In this section, we introduce a family of Bargmann transform{βt}t>0 parametrized by the pa-rametert >0. Using Bargmann’s result [12], we show that eachβtmapsL2(Rn)isometrically onto H2(Cn). For an integral operator D acting on a dense domain of H2(Cn) we find its corresponding (via the push-forward under βt−1Dβt) integral operator on L2(Rn). We shall also calculate the ,,inverse Bargmann transform” of ezAw whenever A is a Hermitian matrix satisfying some conditions. This result will be used in Section 4.3 to calculate the heat ker-nel of a certain type of sub-elliptic operators. Finally, we introduce the sub-Laplacian on the (2N + 1)-dimensional Heisenberg group.
Throughout this chapter and to simplify the notation used here we simply writeh·,·i (re-spectivelydµ) for the inner producth·,·i(1)(respectively for the Gaussian measuredµ(1)) on the Segal-Bargmann spaceH2(Cn) := H21(Cn). For a matrixA = (ajk) ∈ Mn(C)with complex entries we denote by A := (ajk), AT := (akj)∈ Mn(C)the complex conjugate and the trans-pose ofA, respectively. Moreover, we writeA? :=AT = (akj)for the conjugate transpose of A. Forz, w ∈Cnwe will sometimes denote byz·wthe usual productzw =P
zjwj. For each z ∈ Cn we write Kz(u) := euz for the reproducing kernelK(1)(u, z)at the pointz and for a fixedt >0we put
Φtz(x) :=tn2(2π)−n4 exp{txz −t2x2 4 −z2
2}.
For each t > 0 one can check that the linear span of{Φtz}z∈Cn is norm dense inL2(Rn) (c.f.
Chapter 1 in [84]) and that for anyz, u∈Cnthe following equality holds:
hΦtz,ΦtuiL2(Rn) =hKz, Kui=euz. (4.2.1) Let us now introduce the Bargmann transform onL2(Rn).
Definition 4.2.1. For each fixedt >0, the Bargmann transform off ∈L2(Rn)denoted byβtf is the entire function defined onCnby:
βtf(z) := hf,ΦtziL2(Rn) = Z
Rn
f(x)Φtz(x)dx.
The Bargmann transformβ√2 was first introduced by S. Bargmann in his paper [12] on the structure of the Segal-Bargmann space. Bargmann proved thatβ√2 mapsL2(Rn)isometrically ontoH2(Cn)(c.f. also [84]). However, for anys, t >0on can easily check that
βs =βt◦Ut
s, (4.2.2)
where Ut
s is the unitary operator defined on L2(Rn)by [Ut
sf](x) := (st)n2f(tsx). This shows that for anyt > 0the Bargmann transform βtmapsL2(Rn)isometrically ontoH2(Cn). Using Equation (4.2.2) together with Equation (2.11) in [12] for the inverse Bargmann transformβ√−1
2
one can easily calculateβt−1 for allt > 0. In the next theorem, we summarize the above facts and give the explicit expression for the inverse Bargmann transformβt−1.
4.2. THE BARGMANN TRANSFORM 97 Theorem 4.2.1. For eacht > 0 the Bargmann transformβtis an isometry fromL2(Rn)onto H2(Cn)and the inverse Bargmann transform is given by:
[βt−1F](x) = Z
Cn
F(z)Φtz(x)dµ(z) for allF ∈H2(Cn). (4.2.3) Denote byS(Rn) ⊂L2(Rn)the Schwartz space overRn. In the following we characterize the entire functions overCnwhich are in the range of the Bargmann transformβt
S(Rn) . Proposition 4.2.1. Letf(z) =P
α∈Nn0 cαzαbe an entire function overCn. Then for everyt >0
is invariant under the change of variable z 7−→ P z for any unitary matrixP.
Proof. We refer the reader to Chapter 4 in [139] for the caset = √
2. Now for anys, t > 0it is easy to see that the operatorUt
s is bijective onS(Rn). Using (4.2.2) we obtainβt
Remark 4.2.1. The above proposition shows that iff ∈βt
S(Rn) Via the push-forward under βt each operator D on H2(Cn) corresponds to the operator βt−1Dβt on L2(Rn). The next proposition exhibit the push-forward under βt of a family of integral operators acting onβt
S(Rn)
⊂H2(Cn).
Proposition 4.2.2. Let(As)s≥0be a family of integral operators acting onβt
S(Rn)
98 CHAPTER 4. ,,HEAT KERNEL” FOR TOEPLITZ OPERATORS Furthermore, we suppose that for eachx∈Rnthere is a constantCxdepending onxsuch that
Z
Equations (4.2.5) and (4.2.6) allow us to apply Fubini theorem to the above integral. Indeed, Z
Equation (4.2.8) follows from (4.2.9) by an application of the Lebesgue dominated convergence theorem together with the fact thatK(0, z, w) =ezw.
4.2. THE BARGMANN TRANSFORM 99 Remark 4.2.2. With the notation used in the above proposition we remark that if f(x, y) ∈ L2(Rn×Rn)andβt,2ndenotes the2n-dimensional Bargmann transform (i.e. onL2(R2n)then
Using this observation together with above proposition one can easily show that if T is an integral operator onL2(Rn)with a kernelk(x, y)∈ L2(R2n)then the corresponding operator
whereK is the (2n-dimensional) Bargmann transform ofk(c.f. Chapter 1 in [84] for the case t =√
2).
