THE TWO-PARAMETER SEGAL-BARGMANN TRANSFORM: UNITARITY AND UNIQUENESS

AMBAR N. SENGUPTA

Abstract. A proof is presented of the fact that the two-parameter Segal-Bargmann transform is a unitary isomorphism. The extent to which S_{s,t} is specified by its fundamental properties is determined exactly.

1. Introduction

In the paper [5], Hall has generalized the classical Segal-Bargmann transform and the Hall transform [4] to a unitary transform S_{s,t}, dependent on two positive parameters s and t satisfying s > t/2, in the context of Lie groups of compact type. The transform S_{s,t} maps an L^{2}– space unitarily onto a space of holomorphic square-integrable functions over a complex space. In this article we deal with S_{s,t} in the context of linear spaces, and we

• present a different proof of the fact that S_{s,t} is a unitary isomorphism

• determine the exact extent to which S_{s,t} is specified by its fundamental properties (unitarity, naturality, and a factorization property)

This paper is essentially self-contained.

In the elegant paper [3], Driver and Hall have studied the transform S_{s,t} on the infinite-dimensional Wiener space of paths on a Lie algebra and have deduced from it the corresponding unitary transform on a compact Lie group. The case s = t, for linear spaces, corresponds to the classical Segal-Bargmann transform ([7] and [2]). In [5] Hall has shown that there is a special string of transformations which can be used to reduce the two-parameter case for finite-dimensional linear spaces to the classical one-parameter case. The reduction does not work in the infinite-dimensional case, nor for groups. Thus we may justifiably view the transforms S_{s,t} as constituting a larger family of transforms of which the classical case is a special instance. The approach in [3], and one of the approaches used in [5], uses Hermite polynomials to prove unitarity of S_{s,t}.

Date: July 10, 1999.

In this article we present a direct proof of the fact that S_{s,t} is a unitary isomorphism, without using Hermite polynomials or using a reduction to the s = t case. We also determine the extent of “rigidity” of the unitary transform S_{s,t}, i.e. we determine the groups of unitary isomorphisms on the domain and on the range of S_{s,t} which, when composed with S_{s,t}, result in transforms which have the same fundamental properties as S_{s,t}.

In the last section we recast several of the results in category-theoretic language, which permits clean and efficient formulations of the results.

2. Results

Let s, t be two positive real numbers with s > t/2, and let

r = 2s − t and ρ = \frac{s − t}{s} (2.1)

The following relations are useful

1 − ρ = \frac{t}{s}, 1 + ρ = \frac{r}{s}, and \sqrt{rt} = s(1 − ρ^{2})^{1/2} (2.2)

Note also that the conditions s > t/2 > 0 imply that −1 < ρ < 1.

Let V be a real vector space, equipped with an inner-product, of finite (positive) dimension dim V. The inner-product of x, y ∈ V will be denoted xy (thus x^{2} is the inner product of x with itself). Let P_{s} be the Gaussian probability measure on V specified by

dP_{s}(x) = (2πs)^{− dim V/2}e^{− \frac{x^{2}}{2s}} dx (2.3)

where dx denotes the lebesgue measure on V.

Next let V_{c} be the complexification of V, and let M_{s,t} be the Gaussian measure on V_{c} given by

dM_{s,t}(z) = \frac{1}{(πr)^{dim V/2}(πt)^{dim V/2}} e^{− \frac{x^{2}}{r} − \frac{y^{2}}{t}} dz = \frac{1}{[πs(1 − ρ^{2})^{1/2}]^{dim V}} e^{− \frac{x^{2}}{r} − \frac{y^{2}}{t}} dz (2.4)

where z = x + iy, with x, y ∈ V, and dz = dxdy is lebesgue measure on V_{c} relative to the inner-product on V_{c} = V^{2}, taken for the moment as a real vector space with the inner-product induced from V.

The inner-product on V induces a complex-bilinear map V_{c} × V_{c} → C : (z, w) ↦ zw. If z = x + iy, with x, y ∈ V, then we have the conjugate

\overline{z} = x − iy

Let HL^{2}(V_{c}, M_{s,t}) be the set of holomorphic functions on V_{c} which are square-integrable with respect to M_{s,t}.

For any r > 0 we have the (analytically continued) Gaussian function

(2.5) ρ_{r}(z) = (2πr)^{− dim V/2}e^{−z^{2}/(2r)} for all z ∈ V_{c}

For f a function on V, and z ∈ V_{c} define

S_{s,t}^{V}f(z) = ∫_{V} (2πt)^{− dim V/2}e^{− \frac{(z−y)^{2}}{2t}} f(y) dy = ∫_{V} f(y)ρ_{t}(z − y) dy (2.6)

When the vector space V is clear from the context we will drop it from the notation S_{s,t}^{V} and simply write S_{s,t}.

It can be readily checked by Cauchy-Schwarz that if f ∈ L^{2}(P_{s}) then S_{s,t}f(z) is meaningful for every z ∈ V_{c}, and then it follows by Morera’s theorem that the function S_{s,t}f is holomorphic. Moreover, by Cauchy-Schwarz,

|S_{s,t}f(z)| ≤ C|f|_{L^{2}(P_{s})} (2.7)

Here C = |ρ_{t}(z − y)/ρ_{s}(y)|_{L^{2}(dP_{s}(y))} is finite because s > t/2.

The inner-product on V induces, naturally, a complex-bilinear form (z, w) ↦ zw on V_{c}. Each vector x ∈ V induces a linear functional

x̂ : V → R : y ↦ ⟨x, y⟩ (2.8)

and each z ∈ V_{c} induces a complex-linear functional

z̃: V_{c} → C : w ↦ zw (2.9)

Let U(1) = {λ ∈ C : |λ| = 1} be the group of unit-modulus complex numbers, forming a group under multiplication.

Let Aut(L^{2}(P_{s}))_{V} be the set of maps U : L^{2}(V, P_{s}) → L^{2}(V, P_{s}) satisfying:

1. \underline{Unitarity}: U is a unitary isomorphism

2. \underline{Functoriality}: if n is any positive integer ≤ dim V, there is a unitary isomorphism L^{2}(R^{n}, P_{s}) → L^{2}(R^{n}, P_{s}) : f ↦ f̂ such that for any u_{1}, ..., u_{n} orthonormal in V, the image of the function F_{f} = f(û_{1}, ..., û_{n}) under U is the function Ff̂ = f̂(û_{1}, ..., û_{n})

3. \underline{Factorizability}: if u_{1}, ..., u_{n+m} (with n, m being any positive integers) are orthonormal vectors in V and f ∈ L^{2}(R^{n}, P_{s}), g ∈

L^{2}(R^{m}, P_{s}), and F_{f,g} =^{def} f(û_{1}, ..., û_{n})g(û_{n+1}, ..., û_{n+m}) then UF_{f,g}

is of the form F_{f̂,ĝ}, where f̂ and ĝ are as given in Condition 2.

It is readily verified that Aut(L^{2}(P_{s}))_{V} is a group under composition.

Theorem 2.1. Suppose V is a finite-dimensional real inner-product space of dimension ≥ 2. Then there is an isomorphism of groups:

U(1) → Aut(L^{2}(P_{s}))_{V} : a ↦ U_{a}^{V} (2.10)

where U_{a}^{V} is the unique unitary isomorphism L^{2}(V, P_{s}) → L^{2}(V, P_{s}) specified by the requirement that its action on the exponential functions ex̂ be given as follows:

U_{a}^{V} ex̂ = e^{ax̂+ \frac{1}{2} (1−a^{2})sx^{2}} (2.11)

for every x ∈ V.

