Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

Transcription

1 Probab. Theory Relat. Fields DOI /s y Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups Bruce K. Driver Maria Gordina Received: 6 August 2008 / Revised: 7 February 2009 The Authors) This article is published with open access at Springerlink.com Abstract We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure µ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the Lie algebra of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the L 2 ν)-closure of holomorphic polynomials by their values on the Cameron Martin subgroup. Keywords Heisenberg group Holomorphic Heat kernel Quasi-invariance Taylor map Mathematics Subject Classification 2000) 58G32 Primary 35K05 43A15; Secondary B. K. Driver s research was supported in part by NSF Grants DMS and DMS and the Miller Institute at the University of California, at Berkeley. M. Gordina s research was supported in part by NSF Grant DMS and the Humboldt Foundation Research Fellowship. B. K. Driver B) Department of Mathematics, 0112, University of California, San Diego, La Jolla, CA , USA driver@euclid.ucsd.edu M. Gordina Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA gordina@math.uconn.edu

2 B. K. Driver, M. Gordina Contents 1 Introduction Statements of the main results Discussion... 2 Complex abstract Wiener spaces The structure of the projections... 3 Complex Heisenberg like groups... 4 Brownian motion and heat kernel measures Heat kernel quasi-invariance properties Finite dimensional approximations... 5 Holomorphic functions on G and G CM Holomorphic functions on Banach spaces Holomorphic functions on G and G CM... 6 The Taylor isomorphism theorem A non-commutative Fock space The Taylor isomorphism... 7 Density theorems Finite rank subspaces Polynomial approximations... 8 The skeleton isomorphism Proof of Theorem The holomorphic chaos expansion Generalities about multiple Itô integrals The stochastic Taylor map Future directions and questions... References... 1 Introduction The aim of this paper is to study spaces of holomorphic functions on an infinitedimensional Heisenberg like group based on a complex abstract Wiener space. In particular, we prove Taylor, skeleton, and holomorphic chaos isomorphism theorems. The tools we use come from properties of heat kernel measures on such groups which have been constructed and studied in [4]. We will state the main results of our paper and then conclude this introduction with a brief discussion of how our results relate to the existing literature. 1.1 Statements of the main results The Heisenberg like groups and heat kernel measures The basic input to our theory is a complex abstract Wiener space, W, H,µ),asin Notation 2.4 which is equipped with a continuous skew-symmetric bi-linear form ω : W W C as in Notation 3.1. Here and throughout this paper, C is a finite dimensional complex inner product space. The space, G := W C, becomes an infinite-dimensional Heisenberg like group when equipped with the following mul-

3 Square integrable holomorphic functions tiplication rule w 1, c 1 ) w 2, c 2 ) = w 1 + w 2, c 1 + c ) 2 ω w 1,w 2 ). 1.1) A typical example of such a group is the Heisenberg group of a symplectic vector space, but in our setting we have an additional structure of an abstract Wiener space to carry out the heat kernel measure analysis. The group G contains the Cameron Martin group, G CM := H C, as a subgroup. The Lie algebras of G and G CM will be denoted by g and g CM respectively which, as sets, may be identified with G and G CM respectively see Definition 3.2, Notation 3.3, and Proposition 3.5 for more details. Let bt) = Bt), B 0 t)) be a Brownian motion on g associated to the natural Hilbertian structure on g CM as described in Eq. 4.1). The Brownian motion {gt)} t 0 on G is then the solution to the stochastic differential equation, dgt) = gt) dbt) with g0) = e = 0, 0). 1.2) The explicit solution to Eq. 1.2) may be found in Eq. 4.2). For each T > 0we let ν T := LawgT )) be the heat kernel measure on G at time T as explained in Definitions 4.1 and 4.2. Analogous to the abstract Wiener space setting, ν T is left right) quasi-invariant by an element, h G, iffh G CM, while ν T G CM ) = 0, see Theorem 4.5, Proposition 4.6, and [4, Proposition 6.3]. In addition to the above infinite-dimensional structures we will need corresponding finite dimensional approximations. These approximations will be indexed by ProjW ) which we now define. Notation 1.1 Let ProjW ) denote the collection of finite rank continuous linear maps, P : W H, such that P H is an orthogonal projection. Explicitly, P must be as in Eq. 2.17) below.) Further, let G P := PW C a subgroup of G CM ) and π P : G G P be the projection map defined by π P w, c) := Pw, c). To each P ProjW ), G P is a finite dimensional Lie group. The Brownian motions and heat kernel measures, {νt P } t>0,ong P are constructed similarly to those on G see Definition We will use {G P,νT P)} P ProjW ) as finite dimensional approximations to G,ν T ) The Taylor isomorphism theorem The Taylor map, T T, is a unitary map relating the square integrable holomorphic functions on G CM with the collection of their derivatives at e G CM. Before we can state this theorem we need to introduce the two Hilbert spaces involved. In what follows, HG CM ) and HG) will denote the space of holomorphic functions on G CM and G respectively. See Sect. 5 for the properties of these function spaces which are used throughout this paper.) We also let T := Tg CM ) be the algebraic tensor algebra over g CM, T be its algebraic dual, J be the two-sided ideal in T

4 B. K. Driver, M. Gordina generated by {h k k h [h, k] :h, k g CM }, 1.3) and J 0 ={α T : αj) = 0} be the backwards annihilator of J see Notation 6.1. Given f HG) we let α := T f denote the element of J 0 defined by α, 1 = f e) and ) α, h 1 h n := h 1... h n f e) where h i g CM and h i is the left invariant vector field on G CM agreeing with h i at e see Proposition 3.5 and Definition 6.2. We call T the Taylor map since T f J 0 g CM ) encodes all of the derivatives of f at e. Definition 1.2 L 2 holomorphic functions on G CM )ForT > 0, let f H 2 T G CM ) = sup f G L P 2 G P,ν P ) for all f H G CM ), and 1.4) P ProjW ) T { } H 2 T G CM) := f H G CM ) : f H 2 T G CM ) <. 1.5) In Corollary 6.6 below, we will see that H 2 T G CM) is not empty and in fact contains the space of holomorphic cylinder polynomials P CM ) on G CM described in Eq. 1.7) below. Despite the fact that ν T G CM ) = 0, H 2 T G CM) should roughly be thought of as the ν T square integrable holomorphic functions on G CM. Definition 1.3 Non-commutative Fock space) Let T > 0 and α 2 JT 0g CM) := T n α, h 1 h n 2 for all α J 0 g CM ), n! n=0 h 1,...,h n S where S is any orthonormal basis for g CM.Thenon-commutative Fock space is defined as } JT 0 g CM) := {α J 0 g CM ) : α 2J 0T gcm) <. It is easy to see that J 0 T g CM ) is a Hilbertian norm on J 0 T g CM) see Definition 6.4 and Eq. 6.8). For a detailed introduction to such Fock spaces we refer to [13]. Remark 1.4 When ω = 0, Gω) is commutative and the Fock space, J 0 T g CM), becomes the standard commutative bosonic Fock space of symmetric tensors over g CM. The following theorem is proved in Sect. 6 see Theorem Theorem 1.5 The Taylor isomorphism) For all T > 0, T H 2 T G CM)) J 0 T g CM) and the linear map, is unitary. T T := T H 2 T G CM ) : H2 T G CM) J 0 T g CM), 1.6)

