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On the Berezin-Toeplitz Calculus
L.A. Coburn
Vienna, Preprint ESI 834 (2000)
Supported by Federal Ministry of Science and Transport, Austria
Available via http://www.esi.ac.at
January 26, 2000
On the Berezin-Toeplitz Calculus*
L.A. Coburn
Abstract
I consider the problem of composing Berezin-Toeplitz operators on the
Hilbert space of Gaussian square-integrable entire functions on complex nspace, Cn . For several interesting algebras of functions on Cn , we have
T'T = T' for all '; in the algebra, where T' is the Berezin-Toeplitz
operator associated with ' and ' is a \twisted" associative product on
the algebra of functions. On the other hand, there is a C 1 function ' for
which T' is bounded but T'T' 6= T for any .
*Research supported by a grant of the NSF and a visiting membership in the
Erwin Schrodinger Institute.
1991 Mathematics Subject Classi cation: Primary 47B35, Secondary 47D25.
1
1. Introduction. For z = (z ; :::; zn) in complex n-space, Cn ; with zj in
C; z w = z w + ::: + znwn ; consider the space L (Cn ; d) of Gaussian squareintegrable complex-valued functions on Cn , with d(z) = expf? jzj =2gdv(z)(2)?n
with dv(z) Lebesgue measure. The entire functions in L (Cn ; d) form a
closed subspace H (Cn ; d) which arises naturally as a representation space
of the Heisenberg group [B, F, BC1; C]. On this (Segal-Bargmann) space,
there are natural operators, formally introduced by Berezin [Be], de ned
densely for '() with '(w)ewa in L (Cn ; d) for all a in Cn ; by
1
1
2
1
2
2
2
2
Z
(T'f )(z) = ezw= '(w)f (w)d(w):
2
C
n
The (possibly unbounded) operator T' is called the Berezin-Toeplitz operator
associated to ': Note that H (Cn ; d) is a Bergman space with reproducing
kernel function eza= for the functional of \evaluation at a" [B]. Note also
that T' = 0 if and only if ' = 0 [F, p. 140].
The operators T' are closely related to pseudodi erential operators on
L (Rn ; dv ). For ' bounded, and somewhat more generally, the relation is
given by
2
2
2
B ? T'B = W '
1
where B is the Bargmann isometry [Gu], W is the Weyl operator on L (Rn; dv)
given by
2
(W g)(x) = (2)?n
Z Z
+
'(; x) = ?n
Z
(
n
'(w)e?jw? x?i j dv (w):
(
C
)
2
RR
n
and
(; x y )ei x?y g(y)dyd;
)
2
n
The operators T' might, therefore be expected to share many of the
properties of pseudodi erential operators. It is not easy to demonstrate a
2
complete equivalence, partly because ' is a \very smoothed" version of
': The analytic structure of H (Cn ; d) also enters the picture so that, for
example,
T'Tz = T'z :
Moreover, the available function-theoretic machinery on H (Cn ; d) is relatively rudimentary, limited primarily to the Bergman space structure and
the structure inherited as a representation space of the Heisenberg group.
In this note, I deal with the composition problem: is there a function
' so that
T'T = T' ?
(*)
As a consequence of representation-theoretic results in [C], we do have (*)
for a reasonably large class of bounded '; and there is an explicit formula
for ' . The same \Moyal-type" formula also holds for a large class
of unbounded '; (with unbounded T'; T ; T' ) { precisely, '; can be
arbitrary polynomials in fzj ; zj : 1 j ng:
On the other hand, I will exhibit a '; (unbounded, but C 1) for which
T' is a bounded operator but T'T' can not be approximated in norm by
bounded Berezin-Toeplitz operators. Thus, there is a genuine limitation on
our ability to compose Berezin-Toeplitz operators.
I thank Don Schack for useful comments.
2
j
j
2
3
2. Composition of Berezin-Toeplitz operators. For C 1 functions
'; we consider the (formal) twisted product
'
=
X
k
(?2)jkj (@ k ')(@k )
k!