We have also the analogous result for HL^{2}(V_{c}, M_{s,t}). Let Aut(HL^{2}(M_{s,t}))_{V} be the set of maps W : HL^{2}(V_{c}, M_{s,t}) → HL^{2}(V_{c}, M_{s,t}) satisfying:

1. \underline{Unitarity}: W is a unitary isomorphism

2. \underline{Functoriality}: if n is any positive integer ≤ dim V, there is a unitary isomorphism HL^{2}(C^{n}, M_{s,t}) → HL^{2}(C^{n}, M_{s,t}) : f ↦ f̃ such that for any u_{1}, ..., u_{n} orthonormal in V, the image of the function F_{f} = f(ũ_{1}, ..., ũ_{n}) under W is the function Ff̃ = f̃(ũ_{1}, ..., ũ_{n})

3. \underline{Factorizability}: if u_{1}, ..., u_{n+m} (with n, m being any positive integers) are orthonormal vectors in V and f ∈ HL^{2}(C^{n}, M_{s,t}),

g ∈ HL^{2}(C^{m}, M_{s,t}), and F_{f,g} =^{def} f(ũ_{1}, ..., ũ_{n})g(ũ_{n+1}, ..., ũ_{n+m}) then

W F_{f,g} is of the form F_{f̃,g̃}, where f̃ and g̃ are as given in Condition 2.

Aut(HL^{2}(M_{s,t}))_{V} is a group under composition.

Theorem 2.2. Suppose V is a finite-dimensional real inner-product space of dimension ≥ 2. Then there is an isomorphism of groups:

U(1) → Aut(HL^{2}(V_{c}, M_{s,t}))_{V} : a ↦ W_{a}^{V} (2.12)

where W_{a}^{V} is the unique unitary isomorphism HL^{2}(V_{c}, M_{s,t}) → HL^{2}(V_{c}, M_{s,t}) specified by the requirement that its action on the exponential functions ew̃ be given as follows:

W_{a}^{V} ew̃ = e^{aw̃+ \frac{1}{2} (1−a^{2})(s−t)} (2.13)

for every w ∈ V.

Our main result shows that S_{s,t} is the unique, up to an action of U(1), unitary isomorphism L^{2}(V, P_{s}) → HL^{2}(V_{c}, M_{s,t}) having certain properties.

Theorem 2.3. Suppose V is a finite-dimensional real inner-product space of positive dimension.

The transform S_{s,t}^{V} : L^{2}(V, P_{s}) → HL^{2}(V_{c}, M_{s,t}) has the following properties:

1. \underline{Unitarity} : S_{s,t}^{V} is a unitary isomorphism of L^{2}(V, P_{s}) onto HL^{2}(V_{c}, M_{s,t})

2. \underline{Naturality} : if n is any positive integer ≤ dim V, and u_{1}, ..., u_{n} orthonormal in V, the image of the function F_{f} = f(û_{1}, ..., û_{n}) under S_{s,t}^{V} is the function (S^{R^{n}}f)(ũ_{1}, ..., ũ_{n})

3. \underline{Factorizability}: for any orthonormal vectors u_{1}, ..., u_{n}, u_{n+1}, ..., u_{n+m} ∈ V and any functions f ∈ L^{2}(R^{n}, P_{s}) and g ∈ L^{2}(R^{m}, P_{s}), the image of the product function f(û_{1}, ..., û_{n})g(û_{n+1}, ..., û_{n+m}) under S_{s,t}^{V} is the product function (S_{s,t}^{R^{n}}f)(ũ_{1}, ..., ũ_{n})(S_{s,t}^{R^{n}}g)(ũ_{n+1}, ..., ũ_{n+m}).

Conversely, assume that dim V ≥ 2, and suppose S̃_{s,t} : L^{2}(V, P_{s}) → HL^{2}(V_{c}, M_{s,t}) is a mapping satisfying the following properties:

1. S̃_{s,t} is a unitary isomorphism of L^{2}(V, P_{s}) onto HL^{2}(V_{c}, M_{s,t})

2. for each f ∈ L^{2}(R, P_{s}) there is a function f̃ on C such that for any unit vector u ∈ V, S̃_{s,t}[f(û)] = f̃(ũ)

3. for any orthonormal vectors u_{1}, u_{2} ∈ V and any functions f, g ∈ L^{2}(R, P_{s}), the operator S̃_{s,t} carries the function f(û_{1})g(û_{2}) to a function of the form f̃(ũ_{1})g̃(ũ_{2}), where f̃ and g̃ are as in Condition 2.

Then there is a unique a ∈ U(1) such that:

S̃_{s,t} = W_{a}^{V} ∘ S_{s,t}^{V} = S_{s,t}^{V} ∘ U_{a}^{V} (2.14)

The fact that S_{s,t} is a unitary isomorphism was first proved by Hall in [5].

Perhaps, in the preceding results, the first two hypotheses imply the third.

We shall also prove that the transform S_{s,t} is a meaningful unitary isomorphism even if V is infinite-dimensional.

In the last section we shall give a reformulation of several of our results in a category-theoretic framework.

Two kernel functions R and K, play an important role in proving unitarity of S_{s,t}. These functions are defined as follows. For x ∈ V and w ∈ V_{c} we set

R(x, w) = (1 − ρ)^{− dim V/2} exp [−\frac{1}{2t}\{ρx^{2} + w^{2} − 2xw\}] = \frac{ρ_{t}(x−w)}{ρ_{s}(x)} (2.15)

and, for z, w ∈ V_{c} we set

K(z, w) = (1 − ρ^{2})^{− dim V/2} exp [−\frac{ρ}{2(1 − ρ^{2})s}(z^{2} + w^{2}) + \frac{1}{(1 − ρ^{2})s}zw] (2.16)

which can also be expressed as

K(z, w) = ∫_{V} \frac{ρ_{t}(z − x)ρ_{t}(w − x)}{ρ_{s}(x)} dx (2.17)

3. Proofs I

Let V be a finite-dimensional real vector space of positive dimension, equipped with an inner-product. The inner-product of v, w ∈ V will be denoted vw. This induces, naturally, a complex bilinear form (w, z) ↦ wz on the complexification V_{c} of V. We write a generic element of V_{c} as z = x + iy, where x, y ∈ V; then the conjugate z̄ is defined to be x − iy.

Let C(·, ·) be the holomorphic quadratic polynomial on V_{c} given by

C(z, w) = A(z^{2} + w^{2}) + 2Bzw (3.1)

where

A = −\frac{ρ}{2(1 − ρ^{2})s} and B = \frac{1}{2(1 − ρ^{2})s} (3.2)

where r, s, t, ρ are as in equations (2.1) and (2.2).

We begin with two observations which may be proven by simple algebra. Recall the functions R and K defined in equations (2.15) and (2.16).

Lemma 3.1. For every z, w ∈ V_{c},

C(z, w) = \frac{1}{4r}(z + w)^{2} − \frac{1}{4t}(z − w)^{2} (3.3)

and

C(z, z̄) = \frac{x^{2}}{r} + \frac{y^{2}}{t} (3.4)

where z = x + iy, with x, y ∈ V. Thus

K(z, w) = (1 − ρ^{2})^{− dim V/2}e^{C(z,w)}, (3.5)

K(z, z̄) = (1 − ρ^{2})^{− dim V/2} exp [\frac{x^{2}}{r} + \frac{y^{2}}{t} ] = (1 − ρ^{2})^{− dim V/2}e^{C(z,z̄)} (3.6)

and

dM_{s,t}(z) = [πs(1 − ρ^{2})^{1/2}]^{− dim V} e^{−C(z,z̄)} dz (3.7)

The following simple algebraic observation will be useful in proving that the integral operator with kernel K(z, w) yields the “restriction functional” on holomorphic functions.

Lemma 3.2. For z, w ∈ V_{c},

C(z, w) + C(z̄, w̄) = C(z, z̄) + C(w, w̄) − 2B|w − z̄|^{2} (3.8)

and

C(z, w̄ + z̄) − C(w + z, w̄ + z̄) = C(z, z̄) − C(z̄, w + z) − 2B|w|^{2} (3.9)

Equation (3.9) follows by writing w̄ + z̄ in place of w in (3.8).

Next we check that K(z, ·) is square-integrable with respect to the measure M_{s,t}.

