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ABOUT COMPOSITION OF TOEPLITZ OPERATORS IN SEGAL-BARGMANN SPACE.

arXiv:0806.1025v1 [math.CV] 5 Jun 2008

ROMINA RAMIREZ AND MARCELA SANMARTINO Abstract. Some positive results about the composition of Toeplitz operators on the Segal-Bargmann space are presented. A Wick symbol where it is not possible to construct its associated Toeplitz operator is given.

2000 AMS Subject Classi?cation: Primary 47B35; Secondary 47G30

1. Introduction Toeplitz operators have been object of study in di?erent disciplines. In physics, these operators (also called anti Wick operators) were introduced by Berezin as a quantization rule in quantum mechanics ( see [1], [2], [7]). In partial di?erential equations , Toeplitz operators and their adjoints, play an important role in extending known results in the space of entire functions to the context of Segal Bargmann spaces. Also, they have been extensively studied as an important mathematical tool in signal analysis ([11], [12], [5]). One of the problems still open is to know how to de?ne the class of the symbols where the composition of Toeplitz operators is closed. In this way, in section 4, we give a positive result for radial symbols. Many authors have studied this problem. In [10], [11], [12], [13] and [14], they have obtained that the composition of Toeplitz operators can be written as a Toeplitz operator plus a controllable remainder term, and in [3] and [8] we can ?nd some exact results.

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ROMINA RAMIREZ AND MARCELA SANMARTINO

2. An overview of Toeplitz operators. 2.1. De?nition of a Toeplitz operator.

Let L2 (Cn , d?) be the Hilbert space of square integrable functions on Cn , with the Gaussian measure d?(z ) = π ?n e?|z| dv (z ) where dv (z ) is the Lebesgue measure on Cn . In this work we will deal with the Segal-Bargmann space F 2 (Cn ), the subspace of L2 (Cn , d?) of all the analytic functions in Cn . This space is a particular case of a Fock space 1. Here the annihilation operator is given by aψ = operator by a? ψ = z.ψ . In quantum ?eld theory, one frequently works with product of creation operators and annihilation operators. Taking into account that this operation is not commutative, we can develop a theory of symbols associated to the order aa? known like theory of Wick symbols, or consider the theory of anti Wick symbols related to the product order a? a. In these two theories, an important role is played by the family of coherent states (or Poisson vectors). These vectors are in F 2 (Cn ) and are given by Kv (z ) = ezv . Some of the main properties of these vectors are: (1) The vector Kv is the eigenvector for the operator a = eigenvalue v . (2) The relation v → Kv is an holomorphic function.

1In the general case, a Fock space F is a Hilbert space in which there exist annihilation and creation operators, satisfying the canonical commutation rules. There exists also a vector Φ0 (called vacumm vector ) annihilated by the annihilations operators, such that the system ? ? ((b a?)α Φ0 ) √ is complete in F . α! ? ?z

2

?ψ and the creation ?z

related to the

ABOUT COMPOSITION OF TOEPLITZ OPERATORS IN SEGAL-BARGMANN SPACE.

3

(3) For all f ∈ F 2 (Cn ), we have that (2.1) f (v ) = f, Kv where , is the usual inner product in L2 (Cn , d?). Let us consider P : L2 (Cn , d?) → F 2 (Cn ), the orthogonal projection operator n the direction of the coherent state Kz P g (z ) = (2.2) =

C

g, Kz g (w) ez.w d?(w)

Now, for a complex function ?(z, z ), we de?ne the operator T? : F 2 (Cn ) → F 2 (Cn ) given by T? f (z ) := (2.3) = ?(w, w ) ezw f (w) d?(w) P [?f ](z )

Note that this operator is well de?ned if ?f ∈ L2 (Cn , d?). De?nition 2.1. If for all z ∈ C, Kz ∈ Dom (T? ) where Dom (T? ) = {f such that ?f ∈ L2 (Cn , d?)}, then T? is a Toeplitz operator (anti Wick operator ) with anti Wick symbol ?. Sometimes we write σ AW (T ) = ?.

Remark 2.2. For example, if ?(z ) satis?es for δ < 1/2 |?(z )| ≤ Ceδr

2

then T? is a Toeplitz operator (for details see [4]).