(**)
where k = (k ; :::; kn) with kj non-negative integers, and
1
= @z@ ; @j = @z@
k
k
k
@ k = @ k1 :::@nk ; @ = @ 1 :::@n
jkj = k + k + ::: + kn
k ! = k !k !:::kn!:
@j
j
j
n
n
1
1
1
1
2
2
In the cases we will consider, the sum in (**) will converge.
The rst case we consider arises from representation-theoretic considerations of the Heisenberg group [C]. We consider '; in the \smooth Bochner
algebra" Ba(Cn ) which consists of all Fourier-Stieltjes transforms of compactly supported, regular, bounded complex-valued Borel measures on Cn :
More precisely, let
a(z ) = expfi
Im(z a)g:
Then Ba(Cn ) consists of all functions
^ (z ) =
Z
C
a (z ) d (a)
n
where is a compactly supported, regular, bounded complex-valued Borel
measure. It is well known that such functions are bounded, uniformly continuous, with bounded derivatives of all orders.
As our rst positive result, we have
Theorem 1. For '; in Ba(Cn ), ' is also in Ba(Cn ) and T'T = T' :
The series in (**) converges uniformly and absolutely.
4
Proof. In [C], I showed that for ' = ^ ; = ^ in Ba(Cn );
= T b .
T'T
(
)
Here, we de ned for all in C (Cn ) by
0
Z
C
(c) d( )(c) =
( )b(z) =
CC
Z Z
n
a b (z )eba= d (a) d (b)
(***)
2
CC
n
Ba (Cn ):
(a + b)eba= d (a) d (b)
2
n
n
so that
Z Z
+
n
is in
Expanding eba= in McLaurin series in (***) gives
2
( )b(z) =
1
X
s
1
X
=0
=
s
1
X
=0
=
s
1
1
1
Z
X
! 2s
ji n
aj1 :::aj a(z ) d (a)
s
X
! 2s
s
bj1 :::bj b (z )d (b)
s
ji n
2s (@j1 :::@j ')(?2)s (@j1 :::@j )
s
s
1
?
s
(
Z
1
X
(@j1 :::@js')(@j1 :::@js
ji n
X (?2)jkj
k
(
@ k ')(@ )
k!
k
s
=0
=
1
2)
s
!
)
1
and it is clear that the series converges uniformly and absolutely. Comparison
with (**) shows that
T'T
= T'
and completes the proof.
Our second case consists of '; arbitrary polynomials in
fzj ; zj : 1 j ng: Here, the operators T'; T are unbounded and we need
to be a little more careful. Nevertheless, we have for ' given by (**),
Theorem 2. For '; polynomials in (z ; :::; zn; z ; :::; zn); we have T'T
de ned on a dense domain consisting of linear combinations of functions of
1
5
1
the form fp(z)eza : a 2 Cn and p(z) polynomial in (z ; :::; zn)g: On this
domain
T'T = T'
and ' is polynomial in the zj ; zj :
1
Proof. Clearly, Tz = 2@j and it is now easy to check that T' p(z)eza =
q (z )eza
j
where p; q are polynomial in z ; :::; zn: The proof of the composition
formula is inductive, in several steps.
We note rst that, for ' polynomial in fzj ; zj : 1 j ng; T'Tz = T'z
implies T'jz j2 = T'Tjz j2 : This is because
1
j
j
j
j
= (T'Tz )Tz
= T 'z z = T'jz j2 :
T'Tjz j2
j
j
(
j
j) j
j
Next, we check inductively that T'Tz = T'z for all ' polynomial in fzj ; zj :
1 j ng: It is enough to consider ' monomial. Assume the result for '
of xed degree (' constant is trivial). The inductive step is:
j
j
T'z Tz
k
j
= T'Tz Tz
= T 'z z
= T'z z k 6= j;
j
j) k
(
j
k
T'z Tz
j
j
=
=
=
=
=
T'(Tz Tz )
T'(Tjz j2 ? 2I )
T'Tjz j2 ? T '
T'jz j2 ? '
T'z z ;
j
k
j
2
j
j
j
j
j
k
2 k( j
j
k
j
2
j
k j
j
j
= Tz (T'Tz ) = Tz
= T'z z ? z @ '
= Tz 'z :
for all '.