Lemma 3.3. For every z ∈ V_{c}, the function K(z, ·) belongs to L^{2}(V_{c}, M_{s,t}).

Proof. From the expressions for K(z, w) and M_{s,t}(w) given in (3.6) and (3.7) we see that the integral ∫_{V_{c}} |K(z, w)|^{2} dM_{s,t}(w) is a finite constant times

∫_{V_{c}} e^{C(z,w)+C(z̄,w̄)−C(w,w̄)} dw (3.10)

where dw is Lebesgue integration over V_{c}. By Equation (3.8), the integrand is constant times e^{−2B|w−z̄|^{2}}, and so, since B > 0, this integral is convergent. (The exact value of the L^{2}–norm of K(z, ·) can be determined easily this way, but we shall see what its value is in a more elegant way below.)

Now we can prove the fact that K(z, w) gives the integral kernel for the evaluation map f ↦ f(z) for holomorphic functions f in L^{2}(M_{s,t}).

Proposition 3.4. For any f ∈ HL^{2}(V_{c}, M_{s,t}), and any z ∈ V_{c},

∫_{V_{c}} K(z, w̄)f(w) dM_{s,t}(w) = f(z) (3.11)

Proof. Since K(z, ·) and f are both in L^{2}(M_{s,t}), the function K(z, ·)f(·) is M_{s,t}–integrable. Then, using the expressions for K(z, w) and M_{s,t}(w) given in (3.6) and (3.7), we have

∫_{V_{c}} K(z, w̄)f(w) dM_{s,t}(w) = [(1 − ρ^{2})^{1/2}πs(1 − ρ^{2})^{1/2}]^{− dim V} ∫_{V_{c}} f(w)e^{C(z,w̄)−C(w,w̄)} dw (3.12)

The constant outside the integral on the right is readily seen to be equal to (\frac{π}{2B} )^{− dim V}. Now we replace w in the integration by w + z, and using Equation (3.9), we have

∫_{V_{c}} K(z, w̄)f(w) dM_{s,t}(w) = (2B/π)^{dim V} ∫_{V_{c}} f(w + z)e^{C(z,z̄)−C(z̄,w+z)−2B|w|^{2}} dw (3.13)

Note that e^{−2B|λ|^{2}|} \frac{2B}{π} dλ is a Gaussian measure on C with total mass one, rotationally invariant about the origin. Thus the right side of the above equation is the (finite) mean of a holomorphic function with respect to a probability measure which, in each component (with respect to an orthonormal basis in V), is rotationally symmetric about the origin. Thus the value of the right side of (3.13) equals the value of the integrand at w = 0, i.e. it equals f(z).

The Proposition has the following consequence:

Corollary 3.5. For any w_{1}, w_{2} ∈ V_{c},

⟨K(·, w_{1}), K(·, w_{2})⟩_{L^{2}(M_{s,t})} = K(w_{1}, w̄_{2}) (3.14)

For any z = x + iy ∈ V_{c}, with x, y ∈ V,

||K(z, ·)||_{L^{2}(M_{s,t})}^{2} = K(z, z̄) = (1 − ρ^{2})^{− dim V/2}e^{\frac{x^{2}}{r} + \frac{y^{2}}{t}} (3.15)

The norm of the functional L^{2}(M_{s,t}) → C : f ↦ K_{z}f, where

K_{z}f =^{def} ∫_{V_{c}} K(z, w̄)f(w) dM_{s,t}(w),

is K(z, z̄)^{1/2} and the inequality

|K_{z}f| ≤ K(z, z̄)^{1/2}|f|_{L^{2}(M_{s,t})} (3.16)

is an equality (even without the absolute values on the left) when f is the holomorphic function K(·, z̄).

Proof. The first equality follows from the Proposition by taking f = K(·, w_{2}) and using the symmetry property K(z, w) = K(w, z). The second follows from the first and Equation (3.6). The results concerning K_{z} now follow directly.

We then have the result:

Proposition 3.6. The set HL^{2}(M_{s,t}) is a closed subspace of L^{2}(V_{c}, M_{s,t}).

Proof. Let f ∈ HL^{2}(M_{s,t}). Then K_{z}f = f(z). So, by (3.15) and (3.16), we have

|f(z)| ≤ (1 − ρ^{2})^{− dim V/4}e^{(\frac{x^{2}}{r} + \frac{y^{2}}{t} )/2}|f|_{L^{2}(M_{s,t})} (3.17)

where z = x + iy, with x and y in V. This shows that if f_{n} is Cauchy in HL^{2}(M_{s,t}) then f_{n}(z) is uniformly Cauchy for z running over any compact subset of V_{c}, and so the L^{2}–limit of f_{n} is a holomorphic function.

We also have :

Proposition 3.7. The functions K(z, ·), as z runs over V_{c}, span a dense subspace of HL^{2}(M_{s,t}).

Proof. Any function in HL^{2}(M_{s,t}) orthogonal to the function K(z, ·) must, by Equation (3.11), be zero at z̄; hence if f ∈ HL^{2}(M_{s,t}) orthogonal to all the functions K(z, ·), with z running over V_{c}, then f = 0.

We turn now to studying the kernel R(x, w).

Lemma 3.8. For every w ∈ V_{c}, the function R(·, w) belongs to L^{2}(V, P_{s}). If w_{1}, w_{2}, ∈ V_{c}, then

⟨R(·, w_{1}), R(·, w_{2})⟩_{L^{2}(P_{s})} = K(w_{1}, w̄_{2}) (3.18)

Proof. Recall that ρ_{r}(z) = (2πr)^{− dim V/2}e^{−z^{2}/(2r)}, for z ∈ V_{c} and r > 0. Examining

\frac{ρ_{t}(x − w_{1})}{ρ_{s}(x)} \frac{ρ_{t}(x − w_{2})}{ρ_{s}(x)}ρ_{s}(x)

we see, by completing the squares, that it is equal to K(w_{1}, w_{2}) times the normalized Gaussian factor

(2πts/r)^{− dim V/2} exp [−\frac{r}{2ts} \{x − \frac{s}{r}(w_{1} + w_{2})\}^{2}]

Thus

∫_{V} R(x, w_{1})R(x, w_{2}) dP_{s}(x) = K(w_{1}, w_{2}) (3.19)

In particular, R(·, w) ∈ L^{2}(P_{s}).

Next we prove that S_{s,t}^{V} takes the function R(·, w) to K(·, w):

Proposition 3.9. For every w ∈ V_{c}

[S_{s,t}^{V}R(·, w)](z) = K(z, w) (3.20)

Proof. Since R(x, w) = ρ_{t}(x − w)/ρ_{s}(x), we have

(S_{s,t}R(·, w))(z) = ∫_{V} \frac{ρ_{t}(x − w)}{ρ_{s}(x)}ρ_{t}(x − z) dx
= ∫_{V} R(x, w)R(x, z) dP_{s}(x)
= K(z, w)

Before proving unitarity of S_{s,t} we need one last observation:

Lemma 3.10. The linear span of the set {R(·, w) : w ∈ V } is dense in L^{2}(P_{s}).

Proof. Consider a function f ∈ L^{2}(P_{s}) which is orthogonal to each function R(·, w), as w runs over V ⊂ V_{c}. The form of the function R(·, w) then shows that this means, for all k ∈ V,

∫ e^{kx}e^{− \frac{ρ}{2t} x^{2}}f(x)e^{− \frac{x^{2}}{2s}} dx = 0

It can be verified by Cauchy-Schwarz that the function f(x)e^{− \frac{x^{2}ρ}{2t}} is in L^{2}(V, dP_{s}(x)). Thus the function f(x)e^{− \frac{x^{2}ρ}{2t}} is orthogonal to the subspace spanned by the functions e^{kx}, as k runs over V, in L^{2}(dP_{s}(x)). The latter subspace is known to be dense in L^{2}(V, P_{s}), and so f(x)e^{− \frac{x^{2}ρ}{2t}} = 0, and so f = 0. Thus {R(·, w) : w ∈ V } spans a dense linear subspace of L^{2}(P_{s}).

