iϑU for someϑ ∈Rwhich completes the proof.

Chapter 3

Stone-type theorem on

automorphisms of multilinear functionals

Throughout this chapter, as introduced in Example 2.4.13, let H,H⁽¹⁾, . . . ,H^(N) be arbi-trarily fixed complex Hilbert spaces. Our chief aim will be to study the structure of the strongly continuousone-parameter automorphism groups of the spaceB=B(H⁽¹⁾, . . . ,H^(N)) of all bounded N-linear functionals H⁽¹⁾× · · · ×H^(N)→C that is the maps

U:R→A:={

surjective linear isometries B → B}

with the group property U(t+h) =U(t)U(h) (t, h∈ R) and being such that the func-tions t 7→ U(t)Φ are continuous for all fixed Φ ∈ B. The case N = 1 is covered by Stone’s classical theorem [73, 114]: given a strongly continuous one-parameter subgroup U:R→ U(H) :={unitary operatorsH→H} ≃A, there exists a possibly unbounded self-adjoint linear operatorAon some dense linear submanifold ofHsuch thatU(t) = exp(itA) (t ∈ R). In the case N = 2, as a simple consequence of the theory of unbounded C^∗ -algebra derivations [9], in L(H) ≃ B(H,H) we have a precise abstract description of the special one-parameter isometry groups of the form U(t)X = exp(itA)Xexp(−itA) with a suitable possibly unbounded self-adjoint operator A. Our problems with N = 2 and B(H⁽¹⁾,H⁽²⁾) ≃ L(H⁽¹⁾,H⁽²⁾) are naturally associated with Jordan triple derivations [49, 102], and may have far reaching importance even for the description of all strongly continuous one-parameter automorphism groups of general JB*-triples (complex Banach spaces with symmetric unit ball). Namely, by the Hille-Yosida theorem [19, 114] the in-finitesimal generator of a strongly continuous one-parameter group of automorphisms of a JB*-triple is a possibly unbounded Jordan triple derivation. As far as we know, thebounded JB*-triple derivations are well-understood [3]. However, no results seem to be concerned with the unbounded case even for Cartan factors. From a Jordan theoretical view point, L(H⁽¹⁾,H⁽²⁾) is a typical Cartan factor of type I where the connected component of the automorphism group containing the identity consists of mappings of the form X 7→U XV with suitable unitary operatorsU∈U(H⁽²⁾) andV∈U(H⁽¹⁾) [38, 105]. Hence the structure

of all norm-continuous one parameter groups W : R → Aut(L(H⁽¹⁾,H⁽²⁾)) ≃ A is im-mediate: in this case W(t)X = [exp(itA₂)]X[exp(itA₁)] for a suitable couple of bounded self-adjoint operatorsA₁ ∈ L(H⁽¹⁾), B ∈ L(H⁽²⁾). Our main result below is a natural gen-eralization of Stone’s Theorem on strongly continuous one-parameter unitary groups. Its simple (but non-trivial) finite-dimensional special case along with the Projection Principle is used in Step (2) of the proof of Theorem 2.4.17, which settles the structure description of the group of holomorphic automorphisms of the unit ball of B(H⁽¹⁾, . . . ,H^(N)) for N ≥3.

Theorem 3.1.1. Let U:R →A(H⁽¹⁾, . . . ,H^(N)) be a strongly continuous one-parameter group such that

U(t) = U_1,t⊗ · · · ⊗U_N,t (t∈R)

with suitable unitary operators U_k,t ∈ U(H^(k)). Then there are possibly unbounded self-adjoint operators A_k : dom(A_k) → H^(k) (k = 1, . . . , N) defined on dense linear submani-folds in the respective spaces such that that

U(t) =[

exp(itA₁)]

⊗ · · · ⊗[

exp(itA_N)]

(t∈R).

Corollary 3.1.2. If W :R→Aut(

L(H⁽¹⁾,H⁽²⁾))

is a strongly continuous one-parameter group then W(t)X = exp(tA₁)Xexp(tA₂) for a suitable couple of possibly unbounded self-adjoint operators Ak : dom(Ak)→H^(k).