For eachi ∈ {1,· · · , n}letxi (respectively ∂x∂
i) be the operators of multiplication (respec-tively differentiation) with respect to the i-th coordinate ofRn. Under the Bargmann transform βt these differential operators correspond to the following Toeplitz operators densely defined onH2(Cn):
We only prove (4.2.13). Let C0∞(Rn)denotes the space of all smooth functions with compact support inRn. Then for anyf ∈C0∞(Rn)we have
100 CHAPTER 4. ,,HEAT KERNEL” FOR TOEPLITZ OPERATORS
In the following we aim to calculate h AA)(Id−AA)−1 is positive definite. The reason for this calculation is motivated by the fol-lowing observation: let N ∈ N be fixed and put n = 2N. If L is the operator defined on L2(Rn)obtained by applying the partial Fourier transform to the sub-Laplacian on the(2N +1)-dimensional Heisenberg group then the ,,heat kernel” of its corresponding Toeplitz onH2(Cn) (c.f. Definition 4.3.3) is of the form ezAw where(z, w) ∈ Cn×Cn andA has the previously mentioned properties. Hence by Proposition 4.2.2 we get the heat kernel ofLby (4.2.14). The next proposition is essential in calculating (4.2.14) and its proof is given in Appendix A.3.
Proposition 4.2.3. LetE ∈ Mn(C)be a Hermitiann×ncomplex matrix such thatkEk< 12. Denote by F the real matrix F := E +E and assume that (Id−F)(Id+F)−1 is positive definite. We writeβt,x→z−1 for then-dimensional inverse Bargmann transform onH2(Cn). Then for fixed vectoru∈Cnwe have
βt,x→z−1 exp{u·z+zEz}(x) =cexpn
u(Id+F)−1tx−t2
4x(Id−F)(Id+F)−1xo
, (4.2.15) wherecis the constant given by:
c=tn2(2π)−n4det(Id−F)(Id+F)−1
The next proposition gives the explicit expression of (4.2.14).
Proposition 4.2.4. Denote by(x, y),(z, w)the coordinates inRn×Rn,Cn×Cnrespectively.
Let A ∈ Mn(C) be a Hermitian matrix and write F := −AA. Suppose that AA = AA, kAAk<1and the matrix(Id−F)(Id+F)−1 is positive definite then
4.2. THE BARGMANN TRANSFORM 101 Proof. Using Equation (4.2.1) we obtain:
h Substituting (4.2.19) and (4.2.20) into (4.2.18) we then obtain:
h
102 CHAPTER 4. ,,HEAT KERNEL” FOR TOEPLITZ OPERATORS Using a similar calculation together with the fact that y(Id+F)−1Ax = yA(Id+F)−1xfor all(x, y)∈Rn×Rnone can show that
h
βt,y→w−1
βt,x→z−1 ezAw (x)
i (y) is also given by (4.2.17).
Remark 4.2.3. Let A ∈ Mn(C)be a Hermitian matrix and denote by ρ(A) ≥ 0its spectral radius. Then the functionezAw ∈H2(C2n)if and only ifρ(A)<1. Indeed, sinceAis Hermitian there is a unitary matrixP such that
A=P D(λ1,· · · , λn)P−1,
whereD(λ1,· · · , λn)is the diagonal matrix whose entriesλj are the eigenvalues of A(λj are real). Since the Segal-Bargmann space is invariant under a unitary transformation of coordi-nates it follows that ezAw ∈ H2(C2n) if and only if ePλjzjwj ∈ H2(C2n) (here we used the transformation(z, w)7−→(P z, P w)). This is equivalent to say that for eachj = 1,· · · , n
eλjzjwj = X
α∈N0
λαj (zjwj)α
α! ∈H2(C2)⇐⇒ X
α∈N0
|λj|2α <∞ ⇐⇒ |λj|<1.
Hence by Remark 4.2.2 it follows that if A ∈ Mn(C) is such that ρ(A) < 1 and satisfies the conditions in the above proposition then the 2n-dimensional inverse Bargmann transform βt,2n−1 ezAw ∈L2(R2n)exists and is given by (4.2.17).
The next example is an application of Proposition 4.2.2 for a certain class of integral opera-tors. This example will be essential for obtaining the heat kernel of a class of elliptic operators from the ,,heat kernel” of their corresponding Toeplitz operators (c.f. Corollary 4.3.2).
Example 4.2.1. Let A ∈ Mn(C)be a positive semidefinite matrix such that (A+A)is pos-itive definite and AA = AA. We consider a family (Ts)s≥0 of integral operators acting on βt
S(Rn)
⊂H2(Cn)defined by (TsF)(z) :=
Z
Cn
eze−sAwF(w)dµ(w), for all F ∈βt
S(Rn) .
Then for each fixed couples, t >0the corresponding operatorβt−1TsβtonS(Rn)is an integral operator given by:
βt−1Tsβtf (x) =
Z
Rn
k(s, x, y)f(y)dy,
where k(s, x, y)is given by an application of (4.2.17) to the matrix e−sA. Moreover, for each x∈Rnwe have
lims→0
Z
Rn
k(s, x, y)f(y)dy=
βt−1T0βtf
(x) =f(x), for all f ∈ S(Rn).
Indeed, this is an application of Proposition 4.2.2 to the operators defined by the kernels K(s, z, w) = eze−sAwtogether with another application of Proposition 4.2.4 to the matrixe−sA.
4.3. HEAT KERNEL BY TOEPLITZ OPERATOR THEORY TECHNIQUES 103