Finally, we can prove that the two-parameter Segal-Bargmann transform is a unitary isomorphism:

Theorem 3.11. For every f ∈ L^{2}(P_{s}), the function S_{s,t}f is in HL^{2}(V_{c}, M_{s,t}), and the operator S_{s,t} is a unitary isomorphism

S_{s,t} : L^{2}(V, P_{s}) → HL^{2}(V_{c}, M_{s,t}) (3.21)

Proof. We know that: (i) S_{s,t} carries each R(·, w) to K(·, w), (ii) the L^{2}(P_{s})–inner-product of R(·, w_{1}) and R(·, w_{2}) is equal to the L^{2}(M_{s,t})– inner-product of K(·, w_{1}) and K(·, w_{2}), and (iii) the functions R(·, w) span a dense subspace of L^{2}(P_{s}) and the functions K(·, w) span a dense subspace of HL^{2}(M_{s,t}). Therefore, there is a unique unitary isomorphism S_{s,t}^{1} : L^{2}(V, P_{s}) → HL^{2}(V_{c}, M_{s,t}) which agrees with S_{s,t} on the linear span of the functions R(·, w), as w runs over V. On the other hand, for each fixed z ∈ V_{c}, the linear functional A_{z} : L^{2}(P_{s}) → C : f ↦ S_{s,t}f(z) is bounded (see Equation (2.7)) and the functional B_{z} : L^{2}(P_{s}) → C : f ↦ (S_{s,t}^{1}f)(z), being the composite of the unitary map S_{s,t}^{1} and the bounded evaluation functional F ↦ K_{z}F = F (z), is also bounded. Since A_{z} and B_{z} agree on the dense subspace spanned by the functions R(·, w), with w running over V, it follows that they agree everywhere. Thus S_{s,t} is, in fact, the same as S_{s,t}^{1}, and so, in particular, is a unitary isomorphism.

Since we will encounter numerous Gaussian integrals, we record here a simple observation which is very useful in computing such integrals. If X is a real Gaussian random variable and λ ∈ C then

E(e^{λX}) = e^{λE(X)+ \frac{λ^{2}}{2} var(X)} (3.22)

where E(·) is the expectation and var(·) the variance.

We know that for each x ∈ V there is x̂ ∈ L^{2}(V, P_{s}). So for z = x+iy, with x, y ∈ V, we have the function

ẑ =^{def} x̂ + iŷ

which is also an element of L^{2}(V, P_{s}).

The following observation will be useful:

Proposition 3.12. For x ∈ V,

S_{s,t}ex̂ = e^{x̃+ \frac{1}{2} tx^{2}} (3.23)

More generally, for z ∈ V_{c},

S_{s,t}eẑ = e^{z̃+ \frac{1}{2} tz^{2}} (3.24)

The set {ex̂ : x ∈ V } spans a dense linear subspace of L^{2}(V, P_{s}), and the set {ex̃ : x ∈ V } spans a dense linear subspace of HL^{2}(V_{c}, M_{s,t}).

Proof. The formula (3.24) is obtained from the definition of S_{s,t}eẑ using the integration formula (3.22). The fact that {ex̂ : x ∈ V } spans a dense linear subspace of L^{2}(V, P_{s}) is well known. Applying the unitary isomorphism S_{s,t}, which maps onto HL^{2}(V_{c}, M_{s,t}), and using equation (3.23), we see that {ex̃ : x ∈ V } spans a dense linear subspace of HL^{2}(V_{c}, M_{s,t}).

We list some fundamental properties of S_{s,t} which are verified directly from the definition of S_{s,t}, using simply properties of the Gaussian functions.

Proposition 3.13. Suppose V is a finite-dimensional real inner-product space of positive dimension.

The mapping S_{s,t}^{V} : L^{2}(V, P_{s}) → HL^{2}(V_{c}, M_{s,t}) has the following properties:

1. \underline{Unitarity} : S_{s,t}^{V} is a unitary isomorphism of L^{2}(V, P_{s}) onto HL^{2}(V_{c}, M_{s,t})

2. \underline{Naturality} : if n is any positive integer ≤ dim V, f ∈ L^{2}(R^{n}, P_{s}) and u_{1}, ..., u_{n} orthonormal in V, the image of the function F_{f} = f(û_{1}, ..., û_{n}) under S_{s,t}^{V} is the function (S_{s,t}^{R^{n}}f)(ũ_{1}, ..., ũ_{n})

3. \underline{Factorizability}: for any orthonormal vectors u_{1}, ..., u_{n}, u_{n+1}, ..., u_{n+m} ∈ V and any functions f ∈ L^{2}(R^{n}, P_{s}) and g ∈ L^{2}(R^{m}, P_{s}), the image of the product function f(û_{1}, ..., û_{n})g(û_{n+1}, ..., û_{n+m}) under S_{s,t}^{V} is the product function (S_{s,t}^{R^{n}}f)(ũ_{1}, ..., ũ_{n})(S_{s,t}^{R^{n}}g)(ũ_{n+1}, ..., ũ_{n+m}).

4. Proofs II

We turn now to proving the results which show that S_{s,t} is unique up to a factor depending on a unit modulus complex number. In this section the vector spaces V and W have positive but finite dimension.

Let Aut(HL^{2}(M_{s,t}))_{V} be the set of maps W : HL^{2}(V_{c}, M_{s,t}) → HL^{2}(V_{c}, M_{s,t}) satisfying:

1. \underline{Unitarity}: W is a unitary isomorphism

2. \underline{Functoriality}: if n is any positive integer ≤ dim V, there is a unitary isomorphism HL^{2}(C^{n}, M_{s,t}) → HL^{2}(C^{n}, M_{s,t}) : f ↦ f̃ such that for any u_{1}, ..., u_{n} orthonormal in V, the image of the function F_{f} = f(ũ_{1}, ..., ũ_{n}) under W is the function Ff̃ = f̃(ũ_{1}, ..., ũ_{n})

3. \underline{Factorizability}: if u_{1}, ..., u_{n+m} (with n, m being any positive integers) are orthonormal vectors in V and f ∈ HL^{2}(C^{n}, M_{s,t}),

g ∈ HL^{2}(C^{m}, M_{s,t}), and F_{f,g} =^{def} f(ũ_{1}, ..., ũ_{n})g(ũ_{n+1}, ..., ũ_{n+m}) then

W F_{f,g} is of the form F_{f̃,g̃} = f̃(ũ_{1}, ..., ũ_{n})g̃(ũ_{n+1}, ..., ũ_{n+m}), where f̃ and g̃ are as given in Condition 2.

It is clear that Aut(HL^{2}(M_{s,t}))_{V} is a group under composition.

Let U(1) = {λ ∈ C : |λ| = 1} be the group of unit modulus complex numbers.

Proposition 4.1. For each complex number a ∈ U(1) there is a unique unitary isomorphism

W_{a}^{V} : HL^{2}(V_{c}, M_{s,t}) → HL^{2}(V_{c}, M_{s,t}) (4.1)

such that for any z ∈ V

W_{a}^{V} (ez̃) = e^{az̃+bz^{2}} (4.2)

where

b = \frac{1}{2}(1 − a^{2})(s − t) (4.3)

The formula (4.2) is also valid for all z ∈ V_{c}.

Each W_{a}^{V} is in Aut(HL^{2}(M_{s,t}))_{V} and the map W^{V} : U(1) → Aut(HL^{2}(M_{s,t}))_{V} : a ↦ W_{a}^{V} is an injective group homomorphism.