The main technical obstacle for the proof arises from the fact that an operatorU₁⊗· · ·⊗U_N admits alternative representations as [

(κ₁U₁)]

⊗ · · · ⊗[

(κ_NU_N)]

with κ₁, . . . , κ_N ∈ T :=

{κ∈C: |κ|= 1} and ∏_N

k=1κk = 1. Our considerations, which rely heavily upon complex Hilbert space structure, can be divided into three main steps. First we establish that, under the hypothesis of Theorem 3.1.1, there are multiplier functions κ_k : R → T with

∏_N

k=1κk(t) = 1 (t ∈ R) such that each component t7→κk(t)Uk,t is strongly continuous;

that is, all the functions t7→ κ_k(t)U_k,th_k (h_k ∈H^(k); k = 1, . . . , n) are continuous from R into H^(k) with norm topology. Assuming then without loss of generality the strong continuity of the components t 7→ Uk,t, we show that the families {

Uk,t : t ∈ R} are Abelian and then, by means of their Gelfand representations we can choose the multipliers κ_k : R→T even in a manner such that we have U_k,t = κ_k(t)U_k^t with some not necessarily strongly continuous one-parameter groups t 7→ U_k^t. We finish the proof after a series of probabilistic arguments where we establish that this representations can be improved to the formU_k,t =χ_k(t)Ue_k^twith strongly continuous one-parameter groupst 7→Ue_k^tand continuous functions χk:R→T, respectively.

3.2 Adjusted strong continuity

We shall use the tensorial notations established in 2.4.13. Notice that the factorization of non-trivial composition operators is unique up to constant coefficients: if A1, . . . , AN ̸= 0

we have A₁ ⊗ · · · ⊗ A_N = B₁ ⊗ · · · ⊗ B_N if and only if B_k = β_kA_k for constants with

∏_N

k=1β_k= 1.¹ In particular for unitary operators U_k, V_k ∈ U(H^(k)), U₁⊗ · · · ⊗U_N =V₁⊗ · · · ⊗V_N ⇐⇒ V_k =κ_kU_k, κ_k ∈Twith

∏N k=1

κ_k= 1.

Lemma 3.2.1. Assume Φ−Ψ≤ ε where Φ :=g₁^∗⊗ · · · ⊗g_N^∗ and Ψ :=h^∗₁⊗ · · · ⊗h^∗_N with unit vectors g_k,h_k∈H^(k). Then

dist(Tgk,Thk) := min

κ,µ∈T∥κgk−µhk∥ ≤2^N−1ε (k= 1, . . . , N).

Proof. Fix any 1 ≤ k ≤ N. Define κk := ∏

ℓ:ℓ̸=kκℓ where for ℓ ̸= k we set κℓ :=

⟨g_ℓ|h_ℓ⟩/|⟨g_ℓ|h_ℓ⟩| if g_ℓ ̸⊥h_ℓ and κ_ℓ := 1 if g_ℓ ⊥h_ℓ. With this choice κ₁, . . . , κ_N ∈T,

∏N m=1

κ_m = 1 and ⟨g_ℓ|κ_ℓh_ℓ⟩=|⟨g_ℓ|h_ℓ⟩| ≥0 (ℓ̸=k).

Observe that for the vectors xℓ :=gℓ+κℓhℓ with the values ϱℓ:= 1 +|⟨hℓ|gℓ⟩| we have

⟨x_ℓ|g_ℓ⟩=⟨x_ℓ|κ_ℓh_ℓ⟩=ϱ_ℓ∈[1,2], ∥x_ℓ∥=⟨x_ℓ|x_ℓ⟩^1/2= (2ϱ_ℓ)^1/2∈[√

2,2] (ℓ̸=k).