5 Square integrable holomorphic functions Associated to this theorem is an analogue of Bargmann s pointwise bounds which appear in Theorem 6.11 below The skeleton isomorphism theorem Similarly to how it has been done on a complex abstract Wiener space by Sugita [26,27], the quasi-invariance of the heat kernel measure ν T allows us to define the skeleton map from L p G,ν T ) to a space of functions on the Cameron Martin subgroup G CM,asetofν T -measure 0. Definition 1.6 A holomorphic cylinder polynomial on G is a holomorphic cylinder function see Definition 4.3) of the form, f = F π P : G C, where P ProjW ) and F : PW C C is a holomorphic polynomial. The space of holomorphic cylinder polynomials will be denoted by P. The Gaussian heat kernel bounds in Theorem 4.11 easily imply that P L p ν T ) for all p < see Corollary Definition 1.7 Holomorphic L p functions) For T > 0 and 1 p <,leth p T G) denote the L p ν T )-closure of P L p ν T ). From Corollary 4.8 below, if T > 0, p 1, ], f L p G,ν T ), and h G CM, then G f h g) dν T g) <. Thus, if f H 2 T G) we may define the skeleton map see Definition 4.7) by S T f )h) := f h g)dν T g). It is shown in Theorem 5.12 that S T H 2 T G)) H2 T G CM) for all T > 0. Theorem 1.8 The skeleton isomorphism) For each T S T : H 2 T G) H2 T G CM), is unitary. G > 0, the skeleton map, Following Sugita s results [26,27] in the case of an abstract Wiener space, we call S T H 2 T G) the skeleton map since it characterizes f H2 T G) by its values, S T f, on G CM. Sugita would refer to G CM as the skeleton of Gω) owing to the fact that ν T G CM ) = 0 as we show in Proposition 4.6. Theorem 1.8 is proved in Sect. 8 and relies on two key density results from Sect. 7. The first is Lemma 7.3 an infinite-dimensional version of [7, Lemma 3.5]) which states that the finite rank tensors see Definition 7.2) are dense inside of J 0 T g CM). The second is Theorem 7.1 which states that P CM := { p GCM : p P } 1.7) is a dense subspace of H 2 T G CM).MattCecil[2] has modified the arguments presented in Sect. 7 to cover the situation of path groups over graded nilpotent Lie groups. Cecil s arguments are necessarily much more involved because his Lie groups have nilpotency of arbitrary step.

6 B. K. Driver, M. Gordina The holomorphic chaos expansion So far we have produced for each T > 0) two unitary isomorphisms, the skeleton map S T and the Taylor isomorphism T T, H 2 T G) S T = H 2 T G CM) T T = J 0 T g CM). The next theorem gives an explicit formula for T T S T ) 1 : JT 0g CM) H 2 T G). Theorem 1.9 The holomorphic chaos expansion) If f H 2 T G) and α f := T T S T f, then f gt )) = α f, dbs 1 ) dbs n ) 1.8) n=0 0 s 1 s 2 s n T where bt) and gt) are related as in Eq. 1.2) or equivalently as in Eq. 4.2). This result is proved in Sect. 9 and in particular, see Theorem The precise meaning of the right hand side of Eq. 1.8) is also described there. 1.2 Discussion As we noticed in Remark 1.4 when the form ω 0 the Fock space JT 0g CM) is the standard commutative bosonic Fock space [9]. In this case the Taylor map is one of three isomorphisms between different representations of a Fock space, one other being the Segal Bargmann transform. The history of the latter is described in [13] beginning with works of Bargmann [1] and Segal [24]. For other relevant results see [8,14]. To put our results into perspective, recall that the classical Segal Bargmann space is the Hilbert space of holomorphic functions on C n that are square-integrable with respect to the Gaussian measure dµ n z) = π n e z 2 dz, where dz is the 2n-dimensional Lebesgue measure. One of the features of functions in the Segal Bargmann space is that they satisfy the pointwise bounds f z) f L 2 µ n ) exp z 2 /2) compare with Theorem 6.11). As it is described in [13], if C n is replaced by an infinite-dimensional complex Hilbert space H, one of the first difficulties is to find a suitable version of the Gaussian measure. It can be achieved, but only on a certain extension W of H, which leads one to consider the complex abstract Wiener space setting. From Sugita s [26,27] work on holomorphic functions over a complex abstract Wiener space, it is known that the pointwise bounds control only the values of the holomorphic functions on H. This difficulty explains, in part, the need to consider two function spaces: one is of holomorphic functions on H or G CM in our case) versus the square-integrable weakly) holomorphic functions on W or G in our case). The Taylor map has also been studied in other non-commutative infinitedimensional settings. Gordina [10 12] considered the Taylor isomorphism in the context of Hilbert-Schmidt groups, while Cecil [2] considered the Taylor isomorphism for path groups over stratified Lie groups. The nilpotentcy of the Heisenberg

7 Square integrable holomorphic functions like groups studied in this paper allow us to give a more complete description of the square integrable holomorphic function spaces than was possible in [10 12] for the Hilbert Schmidt groups. Complex analysis in infinite dimensions in a somewhat different setting has been studied by Lempert e.g.[20]), and for more results on Gaussian-like measures on infinite-dimensional curved spaces see papers by Pickrell e.g.[22,23]). For another view of different representations of Fock space, one can look at results in the field of white noise, as presented in the book by Obata [21]. The map between an L 2 -space and a space of symmetric tensors sometimes is called the Segal isomorphism as in [18,19]. For more background on this and related topics see [16]. 2 Complex abstract Wiener spaces Suppose that W is a complex separable Banach space and B W is the Borel σ algebra on W.LetW Re denote W thought of as a real Banach space. For λ C,letM λ : W W be the operation of multiplication by λ. Definition 2.1 A measure µ on W, B W ) is called a mean zero, non-degenerate) Gaussian measure provided that its characteristic functional is given by ˆµu) := W e iuw) dµw) = e 1 2 qu,u) for all u W Re, 2.1) where q = q µ : WRe W Re R is an inner product on W Re. If in addition, µ is invariant under multiplication by i, that is, µ Mi 1 = µ, we say that µ is a complex Gaussian measure on W. Remark 2.2 Suppose W = C d and let us write w W as w = x + iy with x, y R d. Then the most general Gaussian measure on W is of the form dµw) = 1 Z exp 1 [ ] [ ]) x x 2 Q dx dy y y where Q is a real positive definite 2d 2d matrix and Z is a normalization constant. The matrix Q may be written in 2 2 block form as Q = [ ] A B B tr. C A simple exercise shows µ = µ Mi 1 iff B = 0 and A = C. Thus the general complex Gaussian measure on C d is of the form