Proof. Unless there is possibility for confusion, we will simply write W_{a} to mean W_{a}^{V}. We know that the unitary transformation S_{s,t}^{V} carries the function ex̂ ∈ L^{2}(V, P_{s}) to e^{x̃+ \frac{1}{2} tx^{2}}, for every x ∈ V, and, further, that the linear span of {ex̃ : x ∈ V } spans a dense subspace of HL^{2}(V_{c}, M_{s,t}). So, to prove that W_{a}^{V} is a unitary isomorphism, it will suffice to prove that ⟨W_{a}(ew̃_{1}), W_{a}(ew̃_{2})⟩_{HL^{2}(M_{s,t})} equals ⟨ew̃_{1}, ew̃_{2}⟩_{HL^{2}(M_{s,t})}, for all w_{1}, w_{2} ∈ V_{c}. Now, using the Gaussian integration formula (3.22), we have

⟨ew̃_{1}, ew̃_{2}⟩_{HL^{2}(M_{s,t})} = ∫_{V^{2}} e^{(w_{1}+\overline{w}_{2})x+i(w_{1}−\overline{w}_{2})y− \frac{x^{2}}{r} − \frac{y^{2}}{t}} \frac{dxdy}{(\sqrt{πr}\sqrt{πt})^{dim V}}
(4.4)  =^{(3.22)} e^{\frac{1}{2} (w_{1}+\overline{w}_{2})^{2} \frac{r}{2} − \frac{1}{2} (w_{1}−\overline{w}_{2})^{2} \frac{t}{2}}

On the other hand,

⟨W_{a}(ew̃_{1}), W_{a}(ew̃_{2})⟩ = ⟨e^{aw̃_{1}+bw_{1}^{2}}, e^{aw̃_{2}+bw_{2}^{2}}⟩_{HL^{2}(M_{s,t})}
(4.5)  = e^{w_{1}^{2}b+\overline{w}_{2}^{2}\overline{b}+ \frac{1}{2} (aw_{1}+\overline{a}\overline{w}_{2})^{2} \frac{r}{2} − \frac{1}{2} (aw_{1}−\overline{a}\overline{w}_{2})^{2} \frac{t}{2}}

To see that the exponents in (4.4) and (4.5) are equal, observe first that the difference (w_{1} ±\overline{w}_{2})^{2} −(aw_{1} ±\overline{a}\overline{w}_{2})^{2} equals (1−a^{2})w_{1}^{2} +(1−\overline{a}^{2})\overline{w}_{2}^{2}, because |a| = 1. So the difference in the exponents is:

(1 − a^{2})\frac{(s − t)}{2}w_{1}^{2} + (1 − \overline{a}^{2})\frac{(s − t)}{2}\overline{w}_{2}^{2} − bw_{1}^{2} − \overline{b}\overline{w}_{2}^{2}

and this is, by definition of b, equal to 0 [of course, this is why b has been defined by (4.3)].

Thus, by the usual density argument, W_{a} extends uniquely to a unitary isomorphism of HL^{2}(V_{c}, M_{s,t}) onto itself, and is uniquely specified by (4.2) for z ∈ V.

Let u_{1}, ..., u_{n} be orthonormal in V. Then the relation (4.2) shows that

W_{a}^{V} (f(ũ_{1}, ..., ũ_{n})) = (W_{a}^{R^{n}}f)(ũ_{1}, ..., ũ_{n})

holds if f is any holomorphic exponential function (i.e. a function on C^{n} of the form eλ̃ for any λ ∈ C^{n}) and so, by the density argument, for all f ∈ L^{2}(R^{n}, P_{s}).

If w_{1}, w_{2} are orthogonal in V then it is clear that

W_{a}^{V} (ew̃_{1}ew̃_{2}) = W_{a}^{V} (ew̃_{1})W_{a}^{V} (ew̃_{2}) (4.6)

Let u_{1}, ..., u_{n+m} be orthonormal in V, f ∈ HL^{2}(C^{n}, M_{s,t}) and g ∈ HL^{2}(C^{m}, M_{s,t}). By the density of holomorphic exponentials (Proposition 3.12), f is the L^{2}(C^{n}, M_{s,t})–limit lim_{k→∞} f_{k}, where each f_{k} is a linear combination of functions of the form eλ̃, with λ ∈ R^{n}. So f(ũ_{1}, ..., ũ_{n}) is the HL^{2}(V_{c}, M_{s,t})–limit lim_{k→∞} f_{k}(ũ_{1}, ..., ũ_{n}), where each f_{k}(ũ_{1}, ..., ũ_{n}) is a linear combination of functions of the form ew̃_{1}, with w_{1} ∈ V a linear combination of the vectors ũ_{1}, ..., ũ_{n}. Similarly, g(ũ_{n+1}, ..., ũ_{n+m}) is the HL^{2}(V_{c}, M_{s,t})–limit of of functions of the form ew̃_{2}, with w_{2} ∈ V a linear combination of the vectors ũ_{n+1}, ..., ũ_{n+m}. Moreover, orthogonality implies that {ũ_{1}, ..., ũ_{n}} and {ũ_{n+1}, ..., ũ_{n+m}} are mutually independent under M_{s,t}. Using this, it follows that the product f(ũ_{1}, ..., ũ_{n})g(ũ_{n+1}, ..., ũ_{n+m}) is the HL^{2}(V_{c}, M_{s,t})–limit of the corresponding linear combination of products ew̃_{1}ew̃_{2}. Then, by the property (4.6), and continuity of W_{a}, we conclude that W_{a}^{V} has the Factorizability property. Thus:

W_{a}^{V} ∈ Aut(HL^{2}(M_{s,t}))_{V}

Next, for any w ∈ V,

W_{a_{1}}(W_{a_{2}}ew̃) = W_{a_{1}}(e^{a_{2}w̃+b_{2}w^{2}}) = e^{b_{2}w^{2}}W_{a_{1}}(e^{a_{2}w̃}) = e^{a_{1}a_{2}w̃+(b_{1}a_{2}^{2}+b_{2})w^{2}}

and so, since b_{1}a_{2}^{2}+b_{2} is readily verified to be equal to \frac{1}{2}(1−a_{1}^{2}a_{2}^{2})(s−t), which is the b corresponding to a_{1}a_{2}, W_{a_{1}}(W_{a_{2}}ew̃) equals W_{a_{1}a_{2}}(ew̃). So, by the usual density arguments, W_{a_{1}a_{2}} = W_{a_{1}}W_{a_{2}}.

Using the isomorphism S_{s,t} : L^{2}(V, P_{s}) → HL^{2}(V_{c}, M_{s,t}), which satisfies unitarity, naturality, and factorizability, we obtain the following result:

Proposition 4.2. There is an injective homomorphism of groups

U^{V} : U(1) → Aut(L^{2}(P_{s}))_{V} : a ↦ U_{a}^{V}

such that for each a ∈ U(1) and x ∈ V:

U_{a}^{V} (ex̂) = e^{ax̂+ \frac{1}{2} (1−a^{2})sx^{2}} (4.7)

The map U_{a}^{V} : L^{2}(V, P_{s}) → L^{2}(V, P_{s}) is uniquely specified by the requirement that it is a bounded linear map and that it satisfies equation (4.7) for every x ∈ V.

Proof. A straightforward calculation verifies that S_{s,t}^{−1}W_{a}^{V} S_{s,t}(ex̂) is given by (4.7), and so we get the result by defining U_{a}^{V} to be S_{s,t}^{−1}W_{a}^{V} S_{s,t}. Since the exponential functions ex̂ span a dense subspace of L^{2}(V, P_{s}), equation (4.7) and the requirement that U_{a}^{V} is a bounded linear map specify U_{a}^{V} uniquely.

The relationship between W, U, and S_{s,t} is worth recording separately:

Proposition 4.3. For every a ∈ U(1),

S_{s,t}^{−1}W_{a}S_{s,t} = U_{a} (4.8)

Finally we turn to our main result.

