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2. The group of isometries We begin with the notion of holomorphicity in infinite dimensional set up, give a brief introduction of the Carath´eodory metric and describe the form of correspond- ing isometries on the infinite dimensional hyperbolic space which has been taken from [7]. Let E and F be two complex normed spaces. Let U be an open subset of E. A mapping f : U −→ F is called an F -valued holomorphic function if for every x ∈ U, f(x) − f(a) − A(x − a) there exists A ∈ L(E, F ) such that lim =0. Let H be an x→a kx − ak infinite dimensional complex and B be the unit ball in H equipped with the Carath´eodory metric defined as follows. Let D be a domain in a complex normed space E and ∆ denote the open unit ball in C equipped with the Poincar´e metric ρ. Let Hol(D, ∆) denote the set of all holomorphic mappings f : D → ∆. Then

CD(x, y) = sup ρ(f(x),f(y)) for all x, y ∈ D f defines the Carath´eodory pseudo-distance on D. CD becomes a complete metric on every bounded homogeneous domain in a complex . This metric is considered a generalization of ρ as it coincides with the latter on ∆. For a detailed study of this metric, refer [7], [11]. A holomorphic map on B is distance decreasing for the Carath´eodory metric. So, every bi-holomorphic surjection is an isometry for this metric. Let Aut(B) denote the group of all bi-holomorphic surjections on B and this is the group of isometries we will be working with. A general element of Aut(B) is of the form F = U ◦ fx0 , x0 ∈ B, where U is a unitary

x − x0 operator on H and fx0 : Bx0 −→ H defined by fx0 (x) = Tx0 is 1 − x, x0 ! 1 a holomorphic map, B = x ∈ H : kxk < . f restricted to B is a bi- x0 kx k x0  0  holomorphic surjection. Tx0 : H −→ H is a expressed as Tx0 (x) = hx, x0i 2 x0 + 1 −kx0k x. Aut(B) is known to act transitively on B. 2 1+ 1 −kx0k A linear representationp of the elements of Aut(B) is as follows. Let A be a sesquilinearp form on H ⊕ C defined by A((x, λ), (y,µ)) = hx, yi− λµ and Q denote the corresponding quadratic form. Consider the collection of all linear operators on H ⊕ C leaving A invariant. Such linear operators are bounded and injective by definition. Let G be the group consisting of all the bijective linear operators on H ⊕ A ξ C leaving A invariant. General form of an element of G is T = A∗(ξ)  ·, a  a   where ξ is a vector in H satisfying   1 A∗A = I + h·, A∗(ξ)i A∗(ξ) |a|2 and |a|2 =1+ kξk2. iθ The center ZG of G is {e I, θ ∈ R}. Theorem 2.1 ([7] Theorem VI.3.5). The map φ : G → Aut(B) defined by φ(T )= T˜ A ξ A(·)+ ξ is an onto homomorphism where T = A∗(ξ) and T˜ = .  ·, a  A∗(ξ) a ·, + a   a   D E GROUP OF ISOMETRIES OF HILBERT BALL EQUIPPED WITH THE CARATHE´ODORY METRIC3

Proof. Well definedness and homomorphic nature of φ can easily be verified. We will show that φ is an onto map. For x0 ∈ B, let f−x0 ◦ U be a general element of