Thus, since also Ψ = [κ₁h₁]^∗⊗ · · · ⊗[κ_Nh_N]^∗, for any x∈H^(k) we can write [Φ−Ψ]

(x₁, . . . ,x_k−1,x,x_k+1, . . . ,x_N) = ⟨

xg_k−κ_kh_k⟩ ∏

ℓ:ℓ̸=k

ϱ_ℓ . Therefore we have the norm estimate

⟨xgk−κkhk⟩ ∏

ℓ:ℓ̸=k

ϱℓ ≤ϕ−Ψ ∥x∥ ∏

ℓ:ℓ̸=k

|xℓ∥ .

Since here ∥Φ−Ψ∥ ≤ε, 1 ≤ ϱ_ℓ and ∥x_ℓ∥ ≤ 2, it follows |⟨x|g_k−κ_kh_k⟩| ≤2^N−1ε for all vectors x∈H^(k). Hence dist(Tg_k,Th_k)≤ ∥g_k−κ_kh_k∥ ≤2^N−1ε.

In particular, with ε := 0 we see thatg^∗₁⊗ · · · ⊗g^∗_N =h^∗₁⊗ · · · ⊗h^∗_N impliesh_k=κ_kg_k for suitable κ_q, . . . , κ_N ∈ T with ∏_N

k=1κ_k = 1 whenever the vectors g_k,h_k have norm 1.

For later use, notice also that if g,h∈H are unit vectors in a Hilbert space then dist(Tg,Th)= dist(g,Th) = min

|κ|=1

[2−2 Re⟨g|κh⟩]1/2

=

=√ 2[

1− |⟨g|h⟩|]1/2

.

(3.2.2)

1Indeed, evaluated aty^∗₁⊗· · ·⊗y_N^∗, the relationA₁⊗· · ·⊗A_N =B₁⊗· · ·⊗B_N entails∏N

k=1⟨A_kx_k|y_k⟩=

∏N

k=1⟨Bkxk|yk⟩ for any x1, . . .xN. Given any index k, if xℓ,yℓ ∈ H^(ℓ) (ℓ ̸= k) are so chosen that

⟨Bℓxj|yℓ⟩ = 1 then for any x,y ∈ H^(k) we have ⟨Bkx|y⟩ = βk⟨Akx|y⟩ (x,y ∈ H^(k)) with βk :=

∏

ℓ:ℓ̸=k⟨Aℓxℓ|yℓ⟩. The converse implication is trivial.

Lemma 3.2.3. Suppose F : R → P(H) := {Tg : ⟨g|g⟩ = 1} is a continuous mapping with respect to the distance 3.2.2. Then F(t) =Th_t (t ∈R) for some continuous function t7→h_t ∈∂Ball(H) :={g: ⟨g|g⟩= 1}.

Proof. Since the real line R is σ-compact, it suffices to establish the local version of the statement: for every s ∈ R there is an open interval I_s around s where the set valued function F admits a continuous section say I_s ∋ t 7→ h^[s]_t ∈ f(t). [

Proof. In this case there is a strictly increasing double sequence (

T_n)_∞

To prove the local statement, we may assume s = 0 without loss of generality. The continuity of Fentails the continuity of function (t, u) 7→dist(

F(t),F(u))

. Hence we can chooseI₀ to be an open interval around 0 such that√

2>dist(

In particular, with suitable unit vectors u_t ⊥ f₀ and with suitable angle parameters 0 ≤ φ_t< π/2 we can write

Proposition 3.2.4. Assume Ψ : R → B is a continuous function of the form Ψ(t) = h^∗_1,t⊗· · ·⊗h^∗_N,t with suitable unit vectors h_k,t ∈H^(k). Then there are functionsκ₁, . . . , κ_N : R→Tsuch that ∏N

k=1κ_k(t)≡1and the modified components t7→κ_k(t)h_k,t are continuous (as mappings R→[H^(k),norm topology]).