Theorem 4.4. Let V be a real inner-product space of finite dimension dim V ≥ 2. Suppose that S̃_{s,t} : L^{2}(P_{s}) → HL^{2}(M_{s,t}) satisfies the following conditions:

1. S̃_{s,t} is a unitary isomorphism of L^{2}(V, P_{s}) onto HL^{2}(V_{c}, M_{s,t})

2. for each f ∈ L^{2}(R, P_{s}) there is a function f̃ on C such that for any unit vector u ∈ V, S̃_{s,t}f(û) = f̃(ũ)

3. for any orthonormal vectors u_{1}, u_{2} ∈ V and any functions f, g ∈ L^{2}(R, P_{s}), the operator S̃_{s,t} carries the function f(û_{1})g(û_{2}) to the function f̃(ũ_{1})g̃(ũ_{2}), where f̃ and g̃ are as in Condition 2.

Then there is a unique a ∈ U(1) such that:

S̃_{s,t} = W_{a} ∘ S_{s,t} = S_{s,t} ∘ U_{a} (4.9)

Proof. The function f̃ provided by Condition 2 is holomorphic and the mapping f ↦ f̃ a linear unitary map L^{2}(R, P_{s}) → HL^{2}(C, M_{s,t}).

For k > 0, let E_{k} be the function in L^{2}(R, P_{s}) given by E_{k}(x) = e^{kx}, and let A_{k} = Ẽ_{k} be the function on C provided by Condition 2. Then

A_{k}(zu) = (S̃_{s,t}e^{kû})(z)

for every z ∈ V_{c} and every unit vector u ∈ V.

Since k ↦ e^{kû} ∈ L^{2}(V, P_{s}) is continuous and S̃_{s,t} unitary, and since the evaluation map HL^{2}(V_{c}, M_{s,t}) → C : f ↦ f(z) is continuous, it follows that (S̃_{s,t}e^{kû})(z) is continuous in k, for each fixed z. Moreover, (S̃_{s,t}e^{kû})(z) is holomorphic in z for each fixed k. Thus A_{k}(λ) is continuous in k (for fixed λ) and holomorphic in λ ∈ C for fixed k.

Let u_{1}, u_{2} ∈ V be orthogonal unit vectors and k_{1}, k_{2} positive real numbers. Then

e^{k_{1}û_{1}}e^{k_{2}û_{2}} = e^{\sqrt{k_{1}^{2}+k_{2}^{2}}}^{⟨·,_{\frac{k_{1}u_{1}+k_{2}u_{2}}{\sqrt{k^{2}_{1}+k^{2}_{2}}}} ⟩} (4.10)

Applying S̃_{s,t} to (4.10), we have for every z ∈ V_{c}:

A_{k_{1}}(zu_{1})A_{k_{2}}(zu_{2}) = A\sqrt{k_{1}^{2}+k_{2}^{2}} (z · (\frac{k_{1}u_{1} + k_{2}u_{2}}{\sqrt{k_{1}^{2} + k_{2}^{2}}} )) (4.11)

Now consider any λ_{1}, λ_{2} ∈ C. Let

z = λ_{1}u_{1} + λ_{2}u_{2}

Then (4.11) gives:

A_{k_{1}}(λ_{1})A_{k_{2}}(λ_{2}) = A\sqrt{k_{1}^{2}+k_{2}^{2}} (\frac{k_{1}}{\sqrt{k_{1}^{2} + k_{2}^{2}}} λ_{1} + \frac{k_{2}}{\sqrt{k_{1}^{2} + k_{2}^{2}}} λ_{2}) (4.12)

for all λ ∈ C and all k > 0.

It is possible the show that the solution of the functional equation (4.12) is

A_{k}(λ) = e^{kaλ+k^{2}c} (4.13)

16 AMBAR N. SENGUPTA

where a and c are complex constants. As already noted, f ↦ f̃ is unitary. So, since A_{k} = Ẽ_{k}:

⟨E_{k_{1}}, E_{k_{2}}⟩_{L^{2}(R,P_{s})} = ∫_{C} A_{k_{1}}(λ)\overline{A_{k_{2}}(λ)} dM_{s,t}(λ)
= ∫_{R^{2}} e^{(k_{1}a+k_{2}\overline{a})x+i(k_{1}a−k_{2}\overline{a})y− \frac{x^{2}}{r} − \frac{y^{2}}{t} +k_{1}^{2}c+k_{2}^{2}\overline{c}} \frac{dxdy}{\sqrt{πr}\sqrt{πt}}
=^{(3.22)} e^{\frac{1}{2} (k_{1}a+k_{2}\overline{a})^{2} \frac{r}{2} − \frac{1}{2} (k_{1}a−k_{2}\overline{a})^{2} \frac{t}{2} +k_{1}^{2}c+k_{2}^{2}\overline{c}}

Now ⟨E_{k_{1}}, E_{k_{2}}⟩_{L^{2}(R,P_{s})} = ∫_{R} e^{k_{1}x+k_{2}x} dP_{s}(x) = e^{(k_{1}+k_{2})^{2} \frac{s}{2}} (4.14) So we must have:

\frac{(k_{1} + k_{2})^{2}s}{2} = k_{1}^{2}[\frac{a^{2}}{4} (r − t) + c] + k_{2}^{2}[\frac{\overline{a}^{2}}{4} (r − t) + \overline{c}] + \frac{k_{1}k_{2}}{2} (r + t)|a|^{2}

Since this must hold for all k_{1}, k_{2} > 0, it follows that

\frac{s}{2} = \frac{a^{2}}{4} (r − t) + c = \frac{a^{2}}{2} (s − t) + c

and

s = \frac{1}{2}(r + t)|a|^{2} = s|a|^{2}

So |a| = 1, and

c = \frac{1}{2}[s − a^{2}(s − t)]

So

A_{k}(λ) = e^{kaλ+k^{2}c}

for all k > 0 and λ ∈ C, where a, c ∈ C are as above. For any non-zero vector w ∈ V, and z ∈ V_{c},

(S̃_{s,t}eŵ)(z) = (S̃_{s,t}E_{|w|})(z · \frac{w}{|w|})
= e^{|w|az· \frac{w}{|w|} +|w|^{2}c}
= e^{az·w+w^{2}c}
= e^{aw̃(z)+cw^{2}}
(4.15)  = (W_{a}(S_{s,t}^{V}eŵ))(z)

because

W_{a}(S_{s,t}^{V}eŵ) = W_{a}(e^{w̃+ \frac{1}{2} w^{2}t})
= e^{\frac{1}{2} w^{2}t+aw̃+ \frac{1}{2} (1−a^{2})(s−t)w^{2}}
= e^{aw̃+w^{2}[ \frac{1}{2} t+ \frac{1}{2} (1−a^{2})(s−t)]}
= e^{aw̃+ \frac{w^{2}}{2} [s−a^{2}(s−t)]}
(4.16)  = e^{aw̃+w^{2}c}

Now w ↦ eŵ and w ↦ ew̃ are continuous, and so the equality W_{a}S_{s,t}^{V}eŵ = S̃_{s,t}eŵ holds also for w = 0. Since the linear span of {eŵ : w ∈ V } is dense in L^{2}(V, P_{s}), it follows that S̃_{s,t}^{V} = W_{a} ∘ S_{s,t}^{V}.

We can now sharpen the descriptions obtained for Aut(L^{2}(P_{s}))_{V} and Aut(HL^{2}(M_{s,t}))_{V}.

Proof of Theorem 2.1 and Theorem 2.2 . We have the real innerproduct space V with 2 ≤ dim V < ∞. Let U ∈ Aut(L^{2}(P_{s}))_{V}. Con-

sider the map S̃_{s,t} =^{def} S_{s,t}^{V} ∘ U : L^{2}(V, P_{s}) → HL^{2}(V_{c}, M_{s,t}). Then

S̃_{s,t} satisfies the three conditions of Theorem 4.4. Therefore there is a a ∈ U(1) such that S̃_{s,t} = S_{s,t}^{V} ∘ U_{a}^{V} and so U = U_{a}^{V}. Thus the injective group homomorphism U(1) → Aut(L^{2}(P_{s}))_{V} : a ↦ U_{a}^{V} is also surjective. A similar argument works for Aut(HL^{2}(M_{s,t}))_{V}.