8 dµw) = 1 Z exp 1 ) 2 Ax x + Ay y) dx dy = 1 Z exp 1 ) 2 Aw w dx dy, B. K. Driver, M. Gordina where A is a real positive definite matrix. Given a complex Gaussian measure µ as in Definition 2.1, let w H := sup u W Re \{0} and define the Cameron Martin subspace, H W,by uw) qu, u) for all w W, 2.2) H = {h W : h H < }. 2.3) The following theorem summarizes some of the standard properties of the triple W, H,µ). Theorem 2.3 Let W, H,µ) be as above, where µ is a complex Gaussian measure on W, B W ). Then 1) H is a dense complex subspace of W. 2) There exists a unique inner product,, H, on H such that h 2 H = h, h for all h H. Moreover, with this inner product H is a complete separable complex Hilbert space. 3) There exists C < such that h W C h H for any h H. 2.4) 4) If {e j } j=1 is an orthonormal basis for H and u,v W Re, then qu,v)= u,v H Re = [ ) u e j ve j ) + u ) )] ie j v iej. 2.5) j=1 5) µ Mλ 1 = µ for all λ C with λ =1. Proof We will begin with the proof of item 5. From Eq. 2.1), the invariance of µ under multiplication by i µ Mi 1 = µ) is equivalent to assuming that qu M i, u M i ) = qu, u) for all u WRe. By polarization, we may further conclude that q u M i,v M i ) = q u,v) for all u,v W Re. 2.6) Taking v = u M i in this identity then shows that qu M i, u) = qu, u M i ) and hence that q u, u M i ) = 0 for any u W Re. 2.7)

9 Square integrable holomorphic functions Therefore if λ = a + ib with a, b R, we see that q u M λ, u M λ ) = q au + bu M i, au + bu M i ) = a 2 + b 2 )qu, u) = λ 2 qu, u), 2.8) from which it follows that qu M λ, u M λ ) = qu, u) for all u WRe and λ =1. Coupling this observation with Eq. 2.1) implies µ Mλ 1 = µ for all λ =1. If λ =1, from Eqs. 2.2) and 2.8), it follows that λw H = u λw) sup = u WRe \{0} qu, u) sup u W Re \{0} u M λ w) q u Mλ, u M λ ) uw) = sup = w H for all w W. u WRe \{0} qu, u) In particular, if h H < and λ =1, then λh H = h H < and hence λh H which shows that H is a complex subspace of W.From[4, Theorem 2.3] summarizing some well-known properties of Gaussian measures, we know that item 3. holds, H is a dense subspace of W Re, and there exists a unique real Hilbertian inner product,, HRe,onH such that h 2 H = h, h H Re for all h H. Polarizing the identity λh H = h H implies λh,λk HRe = h, k HRe for all h, k H. Taking λ = i and k = ih then shows ih, h Re = h, ih Re, and hence that ih, h Re = 0 for all h H. Using this information it is a simple matter to check that h, k H := h, k HRe + i h, ik HRe for all h, k H, 2.9) is the unique complex inner product on H such that Re, H =, HRe. So it only remains to prove Eq. 2.5). For a proof of the first equality in Eq. 2.5), see [4, Theorem 2.3]. To prove the second equality in this equation, it suffices to observe that {e j, ie j } j=1 is an orthonormal basis for H Re,, HRe ) and therefore, u,v H Re = [ ) ) ) )] u e j v e j + u iej v iej j=1 for any u,v H Re. Notation 2.4 The triple, W, H,µ), appearing in Theorem 2.3 will be called a complex abstract Wiener space Notice that there is redundancy in this notation since µ is determined by H, and H is determined by µ). Lemma 2.5 Suppose that u,v WRe and a, b C, then 1 e au+bv dµ = exp a 2 q u, u) + b 2 qv, v) + 2abqu,v)) ). 2.10) 2 W

10 B. K. Driver, M. Gordina Proof Equation 2.10) is easily verified when both a and b are real. This suffices to complete the proof, since both sides of Eq. 2.10) are analytic functions of a, b C. Lemma 2.6 Let W, H,µ)be a complex abstract Wiener space, then for any ϕ W, we have W W e ϕw) dµw) = 1 = e ϕw) dµw), 2.11) W Re ϕw) 2 dµw) = Im ϕw) 2 dµw) = ϕ 2 H, 2.12) W and W ϕw) 2 dµw) = 2 ϕ 2 H. 2.13) More generally, if C is another complex Hilbert space and ϕ LW, C), then W ϕw) 2 C dµw) = 2 ϕ 2 H C. 2.14) Proof If u = Re ϕ, then ϕw) = uw) iuiw). Therefore by Eqs. 2.6), 2.7), and 2.10), W e ϕ dµ = e u iu M i dµ W = exp ) 1 2 qu, u) q u M i, u M i ) 2iq u, u M i )) = 1. Taking the complex conjugation of this identity shows W eϕw) dµw) = 1. Also using Lemma 2.5, wehave W W Re ϕw) 2 dµw) = qu, u) and Im ϕw) 2 dµw) = uiw) 2 dµw) = q u M i, u M i ) = q u, u). W

11 Square integrable holomorphic functions To evaluate qu, u), let{e k } k=1 be an orthonormal basis for H so that {e k, ie k } k=1 is an orthonormal basis for H Re, Re, H ). Then by Eq. 2.5), q u, u) = k=1 [ u e k ) 2 + u ie k ) 2] = To prove Eq. 2.14), apply [4, Eq. 2.13)] to find ϕ e k ) 2 = ϕ 2 H. k=1 W ϕw) 2 C dµw) = = 2 k=1 [ ] ϕe k ) 2 C + ϕie k) 2 C ϕe k ) 2 C = 2 ϕ 2 H C. k=1 Remark 2.7 Heat kernel interpretation of Lemma 2.6) The measure µ formally satisfies W f w)dµw) = ) e 2 1 H Re f 0), where HRe = j=1 2 e j and {e j } j=1 is an orthonormal basis for H Re. If f is holomorphic or anti-holomorphic, then f is harmonic and therefore W f w)dµw) = ) e 2 1 H Re f 0) = f 0). Applying this identity to f w) = e ϕw) or f w) = e ϕw) with ϕ W gives Eq. 2.11). If u W Re,wehave W u 2 w)dµw) = = 1 2 = e 1 2 H Re u 2) 0) = n=0 HRe u 2) 0) = 1 2 u ) 2 e j = u 2 HRe. j=1 1 2 n n H n! Re u 2) 0) j=1 2 e j u 2) 0) Eqs. 2.12) and 2.13) now follow easily from this identity.