Aut(B). We will find preimages of f−x0 and U separately. For a unitary element U 0 U ∈ Aut(B), T = a will work where |a| = 1. Let x ∈ B \{0}. To find 0 1 0   A(x)+ ξ suitable A, ξ and a such that = f−x0 (x) for all x ∈ B. Suppose such A∗(ξ) x, a + a D ξE A, ξ and a exist. This gives T˜(0) = = f (0) = x . Choose a = 1+ kξk2. a −x0 0 2 2 2 kx0k p Then kξk = (1+kξk )kx0k and kξk = . Hence, choose ξ = ax0. Now, 2 1 −kx0k we will choose a suitable self adjoint operator A. Construct an operator S ∈ B(H) p such that S(ξ)= a2ξ and S = I on hξi⊥. Then S = I +h·, ξi ξ. For, x = kξ+y ∈ H, for some k ∈ C and y ∈ hξi⊥, S(x)= a2kξ + y + kξ − kξ = x +(1+ kξk2 − 1)kξ = x + kξk2kξ + hy, ξi ξ = x + hkξ,ξi + hy, ξi ξ = x + hx, ξi ξ. Also, S is a positive operator such that sp(S) = {a2, 1}. Hence, S is invertible. Choose A to be the positive square root of S. Then A(ξ)= aξ and A = I on hξi⊥. Now, for such choice ˜ of a, ξ and A, it can be verified that T (x)= f−x0 (x).  3. Some special subclasses of G We begin by describing a simplified form of elements of G. UAU(ξ) Proposition 3.1. A general element of G is of the form eiθ , θ ∈ h·, ξi a   R, where ξ ∈ H, a = 1+ kξk2, U is a on H and A is a positive ⊥ operator on H such thatp A = I on hξi and A(ξ)= aξ. Proof. It can be observed from the proof of Theorem 1.1 that for every element A ξ f ∈ Aut(B), there exists a self adjoint element T = ∈ G satisfying x0 1 h·, ξi a   2 ⊥ a = 1+ kξk , A(ξ) = aξ and A = I on hξi such that φ(T1) = fx0 . Hence for U 0 UAU(ξ) any elementp U ◦ f ∈ Aut(B), we have ◦ T = ∈ G such x0 0 1 1 h·, ξi a     UAU(ξ) that φ = U ◦ f . This gives the pre image of U ◦ f as the set h·, ξi a x0 x0   UAU(ξ) eiθ , θ ∈ R .  h·, ξi a     UAU(ξ) From now onwards, T ∈ G will denote an element of the form eiθ . h·, ξi a   The sesquilinear form A defined above is identified with the linear operator A′ = I 0 , i.e. A((x, λ), (y,µ)) = hA′(x, λ), (y,µ)i for all (x, λ), (y,µ) ∈ H ⊕ C. 0 −1   (UA)∗ ξ This gives T ∗ = e−iθ . Also, for T of the above form, T −1 = h·,U(ξ)i a   (UA)∗ −ξ e−iθ . − h·,U(ξ)i a   The following theorem by Hayden and Suffridge gives the fixed point classifica- tion of isometries in Aut(B). Theorem 3.2 ( [9]). If g ∈ Aut(B) has no fixed point in B, then the fixed point set in B consists of one or two points. 4 MUKUNDMADHAVMISHRAANDRACHNAAGGARWAL

We call an isometry in Aut(B) elliptic if has a fixed point in B, hyperbolic or parabolic if it is not elliptic and has two or one fixed points on ∂B respectively.

We say an element of G is elliptic (resp. hyperbolic or parabolic) if it is in the pre image of an elliptic (resp. hyperbolic or parabolic) isometry of Aut(B) defined by the homomorphism φ.

UA + U(ξ) Observe that for x ∈ B, x is a fixed point for an isometry ∈ Aut(B) h·, ξi + a if and only if (x, 1) is an eigenvector for the corresponding element in G having eigenvalue hx, ξi + a.

Here from, we will assume ξ 6= 0, unless stated otherwise.

We will now investigate a subclass of isometries of G which decomposes H ⊕ C into a two dimensional subspace containing C and its orthogonal complement. UAU(ξ) We know that a general element T of G is of the form eiθ where A h·, ξi a   decomposes H into hξi and hξi⊥. If the unitary operator U is chosen in a manner so as to leave hξi invariant, then all the isometries of G comprising of such unitary operators are precisely the ones decomposing H ⊕ C into hξi⊕ C and its orthogonal complement. Thus we have the following. UAU(ξ) Proposition 3.3. Let T = ∈ G. Then T = T ⊕ T where h·, ξi a 1 2 ⊥  T1 = T ↾hξi⊕C and T2 = U ↾hξi if and only if U(ξ)= rξ, |r| =1.

Proof. Write H ⊕ C = hξi ⊕ hξi⊥ ⊕ C. Let U(ξ) = rξ, |r| = 1. Then for any UAU(ξ) ξ r(a + z)ξ element (ξ,z) ∈ hξi⊕ C, = ∈ hξi⊕ C. Also h·, ξi a z kξk2 + a      UAU(ξ) x U(x) for x ∈ hξi⊥, = ∈ (hξi⊕ C)⊥. Thereby showing h·, ξi a 0 0      C ⊥ ⊥ hξi⊕ is a reducing subspace for T and T ↾(hξi⊕C) = U ↾hξi . Conversely, if hξi⊕ C is invariant under T , then U(hξi) = hξi and hence hξi⊕ C reduces T .  UA rξ Proposition 3.4. Let T = ∈ G, |r| =1. Then h·, ξi a   a(r + 1) ± a2(r + 1)2 − 4r ⊥ (1) sp(T )= {λ1, λ2}∪sp(U ↾hξi ) where λ1, λ2 = p 2 respectively. The eigenspaces corresponding to the eigenvalues λ1 and λ2 are generated by the eigenvectors (k1ξ, 1) and (k2ξ, 1) where a(r − 1) ± a2(r + 1)2 − 4r k , k = respectively. 1 2 ξ 2 p2k k 1 1 (2) |λ1| = and kk1ξk = . |λ2| kk2ξk