Tz ' Tz
Thus, T'Tz = T'z
k
6
)
'zj
k(
)
Next, for arbitrary ' we consider T'T and do induction on the degree
of . We can assume is monomial. Assume the result for all ' and for
of xed degree ( constant is trivial). The inductive step is, rst,
T'T z
j
= (T'T )Tz = T '
(
j
z
) j
= T' z :
j
We must also consider
T'Tz
= (T'Tz )T :
j
j
By the rst part of the proof,
T'Tz
j
= T'z
j
and by the inductive hypothesis
T'z T
j
= T 'z
(
:
j)
Thus, we need only check that
' zj
= (' zj ) :
This is a direct calculation. We note that
' zj
= 'zj ? 2(@j ')
so
(' zj ) = 'zj ? 2(@j ') X (?2)jkj
k
=
zj (@ k ')(@ )
k!
k
jkj
X
?2 (?k2)! (@ k @j ')(@k ):
k
Using
k
@ (z j
) = zj (@ k ) + kj (@ k? )
j
7
where
k ? j
= (k ; k ; :::; kj ? 1; kj ; :::; kn);
1
2
+1
we see that
(?2)jkj (@ k ')(@k z )
j
k!
k
X (?2)jkj
k
z j (@ k ')(@ )
=
k!
k
jkj
X
+ (?k2)! (@ k ')kj (@ k? ):
k
' zj
=
X
j
Thus, we need only check that
X
k
(?2)jkj (@ k ')k (@ k? ) = ?2 X (?2)jkj (@ k @ ')(@k ):
j
j
k!
k!
k
j
Reindexing the sum on the left by ` = k ? j completes the proof.
Remark. Since zj = zj ; the identity
' zj
= (' zj ) follows from the reasonably well-known associativity of [G]. Our computational proof has the advantage of giving associativity of as an immediate
corollary of Theorem 2 since
T' (
)
= T'(T T ) = (T'T )T = T '
(
8
:
)
3. T' with T'T' 6= T for any : In this section, I produce the
promised obstruction to composition of Berezin-Toeplitz operators. I use
some calculations from [BC2]. I begin with a needed improvement of [BC2:
Theorem 17]. In this section, we work on H (C; d) (n = 1): Here, the
Bergman reproducing kernel function for evaluation at z is just
2
K (w; z ) = ewz=
2
and it follows that
q
2
kz (w) = K (w; z )= K (z; z ) = ewz= ?jzj =
2
4
is a unit vector in H (C; d): We consider the unitary operator
2
(Raf )(z) = f (az)
on H (C; d) for jaj = 1:
Theorem 3. For jaj = 1 and Re a < 0, we have
2
kRa ? T k 1
for all such that K (; z) is in L (C; d) for every z in C:
Proof. We consider
2
kT ? Rak jhT
jhT
kz ; Ra kz i ? hRa kz ; Rakz ij
kz ; Ra kz i ? 1j :
Now,
hT
2
kz ; Ra kz i = h z ; K (; (1 + a)z )i e?jzj =
2
so we have
jhT
kz ; Rakz ij
q
e?jzj = k k K ((1 + a)z; (1 + a)z)
k k e?jzj = ej aj jzj =
k k ejzj a= :
2
2
2
2
2 Re
9
1+
2
2
2
4
Since Re a < 0, we see that
jhT kz ; Rakz ij ! 0
as jzj ! 1. Thus, kT ? Rak 1:
The function ' will be chosen to have the form '(z) = ejzj2 where
Re < so that T' makes sense.
1
4
Lemma. For = + i and '(z) = ejzj ; we have T' unitary with
1
2
5
5
2
T'T' = aRa
for a = (1 ? 2) = ? + i .
Proof. Re < and calculations outlined in [BC2; p. 582] show that
T' is diagonal in the basis
ek = (2k k !)? = z k ; k = 0; 1; :::
for H (C; d); with
T'ek = (1 ? 2)? k ek :
Now
= +i
and so
2
7
24
25
25
1
4
1 2
2
( +1)
T'T'ek
1
2
5
5
= (1 ? 2)
= ak ek :
k
2( +1)
ek
+1
But
aRaek
= ak ek
+1
and we are done.
We now have the promised
Theorem 4. For = +i and a = (1 ? 2) = ? + i, with '(z) = ejzj2 ,
kT'T' ? T k 1
for all such that K (; z) is in L (C;d) for every z in C.