12 B. K. Driver, M. Gordina 2.1 The structure of the projections Let i : H W be the inclusion map and i : W H be its transpose, i.e. i l := l i for all l W.Alsolet H := { h H :, h H Ran i ) H } 2.15) or in other words, h H is in H iff, h H H extends to a continuous linear functional on W. We will continue to denote the continuous extension of, h H to W by, h H.) Because H is a dense subspace of W, i is injective, and because i is injective, i has a dense range. Since h H, h H H is a conjugate linear isometric isomorphism, it follows from the above comments that H h, h H W is a conjugate linear isomorphism too, and that H is a dense subspace of H. Lemma 2.8 There is a one to one correspondence between ProjW ) see Notation 1.1) and the collection of finite rank orthogonal projections, P, on H such that P H H. Proof If P ProjW ) and u PW H, then, because P H is an orthogonal projection, we have Ph, u H = h, Pu H = h, u H for all h H. 2.16) Since P : W H is continuous, it follows that u H, i.e. PW H. Conversely, suppose that P : H H is a finite rank orthogonal projection such that PH H.Let{e j } n j=1 be an orthonormal basis for PH and l j W such that l j H =, e j H. Then we may extend P uniquely to a continuous operator from W to H still denoted by P) by letting Pw := n l j w)e j = j=1 n w, e j j=1 H e j for all w W. 2.17) From [4, Eq. 3.43], there exists C = CP) < such that Pw H C w W for all w W. 2.18) 3 Complex Heisenberg like groups In this section we review the infinite-dimensional Heisenberg like groups and Lie algebras which were introduced in [4, Section 3]. Notation 3.1 Let W, H,µ) be a complex abstract Wiener space, C be a complex finite dimensional inner product space, and ω : W W C be a continuous skew symmetric bilinear quadratic form on W. Further, let ω 0 := sup { ω w 1,w 2 ) C : w 1,w 2 W with w 1 W = w 2 W = 1 } 3.1)

13 Square integrable holomorphic functions be the uniform norm on ω which is finite by the assumed continuity of ω. Definition 3.2 Let g denote W C when thought of as a Lie algebra with the Lie bracket operation given by [A, a), B, b)] := 0,ωA, B)). 3.2) Let G = Gω) denote W C when thought of as a group with the multiplication law given by g 1 g 2 = g 1 + g [g 1, g 2 ] for any g 1, g 2 G 3.3) or equivalently by Eq. 1.1). It is easily verified that g is a Lie algebra and G is a group. The identity of G is the zero element, e := 0, 0). Notation 3.3 Let g CM denote H C when viewed as a Lie subalgebra of g and G CM denote H C when viewed as a subgroup of G = Gω). We will refer to g CM G CM ) as the Cameron Martin subalgebra subgroup) of g G). For explicit examples of such W, H, C,ω),see[4].) We equip G = g = W C with the Banach space norm w, c) g := w W + c C 3.4) and G CM = g CM = H C with the Hilbert space inner product, A, a), B, b) gcm := A, B H + a, b C. 3.5) The associate Hilbertian norm is given by A, a) gcm := A 2 H + a 2 C. 3.6) As was shown in [4, Lemma 3.3], these Banach space topologies on W C and H C make G and G CM into topological groups. Notation 3.4 Linear differentials) Suppose f : G C, is a Frechét smooth function. For g G and h, k g let f g)h := h f g) = d f g + th) dt 0 and f g) h k) := h k f g). Here and in the sequel a prime on a symbol will be used to denote its derivative or differential.)

14 B. K. Driver, M. Gordina As G itself is a vector space, the tangent space, T g G,toG at g is naturally isomorphic to G. Indeed, if v, g G, then we may define a tangent vector v g T g G by v g f = f g)v for all Frechét smooth functions f : G C. We will identify g with T e G and g CM with T e G CM. Recall that as sets g = G and g CM = G CM.Forg G, let l g : G G be the left translation by g.forh g,let h be the left invariant vector field on G such that hg) = h when g = e. More precisely, if σt) G is any smooth curve such that σ0) = e and σ0) = h e.g. σt) = th), then hg) =, l g h := d dt 0 g σt). 3.7) As usual, we view h as a first order differential operator acting on smooth functions, f : G C, by h f) g) = d f g σt)). 3.8) dt 0 The proof of the following easy proposition may be found in [4, Proposition 3.7]. Proposition 3.5 Let f : G C be a smooth function, h = A, a) g and g = w, c) G. Then hg) :=, l g h = A, a + 12 ) ω w, A) for any g = w, c) G 3.9) and, in particular, A, a) f g) = f g) A, a + 12 ) ωw, A). 3.10) If h, k g, then h k f k h f) = [h, k] f. 3.11) The one parameter group in G, e th, determined by h = A, a) g, is given by e th = th = ta, a). 4 Brownian motion and heat kernel measures This section will closely follow [4, Section 4] except for the introduction of a certain factor of 1/2 into the formalism which will simplify later formulas. Let {bt) = Bt), B 0 t))} t 0 be a Brownian motion on g = W C with the variance determined by E [ Re bs), h gcm Re bt), k gcm ] = 1 2 Re h, k g CM s t 4.1)

15 Square integrable holomorphic functions for all s, t [0, ), h = A, a), and k := C, c), where A, C H and a, c C. Recall the definition of H from Eq. 2.15).) Definition 4.1 The associated Brownian motion on G starting at e = 0, 0) G is defined to be the process gt) = Bt), B 0 t) t 0 ω B τ), dbτ)). 4.2) More generally, if h G,weletg h t) := h gt), the Brownian motion on G starting at h. Definition 4.2 Let B G be the Borel σ algebra on G and for any T > 0, let ν T : B G [0, 1] be the distribution of gt ). We will call ν T the heat kernel measure on G. To be more explicit, the measure ν T is the unique measure on G, B G ) such that ν T f ) := fdν T = E [ f gt ))] 4.3) G for all bounded measurable functions f : G C. Our next goal is to describe the generator of the process {g h t)} t 0. Definition 4.3 A function f : G C is said to be a cylinder function if it may be written as f = F π P for some P ProjW ) and some function F : G P C, where G P is defined as in Notation 1.1. We say that f is a holomorphic smooth) cylinder function if F : G P C is holomorphic smooth). We will denote the space of holomorphic analytic) cylinder functions by A. Proposition 4.4 Generator of g h ) If f : G C is a smooth cylinder function, let Lf := j=1 [ e j, 0 ) 2 ie + j, 0 ) 2 ] f + d j=1 [ )2 0, )2 ] 0, f j + if j f, 4.4) where {e j } j=1 and { f j} d j=1 are complex orthonormal bases for H,, H ) and C,, C ) respectively. Then L f is well defined, i.e. the sums in Eq. 4.4) are convergent and independent of the choice of bases. Moreover, for all h G, 1 4 Listhe generator for {g h t)} t 0. More precisely, Mt f := f g h t)) 1 t Lf g h τ)) dτ 4.5) 4 0 is a local martingale for any smooth cylinder function, f : G C.

