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Group of Isometries of the Hilbert Ball Equipped with the Caratheodory

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Group of Isometries of the Hilbert Ball Equipped with the Caratheodory GROUP OF ISOMETRIES OF HILBERT BALL EQUIPPED WITH THE CARATHEODORY´ METRIC MUKUND MADHAV MISHRA AND RACHNA AGGARWAL Abstract. In this article, we study the geometry of an infinite dimensional Hyperbolic space. We will consider the group of isometries of the Hilbert ball equipped with the Carath´eodory metric and learn about some special subclasses of this group. We will also find some unitary equivalence condition and compute some cardinalities. 1. Introduction Groups of isometries of finite dimensional hyperbolic spaces have been studied by a number of mathematicians. To name a few Anderson [1], Chen and Greenberg [5], Parker [14]. Hyperbolic spaces can largely be classified into four classes. Real, complex, quaternionic hyperbolic spaces and octonionic hyperbolic plane. The respective groups of isometries are SO(n, 1), SU(n, 1), Sp(n, 1) and F4(−20). Real and complex cases are standard and have been discussed at various places. For example Anderson [1] and Parker [14]. Quaternionic spaces have been studied by Cao and Parker [4] and Kim and Parker [10]. For octonionic hyperbolic spaces, one may refer to Baez [2] and Markham and Parker [13]. The most basic model of the hyperbolic space happens to be the Poincar´e disc which is the unit disc in C equipped with the Poincar´e metric. One of the crucial properties of this metric is that the holomorphic self maps on the unit ball satisfy Schwarz-Pick lemma. Now in an attempt to generalize this lemma to higher dimensions, Carath´eodory and Kobayashi metrics were discovered which formed one of the ways to discuss hyperbolic structure on domains in Cn. It is therefore natural to look for an infinite dimensional counterpart of the finite dimensional hyperbolic spaces. In late 20th century, Franzoni and Vesentini studied infinite dimensional Hilbert ball equipped with the Carath´eodory metric. A description of the group of isometries of the hyperbolic ball has been given in [7]. Motivational sources behind studies carried out in this article are [3], [8] and [12]. In this article, we intend to study dynamical aspects of the group of isometries of infinite dimensional hyperbolic ball. To this arXiv:2108.05703v1 [math.MG] 10 Aug 2021 end, we consider this group, focus on some special subclasses of this group and explore their properties. Sectional detail is as follows. Section 1 is a brief literature survey. Section 2 gives the description of group of holomorphic isometries and its linear representation. In section 3, we will study a class of isometries having a two dimensional reducing subspace, class of normal isometries, self adjoint isometries and involutory isometries. In section 4, we will discuss the condition on an isometry to be unitarily equivalent to its inverse and compute cardinality of the group of isometries, class of self adjoint elements and set of unitary equivalence classes of normal operators on H ⊕ C. Date: May 2021. 2020 Mathematics Subject Classification. 51M10; 51F25. Key words and phrases. Hyperbolic space; isometry group. 1 2 MUKUNDMADHAVMISHRAANDRACHNAAGGARWAL 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 Hilbert space 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 Banach space. 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 linear map 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. UA U(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 unitary operator 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 UA U(ξ) any elementp U ◦ f ∈ Aut(B), we have ◦ T = ∈ G such x0 0 1 1 h·, ξi a UA U(ξ) that φ = U ◦ f . This gives the pre image of U ◦ f as the set h·, ξi a x0 x0 UA U(ξ) eiθ , θ ∈ R . h·, ξi a UA U(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.
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