Proof. According to Lemma 3.2.1, the functions F_k : t 7→ Th_k,t are continuous from R into the metric space [

P(H^(k)),dist]

in the sense of 3.2.2. Thus, by Lemma 3.2.3, we can find functions µ₁, . . . , µ_N : R → T such that the sections t 7→ f_k,t := µ_k(t)h_k,t ∈ F_k(t) (k = 1, . . . , N) are continuous. Then their product t 7→ Ψ(t) :=e f_1,t^∗ ⊗ · · · ⊗f_N,t^∗ is also a continuous map R → B. Observe that Ψ(t) =e µ(t)Ψ(t) (t ∈ R) with the scalar valued function µ(t) := ∏_N

k=1µ_k(t). From the continuity of both Ψ and Ψ we infer the continuitye of µ : R → T.² Hence also the functions t 7→ ef_k,t := µ(t)f_k,t are continuous. Since ef_k,t =∏

ℓ:ℓ̸=kµ_ℓ(t)h_k,t and since Ψ(t) =µ(t)Ψ(t) =e ef_1,t^∗ ⊗f_2,t^∗ ⊗ · · · ⊗f_N,t^∗ , the choice κ₁(t) :=

∏N

j=2µ_j(t) along with κ_k(t) := µ_k(t) for k >1 suits our requirements.

Conventions 3.2.5. To simplify notations for the proof of Theorem 3.1.1, henceforth let U:R→A=A(H⁽¹⁾, . . . ,H^(N)) be a one-parameter subgroup of operators of the form

U(t) = U_1,t^∗ ⊗ · · · ⊗U_N,t^∗ , U_k,t∈ U(H^(k)).

An application of Proposition 3.2.4 to functions of the formU(t)[h^∗₁⊗· · ·⊗h^∗_N] = [U_1,th₁]^∗⊗

· · · ⊗[U_N,th_N]^∗ yields immediately the following.

Corollary 3.2.6. Given any family h_k ∈ H^(k) (k = 1, . . . , N) of unit vectors, there are functions κk : R → T such that ∏N

k=1κk = 1 and the functions t 7→ κk(t)Uk,thk are continuous.

As usual, we say that a net ( V_α)

α∈A of bounded linear operators B → B is strongly convergent to V (notation: V_α −→^s V) if ∥(V_α−V)Φ∥ → 0 for all Φ ∈ B. Accordingly, a function V:R→ L(B) is strongly continuous, ifV(t_α)−→^s V(t) whenever t_α →t in R. Proposition 3.2.7. For some functions κ₁, . . . , κ_N :R→T, the operator-valued functions t7→κ_k(t)U_k,t are strongly continuous.

Proof. Fix any family h₁ ∈ H⁽¹⁾, . . . ,h_N ∈ H^(N) of unit vectors along with a family κ₁, . . . , κ_N : R → T of scalar functions with ∏N

j=1κ_j(t) = 1 such that the functions t7→κk(t)Uk,thk are continuous. This is guaranteed by Corollary 3.2.6. Consider any index k ∈ {1, . . . , N} and let 0 ̸= x ∈ H^(k) be any vector. It suffices to see that the function t7→κ_k(t)U_k,txis continuous.

2In general, ifvα→v̸= 0 andµαvα→µvare convergent nets in a locally convex Hausdorff vector space V, then necessarily µα →µ for the scalar coefficients. Proof: there exists a continuous linear functional ϕ on V such that ϕ(v) = 1. Beyond some index α0 we have ϕ(vα) ̸= 0 and µα = ϕ(µαvα)/ϕ(vα) → ϕ(µv)/ϕ(v) =µ.

Applying Corollary 3.2.6 with the vectorsh₁, . . . ,h_k−1,x/∥x∥,h_k+1, . . . ,h_N, we see the

Hence we deduce the continuity of t 7→ [ ∏

ℓ:ℓ̸=keκ_ℓ(t)κ_ℓ(t)]

Corollary 3.2.8. In the setting of 3.2.7, the functions t 7→ κ_k(t)U_k,t^∗ are also strongly continuous.

Proof. It is a well-known elementary fact [27] that the adjoints of the elements of a strongly convergent net of unitary operators in a Hilbert space form a strongly convergent net.

In document Dissertation for the title DSc (Pldal 22-36)