16 B. K. Driver, M. Gordina Proof After bearing in mind the factor of 1/2 used in defining the Brownian motion bt) in Eq. 4.1), this proposition becomes a direct consequence of Proposition 3.29 and Theorem 4.4 of [4]. Indeed, the Brownian motions in this paper are equal in distribution to the Brownian motions used in [4] after making the time change, t t/2. It is this time change that is responsible for the 1/4 factor rather than 1/2) in Eq. 4.5). 4.1 Heat kernel quasi-invariance properties In this subsection, we are going to recall one of the key theorems from [4]. We first need a little more notation. Let C 1 CM denote the collection of C1 -paths, g :[0, 1] G CM. The length of g is defined as l GCM g) = 1 0 lg 1 s) g s) gcm ds. 4.6) As usual, the Riemannian distance between x, y G CM is defined as } d GCM x, y) = inf {l GCM g) : g CCM 1 g0) = x and g1) = y. 4.7) Let us also recall the definition of kω) from [4, Eq. 7.6]; k ω) = 1 2 sup ω, A) 2 HRe A HRe =1 C Re = sup ω, A) 2 H C ω 2 H H C >, 4.8) A H =1 wherein we have used [4, Lemma 3.17] in the second equality. It is known by Fernique s or Skhorohod s theorem that ω 2 2 = ω 2 H H C <,see[4, Proposition 3.14] for details. Theorem 4.5 For all h G CM and T > 0, the measures, ν T l 1 h and ν T r 1 h absolutely continuous relative to ν T. Let Zh l := dν T lh 1 ) dν T and Zh r := dν T rh 1 ) dν T respective Randon Nikodym derivatives, kω) is given in Eq. 4.8), and,are be the ct) := t e t 1 for any t R with the convention that c0) = 1. Then for all 1 p <, Zh l and Zr h L p ν T ) and satisfy the estimate where =lor =r. are both in ) Z L c k ω) T/2) p 1) h p ν T ) exp dg 2 T CM e, h), 4.9)

17 Square integrable holomorphic functions Proof This is [4, Theorem 8.1] also see [4, Corollary 7.3]) with the modification that T should be replaced by T/2. This is again due to the fact that the Brownian motions in this paper are equal in distribution to those in [4] after making the time change, t t/2. It might be enlightening to note here that we call G CM the Cameron Martin subgroup not only because it is constructed from the Cameron Martin subspace, H, but also because it has properties similar to H. In particular, the following statement holds. Proposition 4.6 The heat kernel measure does not charge G CM, i.e. ν T G CM ) = 0. Proof Note that for a bounded measurable function f : W C C that depends only on the the first component in W C, that is, f w, c) = f w) we have f w)dν T w, c) = E[ f BT ))] = f w)dµ T w). G Note that for the projection π : W C W, πw,c) = w we have π ν T = µ T and therefore ν T G CM ) = ν T π 1 H)) = π ν T H) = µ T H) = 0. For later purposes, we would like to introduce the heat operator, S T := e TL/4, acting on L p G,ν T ). To motivate our definition, suppose f : G C is a smooth cylinder function and suppose we can make sense of ut, y) = e T t)l/4 f )y). Then working formally, by Itô s formula, Eq. 4.5), and the left invariance of L, we expect ut, hgt)) to be a martingale for 0 t T and in particular, E[ f hgt ))] =E[uT, hgt ))] =E[u0, hg0))] =e TL/4 f )h). 4.10) Definition 4.7 For T > 0, p 1, ], and f L p G,ν T ),lets T f : G CM C be defined by S T f )h) = f h g)dν T g) = E[ f hgt ))]. 4.11) G The following result is a simple corollary of Theorem 4.5 and Hölder s inequality along with the observation that p 1 = p 1) 1, where p is the conjugate exponent to p 1, ]. Corollary 4.8 If p > 1, T > 0, f L p G,ν T ),h G CM, and W Z l h L ν T ) := 1 q< L q ν T ) 4.12)

18 B. K. Driver, M. Gordina is as in Theorem 4.5, then S T f is well defined and may be computed as S T f ) h) = G f g)z l h g)dν T g). 4.13) Moreover, we have the following pointwise Gaussian bounds ) c k ω) T/2) S T f ) h) f L p ν T ) exp dg 2 T p 1) CM e, h). 4.14) We will see later that when f is holomorphic and p = 2, the above estimate in Eq. 4.14) may be improved to ) 1 S T f ) h) f L 2 ν T ) exp 2T d2 G CM e, h) for any h G CM. 4.15) This bound is a variant of Bargmann s pointwise bounds see [1, Eq. 1.7)] and [6, Eq. 5.4)]). Lemma 4.9 Let T > 0 and suppose that f : G C is a continuous and in L p ν T ) for some p > 1. Then S T f : G CM C is continuous. Proof For q 1, p) and h G CM we have by Hölder s inequality and Theorem 4.5 that ) E f hgt )) q = v T f q Zh l f q/p L p ν T ) Zh l L p n p q ν T ) f q/p c k ω) T/2) q L p ν T ) exp dg 2 T p q) CM e, h) 4.16) Hence if {h n } n=1 G CM is a sequence converging to h G CM, it follows that sup E f h n gt )) q f q/p c k ω) T/2) q L n p ν T ) exp T p q) ) sup dg 2 CM e, h n ) <, n 4.17) which implies that { f h n gt ))} n=1 is uniformly integrable. Since by continuity of f, lim n f h n gt )) = f hgt )), we may pass to the limit under the expectation to find lim S T f h n ) = lim E f h ngt )) = E [ f hgt ))] = S T f h). n n

19 Square integrable holomorphic functions 4.2 Finite dimensional approximations Notation 4.10 For each P ProjW ), letg P t) denote the G P -valued Brownian motion defined by g P t) = PBt), B 0 t) t 0 ω PBτ), dpbτ)). 4.18) Also, for any t > 0,letνt P := Lawg P t)) be the corresponding heat kernel measure on G P. The following Theorem is a restatement of [4, Theorem 4.16]. Theorem 4.11 Integrated heat kernel bounds) Suppose that ρ 2 : G [0, ) be defined as ρ 2 w, c) := w 2 W + c C. 4.19) Then there exists a δ>0such that for all ε 0,δ)and T > 0 [ ] sup E e T ε ρ2 g P T )) < and e T ε ρ2 g) dν T g) <. 4.20) P ProjW ) Proposition 4.12 Let P n ProjW ) such that P n H I H on H and let g n T ) := g Pn T ). Further suppose that δ > 0 is as in Theorem 4.11, p [1, ), and f : G C is a continuous function such that G f g) Ce ερ2 g)/pt) for all g G 4.21) for some ε 0,δ). Then f L p ν T ) and for all h G we have and lim E f hgt )) f hg nt )) p = 0, 4.22) n lim E f gt )h) f g nt )h) p = ) n Proof If q p, ) is sufficiently close to p so that qp 1 ε<δ, then sup n E f g n T )) q C q sup E [e ] p 1 qερ 2 g)/t n which is finite by Theorem This shows that { f g n T )) p } n=1 is uniformly integrable. As a consequence of [4, Lemma 4.7] and the continuity of f, we also know