⊥ Proof. (1) As T = T1 ⊕ T2, sp(T )= sp(T1) ∪ sp(T2). Clearly, sp(T2)= sp(U ↾hξi ). Also, T1 is an operator on a 2 dimensional space hξi⊕C. So, λ is an eigenvalue of T1 UA rξ kξ kξ with eigenvector (kξ, 1) if = λ which gives r(ak+1)ξ = h·, ξi a 1 1      λkξ and λ = kkξk2 +a. Simplification of these two expressions yield k2kξk2 +a(1− GROUP OF ISOMETRIES OF HILBERT BALL EQUIPPED WITH THE CARATHE´ODORY METRIC5

a(r − 1) ± a2(r + 1)2 − 4r r)k − r = 0. Hence k = (using a2 =1+ kξk2) and ξ 2 p2k k a(r + 1) ± a2(r + 1)2 − 4r λ = . Note that (ξ, 0) cannot be an eigenvector for T . 2 1 p −r 1 λ λ r k k λ k ξ (2) Observe that 1 2 = and 1 2 = 2 . This gives | 1| = and k 1 k = kξk |λ2| 1 respectively.  kk2ξk In general, we have UAU(ξ) Proposition 3.5. Let T = ∈ G and K be a subspace of H con- h·, ξi a taining ξ such that K is invariant under U. Then T decomposes H ⊕ C into K ⊕ C and its orthogonal complement. Proof. Represent a general element y ∈ H as y = kξ + x, where k ∈ C and UAU(ξ) kξ + x x ∈ hξi⊥. Let (kξ + x, z) ∈ K ⊕ C, z ∈ C. Then = h·, ξi a z    U((ak + z)ξ + x) ∈ K ⊕ C as K is invariant under U thereby making K ⊕ C kkξk2 + a   UAU(ξ) y U(y) invariant under T . Also, for y ∈ K⊥, = ∈ (K ⊕ h·, ξi a 0 0      C)⊥. Hence the result.  We are now ready to describe the normal isometries of G. Class of normal isometries forms a subpiece of the above defined subclass of G. In exact terms, if hξi⊕ C reduces T , then hξi reduces U. In specific, the class of all such isometries where U acts as identity on hξi precisely forms the class of normal isometries in G. Theorem 3.6 (Normal elements of G). Let S ∈ G be a . Then UA ξ (1) S = eiθ , θ ∈ R. h·, ξi a   V 0 (2) A unitary element of G is of the form eiθ , where V is a unitary 0 1 operator on H and θ ∈ R.  

Sp S a ξ Sp U ⊥ a ξ (3) ( ) = { ± k k} ∪ hξi , where ± k k are both positive and inverses of each other. Eigenspaces  corresponding to the eigenvalues a±kξk ξ are spanned by the eigenvectors ± , 1 respectively. kξk   (4) Normal isometries are hyperbolic in nature. Proof. (1) For an element T ∈ G, direct computation shows that TT ∗ = T ∗T if and only if U(ξ)= ξ. (2) T ∗ = T −1 if and only if ξ = 0. (3) follows from Proposition 2.4 for r = 1. ξ (4) S has two fixed points ± lying on the unit sphere. Also S can’t be elliptic kξk   as it does not have an eigenvector of the form (x, 1), kxk < 1, for eigenvectors of S either come from hξi⊕ C or (hξi⊕ C)⊥ and a ± ξ 6= 1.  Next we explore the self adjoint elements of G. Prior to that, let us make a simple observation. In the construction of T , A = I on hξi⊥. This implies that every unitary operator U which decomposes H into hξi and hξi⊥ commutes with A. 6 MUKUNDMADHAVMISHRAANDRACHNAAGGARWAL

Proposition 3.7 (Self adjoint elements of G). Let T be a self adjoint element in G. Then UA ξ (1) T is of the form ± , where U is unitary and involutory operator. h·, ξi a   (2) sp(T )= {a ±kξk}∪{±1}. UAU(ξ) Proof. (1) T ∈ G is self adjoint if and only if T = T ∗, i.e eiθ = h·, ξi a   (UA)∗ ξ e−iθ which gives UA is self adjoint, U(ξ)= ξ and θ = nπ, n ∈ Z. h·,U(ξ)i a Now,UA is self adjoint, A is self adjoint and U commutes with A (by the above observation) gives U is self adjoint. Hence U is involutory. ⊥ (2) Clearly, T = T1 ⊕ T2, sp(T2)= sp(U ↾hξi )= {±1}. 