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ROMINA RAMIREZ AND MARCELA SANMARTINO

2.2. Quantization and symbolic calculus. One of the important facts of this kind of operators, is its relation with quantization procedures, essentially this means to associate an operator (quantum observable) to a given function (classical observable). There are to many ways to do that of consistent way to the probabilistic interpretation. In this context, pseudodi?erential operators play a remarkable roll (Toeplitz operators can be understood in this way). Depending on the properties that we want to preserve in the quantization process we can consider a given operator in di?erent symbolic calculus: ? In L2 (Rn , dv ): There are so many symbolic calculus in this space. The most useful in quantization is the Weyl calculus since given a real symbol the operator associated is selfadjoint. An operator A is represented in the Weyl calculus by x+y , ξ ) e2i(x?y)ξ f (y ) dydξ, 2

Af (x) =

σw (

where σw is its symbol. ? In L2 (Cn , d?): The correspondence between symbol and operator can be given by Toeplitz operators, as we have seen, or by Wick operators e(z?ξ)ξ f (v ) σ W (A)(ξ, z ) dv (ξ ) or

(2.4)

Af (z ) =

where σ W (A) is the Wick symbol. All these symbols are related by the kernel of the heat Ht (see for instance [7] or [4])

2

in the following way

(4t)?n e?(x

2The heat di?usion semigroup {H } 2n is de?ned by H f = f ? γ with γ (x, ξ ) = t t>0 on R t t t

2

+ξ2 )/4t .

ABOUT COMPOSITION OF TOEPLITZ OPERATORS IN SEGAL-BARGMANN SPACE.

5

(2.5)

σw (A) σ W (A) σ W (A)

= H1/2 (σ AW (A)) = H1/2 (σw (A)) = H1 (σ AW (A)).

There is another interesting aspect about the Wick and anti Wick symbols to be consider: ? The Weyl and Wick symbols of a given pseudodi?erential operator are always de?ned. By (2.5) we can see that not all these operators have an anti Wick symbol, since in order to ?nd an anti Wick symbol one has to solve -for example- the inverse heat equation with initial condition the Weyl symbol for the time t = 1/2. ? The norm of an operator can be bounded by its anti Wick and Wick symbols: (2.6) σ W (A)

∞

≤ A ≤ σ AW (A)

∞

? The Wick symbol of A can be de?ned by means of its action on the coherent states by (2.7) σ W (A)(v, z ) = AKv , Kz . Kv , Kz

It is not possible to have a formulae like this for anti Wick symbols. ? Toeplitz operators have the properties that for positive anti Wick symbol the associated operator is a positive, and for real anti Wick symbol the operator is selfadjoint (as for Weyl operators). Since the correspondence between symbols and operators, can be interpreted like a quantization process. In this context, it is necessary to have a bilinear operation ? between the symbols, such that if τ = ? ? ψ then Tτ = T? Tψ . This bilinear

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ROMINA RAMIREZ AND MARCELA SANMARTINO

operation, is completely de?ned for Weyl (and the other symbols in Rn ) and Wick symbols, obtaining a closed symbolic calculus in these cases. But the problem for the anti Wick symbols -as we have said in the Introduction- is still open. 3. Preliminary results. Among all the known positive results in the composition of Toeplitz operators, we have the following : If ? and ψ are anti Wick symbols in one of these classes ? Analytic functions ([8]), ? functions in the smooth Bochner algebra Ba (Cn ) ([3]) or ? Polynomial functions ([3]) then T? Tψ = Tτ = T??ψ , where (3.1) ??ψ = (?1)k k k (? ?)(? ψ ). k!

k

By the other way, in [3] we can ?nd an example that show the limitation on the ability to compose Toeplitz operators: considering the one dimensional SegalBargmann space F 2 (C) and ? = e2λ|z| with λ =

2

1+2i 5 .

Although T? T? exists, moreover is a bounded operator, there is not a symbol τ such that T? T? = Tτ on F 2 (C). In the next section, we will give a rather general result that encompassing this example, and a positive result about the composition of Toeplitz operators with radial symbol in F 2 (C).

3.1. Radial symbols.

3

The theory about Toeplitz operators becomes interesting when the symbols are radials, because, these operators are unitary equivalent to multiplication operators.

3All the results in this section, can be founded in Grudsky and Vasilevski, Toeplitz operators on the Fock space: Radial component e?ects. [6]

ABOUT COMPOSITION OF TOEPLITZ OPERATORS IN SEGAL-BARGMANN SPACE.

7

More accurately, given a radial symbol a(|z |) = a(r), the Toeplitz operator Ta is

+ unitary equivalent to the multiplication operator γa I acting on l2 where

(3.2)

γa (n) =

1 n!