5. The Infinite Dimensional Case

In this section V will be a real vector space of positive but possibly infinite dimensions, equipped with an inner-product V^{2} → R : (v, w) ↦ vw (it is customary to also assume that the induced topology is separable). If X is a real vector space, of positive dimension, with an inner-product g(·, ·) then there is a probability space (X′, F, P) and a linear map X → L^{2}(X′, P) : x ↦ x̂ such that :(i) each x̂ is a Gaussian random variable of mean 0 and variance g(x, x), and (ii) the P –almosteverywhere-defined random variables x̂, as x runs over X, generate the completed σ–algebra F. For example, if B is an orthonormal basis of the completion of X then one can take X′ to be the product R^{B}, with each copy of R equipped with standard Gaussian measure, and define x̂ by requiring (linearity, continuity, and) that b̂ : X′ → R is the projection on the b–th coordinate, for each b ∈ B. Applying this procedure to the inner-product on V given by g(v, w) = svw, we obtain a probability measure P_{s} on a space V′, and a linear map V → L^{2}(V′, P_{s}) : x ↦ x̂ such that the random variables x̂, with x running over V, generate the completed σ–algebra on which P_{s} is defined, and

∫_{V′} e^{λx̂} dP_{s} = e^{sλ^{2}x^{2}/2} (5.1)

for every λ ∈ C. Using this mechanism on V_{c} = V^{2}, viewed as a real inner-product space, we also obtain a probability space (V′_{c}, M_{s,t}) and a linear map V′_{c} → L^{2}(V′_{c}, M_{s,t}) : z ↦ ž such that the random variables ž generate the completed σ–algebra on which the measure M_{s,t} is defined and

∫_{V′_{c}} e^{λ_{1}x̌+λ_{2}ˇ(iy)} dM_{s,t} = e^{\frac{λ^{2}_{1}r}{4} + \frac{λ^{2}_{2}t}{4}} (5.2)

for every λ_{1}, λ_{2} ∈ C, and x, y ∈ V.

If z = x + iy ∈ V_{c}, where x, y ∈ V, then we have the complex Gaussian random variable

z̃ = x̌ + iy̌ (5.3)

and the map V_{c} → L^{2}(V′_{c}, M_{s,t}) : z ↦ z̃ is complex-linear.

A cylinder function on V′ is a function which can be expressed in the form f(x̂_{1}, ..., x̂_{n}) for some x_{1}, ..., x_{n} ∈ V and some function f on R^{n}. A cylinder function on V′_{c} is holomorphic if it can be expressed in the form f(z̃_{1}, ..., z̃_{n}), for some z_{1}, ..., z_{n} ∈ V_{c} and holomorphic function f on C^{n}. We denote by HL^{2}(M_{s,t}) the closure in L^{2}(M_{s,t}) of the set of functions in L^{2}(M_{s,t}) which are holomorphic cylinder functions.

We have then the standard exponentials which form dense subspaces in L^{2}(V′, P_{s}) and HL^{2}(V′_{c}, M_{s,t}):

Lemma 5.1. The set {ex̂ : x ∈ V } spans a dense linear subspace of L^{2}(V′, P_{s}). The set {ex̃ : x ∈ V } spans a dense linear subspace of HL^{2}(V′_{c}, M_{s,t}).

Proof. We know, from Proposition 3.12, that any f ∈ L^{2}(R^{n}, P_{s}) is the L^{2}–limit lim_{k→∞} f_{k}, where each f_{k} is a linear combination of functions of the form eλ̂ (with λ running over R^{n}). Let u_{1}, ..., u_{n} be unit vectors in V. Then f(û_{1}, ..., û_{n}) is lim_{k→∞} f_{k}(û_{1}, ..., û_{n}) in L^{2}(V′, P_{s}), and each f_{k}(û_{1}, ..., û_{n}) is a linear combination of fuctions of the form ev̂ for v ∈ V. Thus {ex̂ : x ∈ V } spans a dense linear subspace of L^{2}(V′, P_{s}). The argument for HL^{2}(V′_{c}, M_{s,t}) is analogous.

Theorem 5.2. There is a unique unitary isomorphism

S_{s,t} : L^{2}(V′, P_{s}) → HL^{2}(V′_{c}, M_{s,t}) (5.4) such that for any unit vectors u_{1}, ..., u_{n} ∈ V and any function f ∈ L^{2}(R^{n}, P_{s}),

S_{s,t}[f(û_{1}, ..., û_{n})] = (S_{s,t}^{R^{n}}f)(ũ_{1}, ..., ũ_{n}) (5.5)

Proof. For x, y ∈ V, we can check that

⟨ex̂, eŷ⟩_{L^{2}(P_{s})} = ⟨e^{x̃+ \frac{t}{2} x^{2}}, e^{ỹ+ \frac{t}{2} y^{2}}⟩_{HL^{2}(M_{s,t})} (5.6)

Thus, by the preceding lemma, there is a unique unitary isomorphism S_{s,t} : L^{2}(V′, P_{s}) → HL^{2}(V′_{c}, M_{s,t}) taking ex̂ to e^{x̃+ \frac{t}{2} x^{2}}, for each x ∈ V. This definition of S_{s,t} insures that equation (5.5) holds whenever f is an exponential function on R^{n}, i.e. a function of the form ev̂ for some v ∈ R^{n}. Approximating the general function in L^{2}(R^{n}, P_{s}) by linear combinations of exponentials, and using the observation that L^{2}(R^{n}, P_{s}) → L^{2}(V′, P_{s}) : f ↦ f(û_{1}, ..., û_{n}) is a linear isometry, we obtain equation (5.5).

The unitary isomorphisms U_{a}^{V} and W_{a}^{V}, defined earlier in the finitedimensional context, are meaningful as defined even when V is infinite dimensional. It may be of some interest to understand these isomorphisms in a different way. We focus on U_{a}^{V}, with a ∈ U(1).

Equip T (V_{c}), the tensor algebra over V_{c}, with the inner-product specified on n-tensors by

⟨u_{1} ⊗ ⋯ ⊗ u_{n}, v_{1} ⊗ ⋯ ⊗ v_{n}⟩ = s^{n}n!⟨u_{1}, v_{1}⟩ ⋯ ⟨u_{n}, v_{n}⟩

(The inner-product ⟨·, ·⟩ appearing on the right is the Hermitian innerproduct on V_{c} induced by the inner-product on V.) On 0–tensors, i.e. on C, the inner-product is the usual Hermitian inner-product on C : (λ, μ) ↦ λ\overline{μ}. Let S(V_{c})_{s} be the Hilbert space obtained by completing the subspace of symmetric tensors in T (V_{c}). This is the symmetric Fock space over V (for the inner-product scaled by the parameter s). For each z ∈ V_{c}, we have the exponential Expz =^{def} ∑_{n=0}^{∞} \frac{1}{n!}z^{⊗n}, the series being clearly convergent in S(V_{c})_{s}. It is a fact that {Exp(x) : x ∈ V } span a dense subspace of S(V_{c})_{s}. It is also readily checked that for any z, w ∈ V_{c}, the inner-product ⟨e^{ẑ− \frac{s}{2} z^{2}}, e^{ŵ− \frac{s}{2} w^{2}}⟩_{L^{2}(V′,P_{s})} and ⟨Exp(z), Exp(w)⟩ are equal (to e^{sz\overline{w}}). So, by the usual density argument, there is a unique unitary isomorphism (the Itô iterated multiple integral isomorphism)

I_{s} : S(V_{c})_{s} → L^{2}(V′, P_{s}) (5.7)

satisfying

I_{s}(Exp(x)) = e^{x̂− \frac{s}{2} x^{2}} (5.8) for every x ∈ V, the relation being also valid if x is replaced by any z ∈ V_{c}. The definition, via (4.7), of U_{a} implies that it takes e^{x̂− \frac{s}{2} x^{2}} to e^{âx− \frac{s}{2} (ax)^{2}}. Thus I_{s}^{−1}U_{a}I_{s} is simply the unique unitary isomorphism S(V_{c})_{s} → S(V_{c})_{s} induced by the unitary map V_{c} → V_{c} : z ↦ az.