Let us now discuss the form of involutory elements of G. It is easy to see that V 0 unitary + involutory elements of G are of the form where V is unitary 0 ±1 + involutory.   Proposition 3.8 (Involutory elements of G). Let T be an involutory element in G. Then UA −ξ (1) T = ± where U is involutory. h·, ξi a   ⊥ (2) T = T1 ⊕ T2 where T1 = T ↾hξi⊕C and T2 = U ↾hξi . −a ± 1 (3) sp(T )= {±1}, where sp(T )= ±1 with eigenspace spanned by ξ , 1 . 1 kξk2 (4) Involutory elements are elliptic in nature.   Proof. (1) T is involutory if and only if T = T −1. On comparing respective entries of T and T −1, we get UA is self adjoint, U(ξ) = −ξ and θ = nπ. Following the same argument as in part (1) of the previous proposition, we get U is involutory. (2) and (3) follow from Proposition 2.4 for r = −1. |− a +1| a − 1 (4) Since = < 1, T is elliptic in nature.  kξk kξk

4. Unitarily equivalence condition and cardinality Next we will learn about the condition on an isometry to be unitarily equivalent to its inverse. UAU(ξ) Proposition 4.1. An isometry T = eiθ and its inverse are uni- h·, ξi a   tarily equivalent if and only if there exists a unitary operator V such that (UA)∗ = V −1UAV , V (ξ)= V −1(ξ)= −U(ξ) and θ = nπ. Proof. An isometry T and its inverse are unitarily equivalent to each other if and only if there exists a unitary operator V satisfying V −1 0 UAU(ξ) V 0 (UA)∗ −ξ eiθ = e−iθ 0 1 h·, ξi a 0 1 − h·,U(ξ)i a         i.e. V −1UAV V −1U(ξ) (UA)∗ −ξ eiθ = e−iθ ·, V −1(ξ) a − h·,U(ξ)i a     if and only if (UA )∗ = V −1UAV , V (ξ)= V −1(ξ)= −U(ξ) and θ = nπ.  GROUP OF ISOMETRIES OF HILBERT BALL EQUIPPED WITH THE CARATHE´ODORY METRIC7

A ξ Example of an operator which is unitarily equivalent to its inverse is h·, ξi a where the conjugating operator V can be taken −I. Notice that the way A is defined, A clearly commutes with V .

In finite dimensions, the number of conjugacy classes in a group of isometries used to have implications on the dynamical types of the action of the group on the underlying space. In the infinite dimensional set up, we try to compare the sizes of conjugacy classes through computing the cardinalities.

Lastly, we compute the cardinalities of some sets for H separable. We know that |H| = c. Let U(H) denote the group of unitary operators on H and Gs denote the collection of self adjoint isometries in G. Then

Proposition 4.2. |U(H)| = |G| = |Gs| = c. Proof. As eiθI ∈ U(H), θ ∈ R, we have |U(H)|≥ c. Also, |U(H)| ≤ |L(H)|, where L(H) is the space of linear operators on H. Now, each linear operator has an infinite representation. This gives |L(H)| is same as the cardinality of the set of all complex sequences which is same as c. Hence |U(H)| = c. Also, by the construction of elements of G, it can be seen that |G| = |H|×|U(H)|× |R| = c × c × c = c. A ξ For the collection G , elements of the form where ξ ∈ H lie in G and s h·, ξi a s   cardinality of such elements is |H| = c. This gives |Gs|≥ c. Also, |Gs| < |G| = c. So |Gs| = c. 

Next we will find out the cardinality of the set of unitary equivalence classes of normal operators on H ⊕ C. For this, we will use the following result from Conway [6].

Theorem 4.3. If N is a on a separable Hilbert space H ⊕ C with scalar-valued spectral measure µ, then there is a decreasing sequence {∆n} of Borel subsets of sp(N) such that ∆1 = sp(N) and ∼ N = Nµ ⊕ N ↾µ↾∆2 ⊕N ↾µ↾∆3 ⊕... . Let us introduce some quick notations for the sake of next proposition. Let the set of all normal operators be denoted by N , set of all unitary equivalence classes of normal operators by NU , set of all compact subsets of C by C, Borel σ- algebra of a set A ⊂ C by B(A) and the set of all sequences of all the Borel subsets of a set A by Bs(A). The above theorem gives |N | ≤ | Bs(K)|. K∈K S Proposition 4.4. |NU | = c. Proof. Let K ⊂ C be a compact set. Then there exists a normal operator with K as its spectrum. Also, unitarily equivalent normal operators have the same spectrum. This gives |NU | ≥ |C|. We know that |B(C)| = c. Now, every singleton is a compact set in C and every compact set is a Borel subset of C giving |C| = c and hence |NU |≥ c. We will now show that |NU | ≤ c. |B(K)| ≤ |B(C)| = c which gives |Bs(K)| ≤ |Bs(C)| = c, therefore | Bs(K)|≤ c×c = c. Finally, |NU | ≤ |N | ≤ | Bs(K)|≤ K∈K K∈K c. Hence the result. S S  8 MUKUNDMADHAVMISHRAANDRACHNAAGGARWAL

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Department of Mathematics, Hansraj College, University of Delhi, Delhi, India Email address: [email protected]

Department of Mathematics, University of Delhi, Delhi, India Email address: [email protected]