20 B. K. Driver, M. Gordina that f g n T )) f gt )) in probability as n. Thus we have shown Eqs. 4.22) and 4.23) hold when h = e = 0. Now suppose that g = w, c) and h = A, a) are in G. Then for all α>0, ρ 2 gh) = w + A 2 W + a + c ωw, A) C w 2 W + A 2 W + 2 A W w W + a C + c C ω w, A) C ρ 2 g) + ρ 2 h) + C A W w W ρ 2 g) + ρ 2 h) + C [ ] α 1 A 2 W 2 + α w 2 W 1 + Cα ) ρ 2 g) C ) ρ 2 h), 4.24) 2 2α where C := ω 0). AsEq.4.24) is invariant under interchanging g and h the same bound also hold for ρ 2 hg). By choosing α>0sufficiently small so that 1 + Cα 2 )ε < δ, we see that g f gh) and g f hg) satisfy the same type of bound as in Eq. 4.21)for g f g). Therefore, by the first paragraph, we have now verified Eqs. 4.22) and 4.23) hold for any h G. 5 Holomorphic functions on G and G CM We will begin with a short summary of the results about holomorphic functions on Banach spaces that will be needed in this paper. 5.1 Holomorphic functions on Banach spaces Let X and Y be two complex Banach space and for a X and δ>0let B X a,δ):= {x X : x a X <δ} be the open ball in X with center a and radius δ. Definition 5.1 Hille and Phillips [17, Definition , p. 112.]) Let D be an open subset of X. A function u : D Y is said to be holomorphic or analytic) if the following two conditions hold. 1) u is locally bounded, namely for all a D there exists an r a > 0 such that M a := sup { ux) Y : x B X a, r a )} <. 2) The function u is complex Gâteaux differentiable on D, i.e. for each a D and h X, the function λ ua + λh) is complex differentiable at λ = 0 C. Holomorphic and analytic will be considered to be synonymous terms for the purposes of this paper.)

21 Square integrable holomorphic functions The next theorem gathers together a number of basic properties of holomorphic functions which may be found in [17]. Also see [15].) One of the key ingredients to all of these results is Hartog s theorem, see [17, Theorem ]. Theorem 5.2 If u : D Y is holomorphic, then there exists a function u : D HomX, Y ), the space of bounded complex linear operators from X to Y, satisfying 1) If a D, x B X a, r a /2), and h B X 0, r a /2), then u x + h) u x) u x) h 4M a Y r a r a 2 h X ) h 2 X. 5.1) In particular, u is continuous and Frechét differentiable on D. 2) The function u : D HomX, Y ) is holomorphic. Remark 5.3 By applying Theorem 5.2 repeatedly, it follows that any holomorphic function, u : D Y is Frechét differentiable to all orders and each of the Frechét differentials are again holomorphic functions on D. Proof By [17, Theorem on p. 766.], for each a D there is a linear operator, u a) : X Y such that dua + λh)/dλ λ=0 = u a)h. The Cauchy estimate in Theorem with n = 1) of [17] implies that if a D, x B X a, r a /2) and h B X 0, r a /2) so that x + h B X a, r a )), then u x)h Y M a. It follows from this estimate that { u sup x) } HomX,Y ) : x B X a, r a /2) 2M a /r a. 5.2) and hence that u : D HomX, Y ) is a locally bounded function. The estimate in Eq. 5.1) appears in the proof of the Theorem in [17] which completes the proof of item 1. To prove item 2. we must show u is Gâteaux differentiable on D. We will in fact show more, namely, that u is Frechét differentiable on D. Givenh X, let F h : D Y be defined by F h x) := u x)h. According to [17, Theorem ], F h is holomorphic on D as well. Moreover, if a D and x Ba, r a /2) we have by Eq. 5.2) that F h x) Y 2M a h X /r a. So applying the estimate in Eq. 5.1)toF h, we learn that F h x + k) F h x) F h x) k Y 4 2M a h X /r a ) r a2 ra2 2 k X ) k 2 X 5.3) for x Ba, r a /4) and k X < r a /4, where F h x)k = d dλ 0F h x + λk) = d ) dλ 0u x + λk) h =: δ 2 u x; h, k).

22 B. K. Driver, M. Gordina Againby[17, Theorem ], for each fixed x D, δ 2 u)x; h, k) is a continuous symmetric bilinear form in h, k) X X. Taking the supremum of Eq. 5.3) over those h X with h X = 1, we may conclude that u x + k) u x) δ 2 u x;, k) HomX,Y ) = sup F h x + k) F h x) F h x)k Y h X =1 4 2M a/r a ) r a2 ra2 ) k 2 X 2 k. X This estimate shows u is Frechét differentiable with u x) HomX, HomX, Y )) being given by u x)k = δ 2 u)x;, k) HomX, Y ) for all k X and x D. 5.2 Holomorphic functions on G and G CM For the purposes of this section, let G 0 = G and g 0 = g or G 0 = G CM and g 0 = g CM. Also for g, h g, let as usual) ad g h := [g, h]. Lemma 5.4 For each g G 0,l g : G 0 G 0 is holomorphic in the g0 topology. Moreover, a function u : G 0 C defined in a neighborhood of g G 0 is Gâteaux Frechét) differentiable at g iff u l g is Gâteaux Frechét) differentiable at 0. In addition, if u is Frechét differentiable at g, then ) u lg 0) h = u g) h + 12 ) [g, h]. 5.4) See [13, Theorem 5.7] for an analogous result in the context of path groups.) Proof Since l g h) = gh = g + h [g, h] =g + Id g ad g ) h, it is easy to see that l g is holomorphic and l g is the constant function equal to Id g ad g Endg 0 ).Usingadg 2 = 0 or the fact that l 1 g = l g 1, we see that l g is invertible and that l g 1 = Id g0 + 1 ) 1 2 ad g = Id 1 g0 2 ad g. These observations along with the chain rule imply the Frechét differentiability statements of the lemma and the identity in Eq. 5.4). If u is Gâteaux differentiable at g, h g 0, and k := h [g, h], then d dλ 0u l g λh) = d dλ 0u g λh)) = d dλ 0ug + λk)

23 Square integrable holomorphic functions and the existence of dλ d 0ug+λk) implies the existence of dλ d 0u l g λh). Conversely, if u l g is Gâteaux differentiable at 0, h g 0, and k := h 1 2 [g, h] = Id g ad g) 1 h, then l g λk) = g + λ Id g0 + 1 ) 2 ad g k = g + λh. So the existence of dλ d 0u l g )λk) implies the existence of dλ d 0ug + λh). Corollary 5.5 A function u : G 0 C is holomorphic iff it is locally bounded and h uge h ) = ug h) is Gâteaux Frechét) differentiable at 0 for all g G 0. Moreover, if u is holomorphic and h g 0, then hu) g) = d dλ 0u ge λh) = u g) h + [g, h]) is holomorphic as well. Notation 5.6 The space of globally defined holomorphic functions on G and G CM will be denoted by HG) and HG CM ) respectively. Notice that the space A of holomorphic cylinder functions as described in Definition 4.3 is contained in HG). Also observe that a simple induction argument using Corollary 5.5 allows us to conclude that h 1... h n u HG 0 ) for all u HG 0 ) and h 1,...,h n g 0. Proposition 5.7 If f HG) and h g, then ĩh f = i h f,ĩh f = i h f, [ ĩh ) 2 + h 2] f = 0, and 5.5) h 2 + ĩh 2) f 2 = 4 h f ) Proof The first assertions are directly related to the definition of f being holomorphic. Using the identity ĩh f = i h f twice implies Eq. 5.5). Equation 5.6) is a consequence of summing the following two identities and h 2 f 2 = h f f ) = h 2 f f + f h 2 f + 2 h f h f ĩh 2 f 2 = ĩh f f ) = ĩh 2 f f + f ĩh 2 f + 2ĩh f ĩh f = h 2 f f f h 2 f + 2 h f h f, and using h f = h f.