6. Category theoretic formulation

Some of the results of the preceding two sections can be recast nicely in a category-theoretic framework. We will not write out the proofs of these category-theoretic formulations separately since they are reformulations of results we have already proven.

Let Euc be the category whose objects are real inner-product spaces and whose arrows are linear isometries, let Hilb be the category whose objects are complex Hilbert spaces and whose arrows are again linear isometries. If V is an object of Euc we can associate to it the Hilbert space L^{2}(V′, P_{s}). If j : V → W is a morphism in Euc then there is a naturally induced isometry L^{2}(j, P_{s}) : L^{2}(V′, P_{s}) → L^{2}(W′, P_{s}): it is the unique linear isometry L^{2}(V′, P_{s}) → L^{2}(W′, P_{s}) which takes ex̂ to eĵ(x), for each x ∈ W. It is also readily verified that if u_{1}, ..., u_{n} are orthonormal in W and f ∈ L^{2}(R^{n}, P_{s}) then L^{2}(j, P_{s}) takes the

function f(û_{1}, ..., û_{n}) ∈ L^{2}(W′, P_{s}) to the function f({ \atop j(u_{1})}, ..., { \atop j(u_{n})}) ∈

L^{2}(V′, P_{s}). If V = {0}, we take L^{2}(V, P_{s}) to be the complex line C. Then L^{2}(P_{s}), consisting of the assignments V ↦ L^{2}(V′, P_{s}) and j ↦ L^{2}(j, P_{s}) is a functor

L^{2}(P_{s}) : Euc → Hilb (6.1)

Analogously, we also have the functor

HL^{2}(M_{s,t}) : Euc → Hilb (6.2)

which takes each object V in Euc to the Hilbert space HL^{2}(V′_{c}, M_{s,t}).

The categories Euc and Hilb are monoidal in that they have a product with certain simple properties: in Euc it is the direct sum, and in Hilb it is the tensor product. The functors L^{2}(P_{s}) and HL^{2}(M_{s,t}) are monoidal functors in the following sense. For L^{2}(P_{s}), being a monodial functor means that there is a natural isomorphism θ between the functors Euc^{2} → Hilb : (V, W) ↦ L^{2}(V′, P_{s}) ⊗ L^{2}(W′, P_{s}) and Euc^{2} → Hilb : (V, W) ↦ L^{2}((V ⊕ W)′, P_{s}). Thus, for each V, W ∈ Euc, there is a unitary isomorphism

θ_{V,W} : L^{2}((V ⊕ W)′, P_{s}) → L^{2}(V′, P_{s}) ⊗ L^{2}(W′, P_{s})

The analogous statement holds for HL^{2}(M_{s,t}), and, for simplicity, we use the same notation θ_{V,W} for the corresponding isomorphism.

Now to each object V of Euc we have the unitary isomorphism S_{s,t}^{V} : L^{2}(V′, P_{s}) → HL^{2}(V′_{c}, M_{s,t}). In this language, we can state the basic properties of S_{s,t} as follows:

Proposition 6.1. The assignment V ↦ S_{s,t}^{V} is a natural isomorphism between the functors L^{2}(P_{s}) and HL^{2}(M_{s,t}). Moreover, this natural isomorphism preserves the monoidal character of the functors L^{2}(P_{s}) and HL^{2}(M_{s,t}):

S_{s,t}^{V ⊕W} = θ_{V,W}^{−1} (S_{s,t}^{V} ⊗ S_{s,t}^{W})θ_{V,W} (6.3)

for every V, W ∈ Euc.

Some terminology: Let A and B be monoidal categories [for the present purposes, we may think of a monoidal category simply as any one of the two categories Euc and Hilb] and denote the monoid operation in A by ⊕ and the one in B by ⊗. Consider a functor F : A → B between monoidal categories . Then we have two functors F ∘ ⊕ : A^{2} → B : (V, W) ↦ F (V ⊕ W) and ⊗ ∘ F^{2} : (V, W) ↦ F (V) ⊗ F (W). The functor F is monoidal if there is a natural transformation θ^{F} : F ∘ ⊕ → ⊗ ∘ F^{2}. Note that this means, in particular, that for every objects V and W in A there is an isomorphism θ_{V W}^{F} : F (V ⊕ W) → F (V) ⊗ F (W). If F, G : A → B are two monoidal functors and τ : F → G is a natural transformation, we say that τ is monoidal if

τ_{V ⊕W} = (θ_{V W}^{G} )^{−1}(τ_{V} ⊗ τ_{W})θ_{V W}^{F} (6.4)

Let Mon(F, G) denote the set of all monoidal natural transformations from F to G.

If F : A → B is a monoidal functor between monoidal categories, we denote by Aut(F) the group of monoidal natural isomorphisms F → F. Thus Aut(F) is the group of (monoidal) symmetries of the functor F, and its size describes how “rigid” F is.

It is clear that Aut(F) acts on Mon(F, G) be pre-composition and Aut(G) acts on Mon(F, G) by post-composition.

Let U(1) be the group of unit-modulus complex numbers.

Theorem 6.2. There is a unique isomorphism of groups

U(1) → Aut(L^{2}(P_{s})) : a ↦ U_{a}

such that the unitary isomorphism U_{a}^{R} : L^{2}(R, P_{s}) → L^{2}(R, P_{s}) carries the function e^{kx} (here k is any real constant and x ∈ R the variable) into the function e^{akx+ \frac{1}{2} (1−a^{2})sk^{2}}, for all k ∈ R.

There is a unique isomorphsim of groups

U(1) → Aut(HL^{2}(M_{s,t})) : a ↦ W_{a}

such that the unitary isomorphism W_{a}^{R} : HL^{2}(C, M_{s,t}) → HL^{2}(C, M_{s,t}) carries the function e^{kz} (here k is constant and z ∈ C the variable) into the function e^{akz+ \frac{1}{2} (1−a^{2})(s−t)k^{2}}, for each k ∈ R.

The next result determines the “rigidity” of S_{s,t} and shows that, modulo a U(1)–factor, S_{s,t} is the unique natural isomorphism L^{2}(P_{s}) → HL^{2}(M_{s,t}).

Theorem 6.3. For each S̃_{s,t} ∈ Mon(L^{2}(P_{s}), HL^{2}(M_{s,t})) there is a unique a ∈ U(1) such that

S̃_{s,t} = W_{a} ∘ S_{s,t} = S_{s,t} ∘ U_{a} (6.5)

In particular, Mon(L^{2}(P_{s}), HL^{2}(M_{s,t}))/Aut(L^{2}(P_{s})) has exactly one element, the (class of the) two-parameter Segal-Bargmann transform.

References

[1] Albeverio, S., Hall, B.C., Sengupta, A.N., The Segal-Bargmann Transform for two-dimensional Euclidean Quantum Yang-Mills, Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol.2, Number 1, 27-49(1999)

[2] Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform, Part I. Comm. Pure Appl. Math. 14, 187-214 (1961)

[3] Driver, B., and Hall, B.: Yang-Mills theory and the Segal-Bargmann transform. Comm. Math. Physics, 201, 249-290 (1999)

[4] Hall, B.: The Segal-Bargmann “coherent state” transform for compact Lie groups. J. Funct. Anal. 122, 103-151 (1994)

[5] Hall, B.: A New Form of the Segal-Bargmann Transform for Lie Groups of Compact Type, to appear in Canadian Journal of Mathematics.

[6] Hall, B., and Sengupta, A.: The Segal-Bargmann transform for path-groups. J. Funct. Anal. 152, 220-254 (1998)

[7] Segal, I.: Mathematical problems of relativistic physics, Chap. VI. In: Kac, M. (ed.) Lectures in applied mathematics. Proceedings of the summer seminar, Boulder, Colorado, 1960, Vol. II, pp. 73-84. Providence, Rhode Island: American Math. Soc. 1963

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA

E-mail address: sengupta@math.lsu.edu