Proof. Direct combination of Theorem 3 and the Lemma.
1
2
5
5
2
2
10
7
24
25
25
4. Remarks. There is a considerable space between Theorems 1 and
2 and Theorem 4. It does not seem easy to lift the known much stronger
positive results directly over from the setting of pseudodi erential operators.
It does seem likely that (**) provides a composition formula for BerezinToeplitz operators in a setting substantially larger than those of Theorems
1 and 2. For non-C 1 '; or even for general C 1 '; ; the problem of
determining whether there is a ' with T'T = T' ; as well as the form
of ' remains open.
Theorems 1 and 2 can be extended to the natural family of Gaussian
measures on Cn which provide
representation spaces for the Heisenberg group
[C]. For dr (z) = ( r )ne?rjzj2 dv(z) with r > 0 and H (Cn ; dr ) as before, we
have Bergman kernels
2
Kr (w; z ) = erwz
and Berezin-Toeplitz operators on H (Cn ; dr )
2
Z
(T' r f )(z) = erzw '(w) f (w) dr (w):
( )
C
n
Then minor modi cations yield
Theorem 10: For '; in Ba(Cn); ' r is also in Ba(Cn ) for
X
k
' r = ( ?r )jkj k (@ k ') (@ )
(y)
1
1
!
k
and T' r T r = T'r : The series in (y) converges uniformly and absolutely.
Moreover, for r > 1
X
( ? )jkj (@ k ')(@k )k 1 C ('; ; K )
k' ?
( )
( )
( )
r
r
1
k
jkjK
r
1
1
k
!
rK
+1
for C ('; ; K ) a constant independent of r:
Theorem 20. For '; polynomials in (z ; :::; zn; z ; :::; zn); we have T' r T
1
( )
1
r
( )
de ned on a dense domain consisting of linear combinations of functions of
the form fp(z)eza : a 2 Cn and p(z) polynomial in (z ; :::; zn)g: On this
domain
1
11
T' r T r
( )
( )
= T'r
( )
r
for ' r given by (y) and ' r is polynomial in the zj ; zj :
While Theorems 1 and 2 provide some basis for optimism about the development of a reasonably extensive Berezin-Toeplitz calculus on Cn , the
situation is considerably less promising on the classical Bergman space of
the disc, H (D; dA ) where D = fz 2 C : jzj < 1g and dA is normalized
Lebesgue area measure. In this case, the Bergman kernel function is just
K (z; w) = (1 ? zw )? and the Berezin-Toeplitz operator T' on H (D; dA
) is
given by
Z
(T'f )(z) = K (z; w) '(w) f (w) dAw :
2
2
2
(
)
D
Direct calculation shows, rst, that
Tz Tz
Moreover,
R
where P f = f (z) dA z
0
D
( )
=T
jzj2 :
1+log
=T
jzj2 + P
and P 6= T' for any ':
Tz2 Tz 2
1+2 log
0
0
References
[B]
V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure and Appl. Math. 14 (1961),
187-214.
[Be] F. A. Berezin, Covariant and contravariant symbols of operators,
Math. USSR Izv. 6 (1972), 117-1151.
[BC1] C. A. Berger and L. A. Coburn, Toeplitz operators on the SegalBargmann space, Trans. AMS 301 (1987), 813-829.
, Heat ow and Berezin-Toeplitz estimates, Amer.
[BC2]
J. Math. 116 (1994), 563-590.
12
[C]
L. A. Coburn, The measure algebra of the Heisenberg group, J. Funct.
Analysis 161 (1999), 509-525.
[F] G. B. Folland, Harmonic analysis in phase space, Annals of Math.
Studies, Princeton Univ. Press, Princeton, N.J., 1989.
[G] M. Gerstenhaber, On the deformation of rings and algebras, III, Annals of Math. (2) 88 (1968), 1-34.
[Gu] V. Guillemin, Toeplitz operators in n-dimensions, Integral Equations
and Operator Theory 7 (1984), 145-205.
State University of New York at Bu alo
Bu alo, NY 14214, USA
email: lcoburn@acsu.bu alo.edu
13