R+

√ 2 a( r ) rn e?r dr = n!

a(r) r2n+1 e?r dr,

R+

2

+ + i.e, there exist unitary operators R : L2 (C, d?) → l2 and R? : l2 → F 2 (C) such

that (3.3) RTa R? {cn } = {γa (n).cn }

+ for all sequence {cn } ∈ l2 , where

R? =

1 √ n!

?(z ) z n d?(z )

C n∈Z+

and

(3.4)

R ? { cn }

=

n∈Z+

cn √ z n. n!

+ + RR? = I : l2 → l2 and R? R = P : L2 (C, d?) → F 2 (C), where P is the projector

operator de?ned in (2.2).

Remark 3.1. In order to guarantee the convergence of the integral (3.2) it is con?r sidered, for example a(r) belonging to L∞ ). This space consists of all 1 (R+ , e

2

measurable functions a(r) on R+ such that |a(r)| e?r rn dr < ∞ for all n ∈ Z+ .

2 2

R+

?r Now, given a sequence γ = {γ (n)} ∈ l∞ there exists a symbol a(r) ∈ L∞ ) 1 (R+ , e

such that the operator Ta is unitarily equivalent to the multiplication operator by the sequence γ , i.e γ (n) = γa (n).

?r ) Remark 3.2. For a(r) ∈ L∞ 1 (R+ , e

2

? Ta is a bounded operator if and only if {γa (n)} is a bounded sequence.

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ROMINA RAMIREZ AND MARCELA SANMARTINO

? The spectrum of a bounded Toeplitz operator is given by sp(Ta ) = {γa (n), n ∈ Z+ }

4. Main result.

?r Let us de?ned L∞ ), the linear space of all measurable functions f (r) 2 (R+ , e

2

on R+ such that qf (n) :=

R+

|f (r)|2 e?r rn+1 dr < ∞

2

for all n ∈ Z+ and for these functions we de?ned Af (x) = xn qf (n) , x ∈ R. n! n=0

∞

Theorem 4.1. If T? and Tψ are two Toeplitz operators with radial symbols ?(r),

?r ψ (r) ∈ L∞ )) and 1 (R+ , e

2

(1) Tψ is a bounded Toeplitz operator, (2) {γ? (n).γψ (n)} ∈ l∞ ,

?r (3) ? belongs to L∞ )) and A? (x) converge for all x ∈ R+ , 2 (R+ , e ?r there exists τ ∈ L∞ ) such that Tτ = T? Tψ is a Toeplitz operator. 1 (R+ , e

2 2

Proof : As we have mencioned in the previous section, if {γ? (n).γψ (n)} ∈ l∞

?r there exists τ ∈ L∞ ) such that {γτ (n)} = {γ? (n).γψ (n)}. 1 (R+ , e

2

ABOUT COMPOSITION OF TOEPLITZ OPERATORS IN SEGAL-BARGMANN SPACE.

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Now, we have to see that, for this τ : (1) Tτ = T? Tψ in the domain of T? Tψ (2) Tτ = T? Tψ is indeed a Toeplitz operator. 1) Let us take f (z ) ∈ Dom (T? Tψ ) ? F 2 (C). Then f (z ) =

∞ n=0

a n z n = R ? { cn } ,

√ + where {cn } = {an n!} ∈ l2 and R? is the operator given by (3.4). Using, the de?nition of Toeplitz operator through the projection P = R? R given in (2.3) , and the equation (3.3) we have that T? Tψ f = = = = = = T? P (ψf ) T? R? (RψR? ){cn } T? R? {γψ (n).cn } R? R?R? {γψ (n).cn } R? {γ? (n).γψ (n)cn } =: R? {γτ (n).cn } R? (RTτ R? ){cn } := Tτ f

2) In order to prove that all coherent states belong to the domain of Tτ , we will see that Ka (z ) = eza ∈ Dom (Tτ ). Obviously Ka (z ) ∈ Dom (Tψ ) since Tψ is a Toeplitz operator. Now, it remains to prove ?Tψ Ka (z ) ∈ L2 (C, d?).