24 B. K. Driver, M. Gordina Corollary 5.8 Let L be as in Proposition 4.4. Suppose that f : G C is a holomorphic cylinder function i.e. f A), then L f = 0 and L f 2 = h f 2, 5.7) h Ɣ where Ɣ is an orthonormal basis for g CM of the form Ɣ = Ɣ e Ɣ f = { e j, 0) } j=1 { 0, f j ) } d j=1 5.8) with {e j } j=1 and { f j} d j=1 being complex orthonormal bases for H and C respectively. Proof These assertions follow directly form Eqs. 4.4), 5.5), and 5.6). Formally, if f : G C is a holomorphic function, then e TL/4 f = f and therefore we should expect S T f = f GCM where S T is defined in Definition 4.7. Theorem 5.9 below is a precise version of this heuristic. Theorem 5.9 Suppose p 1, ) and f : G C is a continuous function such that f GCM HG CM ) and there exists P n ProjW ) such that P n H I H, then If we further assume that f L p ν T ) sup f n L p sup f n L p G Pn,ν Pn T G Pn,ν Pn T ). 5.9) ) <, 5.10) then f L p ν T ),S T f = f GCM, and f satisfies the Gaussian bounds ) c kω)t/2) f h) f L p ν T ) exp T p 1) d2 G CM e, h) for any h G CM. 5.11) Proof According to [4, Lemma 4.7], by passing to a subsequence if necessary, we may assume that g Pn T ) gt ) almost surely. Hence an application of Fatou s lemma implies Eq. 5.9). In particular, if we assume Eq. 5.10) holds, then f L p ν T ) and so S T f is well defined. Now suppose that P ProjW ) and h G P. Working exactly as in the proof of Lemma 4.9, we find for any q 1, p) that ) E f hg P T )) q f q/p c kp ω)t/2) q L p G P,νT P ) exp dg 2 T q p) P e, h), 5.12) where d G P, ) is the Riemannian distance on G P and see [4, Eq. 5.13)]), k P ω) := 1 } { ω 2 sup, A) 2 PH) C : A PH, A PH = )

25 Square integrable holomorphic functions Observe that k P ω) kω) and therefore, as c is a decreasing function, ckω)) ck P ω)).letm N be given and h G Pm. Then for n m we have from Eq. 5.12) that c kpn ω) T/2 ) ) q E f hgpn T ) ) q f q/p f q/p L p G Pn,ν Pn T L p G Pn,ν Pn T ) exp T q p) c k ω) T/2) q ) exp T q p) d 2 G Pn e, h) ) dg 2 Pm e, h) wherein in the last inequality we have used ckω)) ck P ω)) and the fact that d 2 G Pn e, h) is decreasing in n m. Hence it follows that sup n m E f hg Pn T )) q < and thus that { f hg Pn T ))} n m is uniformly integrable. Therefore, S T f h) = E f hgt )) = lim E f hg Pn T ) ) = lim n n G Pn f hx)dν P n T x). 5.14) On the other hand by [4, Lemma 4.8] with T replaced by T/2 because of our normalization in Eq. 4.1)), ν P n T is the heat kernel measure on G P n based at e G Pn, i.e. ν P n T dx) = p P n T/2 e, x)dx, where dx is the Riemannian volume measure equal to a Haar measure) on G Pn and p P n T x, y) is the heat kernel on G P n. Since f G Pn is holomorphic, the previous observations allow us to apply [5, Proposition 1.8] to conclude that P f hx)dν n x) = f h) for all n m. 5.15) G Pn T As m N was arbitrary, combining Eqs. 5.14) and 5.15)impliesthatS T f h) = f h) for all h G 0 := m N G Pm. Recall from Lemma 4.9 that S T f : G CM C is continuous and from the proof of [4, Theorem 8.1] that G 0 is a dense subgroup of G CM. Therefore we may conclude that in fact S T f h) = f h) for all h G CM.The Gaussian bound now follows immediately from Corollary 4.8. Corollary 5.10 Suppose that δ > 0 is as in Theorem 4.11 and f : G C is a continuous function such that f GCM is holomorphic and f Ce ερ2 /pt) for some ε [0,δ). Then f L p ν T ),S T f = f, and the Gaussian bounds in Eq. 5.11) hold. Proof By Theorem 4.11, the given function f verifies Eq. 5.10) for any choice of {P n } n=1 ProjW ) with P n H P strongly as n. Hence Theorem 5.9 is applicable. As a simple consequence of Corollary 5.10, we know that P L p ν T ) see Definition 1.6) and that S T p)h) = ph) for all h G CM and p P. Notation 5.11 For T > 0 and 1 p <, leta p T and H p T G) denote the L p ν T ) closure of A L p ν T ) and P, where A and P denote the holomorphic cylinder

26 B. K. Driver, M. Gordina functions see Definition 4.3) and holomorphic cylinder polynomials on G respectively. Theorem 5.12 For all T > 0 and p 1, ), we have S T H p T G)) HG CM). Proof Let f H p T G) and p n h G CM, then by Corollary 4.8 P such that lim n f p n L p ν T ) = 0. If S T f h) p n h) = S T f p n )h) ) c k ω) T/2) f p n L p ν T ) exp dg 2 T p 1) CM e, h). This shows that S T f is the limit of p n GCM HG CM ) with the limit being uniform over any bounded subset of h s contained in G CM. This is sufficient to show that S T f HG CM ) via an application of [17, Theorem ]. Remark 5.13 It seems reasonable to conjecture that A 2 T = H2 T G), nevertheless we do not know if these two spaces are equal! We also do not know if S T f = f for every f A L 2 ν T ). However, Theorem 5.9 does show that S T f = f for all f A P ProjW ) L p νt P) with L p νt P ) norms of f being bounded. 6 The Taylor isomorphism theorem The main purpose of this section is to prove the Taylor isomorphism Theorem 1.5 or Theorem 6.10). We begin with the formal development of the algebraic setup. In what follows below for a vector space V we will denote the algebraic dual to V by V.IfV happens to be a normed space, we will let V denote the topological dual of V. 6.1 A non-commutative Fock space Notation 6.1 For n N let gcm n denote the n fold algebraic tensor product of g CM with itself, and by convention let gcm 0 := C. Alsolet T := T g CM ) = C g CM g 2 CM g 3 CM... be the algebraic tensor algebra over g CM, T be its algebraic dual, and J be the twosided ideal in T generated by the elements in Eq. 1.3). The backwards annihilator of Jis J 0 ={α T : αj) = 0}. 6.1) For any α T and n N {0},weletα n := α g n g n CM CM ). After the next definition we will be able to give numerous examples of elements in J 0.

27 Square integrable holomorphic functions Definition 6.2 Left differentials) For f HG CM ), n N {0}, and g G CM, define fˆ n g) := D n f g) gcm n ) by ) D 0 f g) = f g) and D n ) f g), h 1 h n = h 1... h n f g) 6.2) for all and h 1,...,h n g 0, where h f is given as in Eq. 3.8) oreq.3.10). We will write Df for D 1 f and f ˆg) to be the element of Tg CM ) determined by f ˆ g), β = f ˆn g), β for all β gcm n and n N ) Example 6.3 As a consequence of Eq. 3.11), f ˆg) J 0 for all f HG CM ) and g G CM. In order to put norms on J 0, let us equip gcm n with the usual inner product determined by h 1 h n, k 1 k n g n CM = For n = 0welet z,w g 0 CM, g n CM associated norm on g n n j=1 h j, k j g CM for any h i, k j g CM. 6.4) := z w for all z,w g 0 CM = C. The inner product induces a dual inner product on g n CM ) which we will denote by, n.the CM ) will be denoted by n. We extend n to all of gcm n ) by setting β n = if β gcm n ) \ gcm n ).IfƔ is any orthonormal basis for g CM, then β n may be computed using β 2 gcm n := β, h 1 h n ) h 1,...,h n Ɣ Definition 6.4 Non-commutative Fock space) Given T > 0 and α J 0 g CM ),let α 2 J 0 T g CM) := n=0 T n n! α n 2 n. 6.6) Further let J 0 T g CM) := {α J 0 g CM ) : α 2J 0T gcm) < }. 6.7) The space, J 0 T g CM), is then a Hilbert space when equipped with the inner product α, β J 0 T g CM ) = n=0 T n n! α n,β n n for any α, β J 0 T g CM). 6.8)

28 B. K. Driver, M. Gordina 6.2 The Taylor isomorphism Lemma 6.5 Let f HG CM ) and T > 0 and suppose that {P n } n=1 ProjW ) is a sequence such that P n gcm I gcm as n. Then lim n f ˆ e) J 0 T g Pn ) = f ˆ e) = f J 0 T g H CM ) 2 T G CM) = lim f n L 2 ) G Pn,ν Pn, T 6.9) where H 2 T G CM ) is defined in Eq. 1.4). Proof By Theorem 5.1 of [6], for all P ProjW ), f L 2 G P,νT P ) = f ˆ e), 6.10) J 0 T g P ) where f ˆ e) 2 = JT 0g P ) n=0 T n n! {h j} n j=1 Ɣ P ˆ f e), h 1 h n ) and Ɣ P is an orthonormal basis for g P. In particular, it follows that f H 2 T G CM ) = sup P ProjW ) f ˆ e) J 0 T g P ) 6.12) and hence we must now show sup P ProjW ) f ˆ e) = ˆ f e). 6.13) J 0 T g P ) J 0 T g CM ) If Ɣ is an orthonormal basis for g CM containing Ɣ P,itfollowsthat f ˆ e) 2 JT 0g P ) n=0 T n n! {h j} n j=1 Ɣ f ˆ 2 e), h 1 h n = f ˆ e) 2, JT 0g CM)

29 Square integrable holomorphic functions which shows that sup P ProjW ) f ˆe) J 0 T g P ) f ˆe) J 0 T g CM ). We may choose orthonormal bases, Ɣ Pn,forg Pn such that Ɣ Pn Ɣ as n. Then it is easy to show that lim f n L 2 G Pn,ν Pn T ) = lim ˆ f e) n J 0 T g Pn ) T n = lim n n! = n=0 n=0 T n n! = f ˆ e) {h j} n j=1 Ɣ J 0 T g CM ) {h j} n j=1 Ɣ Pn ˆ f e), h 1 h n 2 ˆ f e), h 1 h n 2 from which it follows that sup P ProjW ) ˆ f e) J 0 T g P ) ˆ f e) J 0 T tg CM ). For the next corollary, recall that P and P CM denote the spaces of holomorphic cylinder polynomials on G and G CM respectively, see Definition 1.6 and Eq. 1.7). Corollary 6.6 If f : G C is a continuous function satisfying the bounds in Proposition 4.12 with p = 2, then f GCM H 2 T G CM) and f ˆe) JT 0g CM). In particular, for all T > 0, P CM H 2 T G CM) and for any p P, ˆpe) JT 0g CM). This shows that H 2 T G CM) and JT 0g CM) are non-trivial spaces. Definition 6.7 For each T > 0, the Taylor map is the linear map, T T : H 2 T G CM) JT 0g CM), defined by T T f := f ˆe). Corollary 6.8 The Taylor map, T T : H 2 T G CM) JT 0g CM), is injective. Moreover, the function H 2 T G CM ) is a norm on H2 T G CM) which is induced by the inner product on H 2 T G CM) defined by u,v H 2 T G CM ) := û e), ˆv e) J 0 T g CM) for any u,v H 2 T G CM). 6.14) Proof If f ˆe) = 0, then f H 2 T G CM ) = 0 which then implies that f G P 0for all P ProjW ). As f : G CM C is continuous and P ProjW ) G P is dense in G CM see the end of the proof of Theorem 5.9), it follows that f 0. Hence we have shown T T injective. Since J 0 T g CM ) is a Hilbert norm and, by Lemma 6.9, f H 2 T G CM ) = T T f J 0 T g CM ),itfollowsthat H 2 T G CM) is the norm on H2 T G CM) induced by the inner product defined in Eq. 6.14). Our next goal is to show that the Taylor map, T T, is surjective. The following lemma motivates the construction of the inverse of the Taylor map.

30 B. K. Driver, M. Gordina Lemma 6.9 For every f HG CM ), f g) = n=0 1 f ˆn e), g n for any g G CM, 6.15) n! where the above sum is absolutely convergent. By convention, g 0 = 1 C. Fora more general version of this Lemma, see Proposition 5.1 in [3].) Proof The function uz) := f zg) is a holomorphic function of z C. Therefore, f g) = u1) = n=0 1 n! un) 0) and the above sum is absolutely convergent. In fact, one easily sees that for all R > 0 there exists CR) < such that 1 n! un) 0) CR)R n for all n N. The proof is now completed upon observing ) d n ) d n u n) 0) = ut) t=0 = f tg) t=0 dt dt ) d n = f e tg ) t=0 = g n f ) e) = f ˆn e), g n. dt The next theorem is a more precise version of Theorem 1.5. Theorem 6.10 Taylor isomorphism theorem) For all T > 0, the space H 2 T G CM) equipped with the inner product, H 2 T G CM ) is a Hilbert space, T H2 T G CM)) JT 0g CM), and T T := T H 2 T G CM ) : H2 T G CM) JT 0g CM) is a unitary transformation. Proof Given Corollary 6.8, it only remains to prove T T is surjective. So let α JT 0g CM). By Lemma 6.9, if f = T 1 α exists it must be given by f g) := n=0 T 1 αn, g n for any g G CM. 6.16) n! We now have to check that the sum is convergent, the resulting function f is in HG CM ), and f ˆe) = α. Once this is done, we may apply Lemma 6.5 to conclude that f H 2 T G CM ) = α JT 0g < and hence we will have shown that CM) f H 2 T G CM) and T T f = α. For each n N {0}, the function u n g) := 1 n! α n, g n is a continuous complex n linear form in g G CM and therefore holomorphic. Since α n, g n α n n g n g CM, then for R > 0 sup { u n g) : g gcm R } α n n R n.