C

|?(z )Tψ Ka (z )|2 e?|z| dv (z ) ≤

2

C

|?(z )|2 Tψ

2

|Ka (z )|2 | e?|z| dv (z )

2

≤

Tψ

2 C

|?|2

(2|a||z |)n ?|z|2 e dv (z ) n! n=0

∞

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ROMINA RAMIREZ AND MARCELA SANMARTINO

Using polar coordinates, we have Tψ

2

|2a|n n! n=0

∞

C

|?|2 |z |n e?|z| dv (z )

2

=

Tψ

2

ω1

|2a|n n! n=0

∞

∞

R+

|?|2 rn+1 e?r dv (z )

2

=

Tψ

2

ω1

|2a|n q? (n) n! n=0

= A? (2|a|). Because Tψ

2

is bounded and A? (2|a|) exists for all a ∈ C, then

C

|?Tψ Ka (z )|2 e?|z| dv (z ) < ∞.

2

So, Ka ∈ Dom (T? Tψ ) ? Dom (Tτ ), and therefore Tτ is indeed a Toeplitz operator.

Remark 4.2. Smooth Bochner Algebra. For radial symbols ? and ψ in the Bochner algebra Ba (C), we know that T? Tψ = T??ψ with ? ? ψ given by (3.1) and T??ψ is a Toeplitz operator. By other way, it is possible to apply the Theorem 4.1, and it can be proved that the symbol obtained τ satis?es τ = ? ? ψ . We give the proof of these facts in the Appendix.

Remark 4.3. Since we have mentioned in section I, there exist results about the composition of Toeplitz operators for analytic symbols and polynomial symbols. The ?rst case is not contemplated by this theorem, because radial symbols cannot be analytic (except by the constants where the theorem is trivial). For polynomial radial symbols the situation is a quite di?erent: When we calculate its sequence {γ (n)} we have that it is unbounded, then we cannot apply the Theorem 4.1. Nevertheless the composition can be made using the formula (3.1).

4.1. About the composition obstruction.

ABOUT COMPOSITION OF TOEPLITZ OPERATORS IN SEGAL-BARGMANN SPACE.

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Theorem 4.4. Let be A a bounded operator with domain in the Fock space F 2 (C), such that its Wick symbol is σ W (A) = c e?θ|z| , with c, θ ∈ C. Then (1) If |θ|2 = 2Re(θ) and Re(θ) > 1, there are not any bounded Toeplitz operator Tτ such that A = Tτ . (2) If |θ|2 > 2Re(θ), there are not any bounded Toeplitz operator Tτ with τ ∈

?r L∞ ) such that A = Tτ . 1 (R+ , e

2 2

Proof : (1) Let us take the operator A such that σ W (A) = e?θ|z| . Then by (2.4) Af (z ) = = If we de?ne en (z ) := e(z?ξ)ξ f (v ) σ W (A)(z, ξ ) dv (ξ ) e(z?ξ)ξ e?θ(zξ) f (v ) dv (ξ ).

zn √ n!

2 2

then {en (z ), n ∈ N}, basis of F 2 (C) and

Aen (z ) = c(1 ? θ)n+1 en (z ). By other way, if we consider the operator Ma f (z ) = c.af (az ), we note that Aen (z ) = c(1 ? θ)n+1 en (z ) = Ma en (z ) where a = 1 ? θ. Then the operators A and Ma agree in the basis and consequently in all the Fock space.

By the hypothesis on θ, we have that |a| = |1?θ| =

1 ? 2Re(θ) + |θ|2 =

1 and Re(a) = 1 ? Re(θ) < 0. By the results in [3], there are not any bounded Toeplitz operator Tτ such that A = Tτ in all the Fock space.

?r (2) Supose that there exists τ ∈ L∞ ) such that Tτ is a bounded 1 (R+ , e

2

Toeplitz operator and Tτ = A.

12

ROMINA RAMIREZ AND MARCELA SANMARTINO

The Wick symbol of Tτ can be calculated by (4.1) σ W (Tτ ) = e?|z|

2

|z |2n γτ (n) n! n=0

∞

where γτ (n) is de?ned by (3.2) (see [6] for more details). By other way (4.2) e?θ|z| = e?|z|

2 2

|z |2n (1 ? θ)n . n ! n=0

∞

Then, taking into account that σ W (Tτ ) = c e?θ|z|

2

we have that γτ (n) = (1 ? θ)n . But by the hypothesis on θ |γτ (n)| = |c.(1 ? θ)|n = |c| 1 ? 2Re(θ) + |θ|2 ≥ |c|(1 + ?)n . Then γτ (n) is an unbounded sequence. This contradicts that the operator can be bounded by the Remark 3.2.

n

Remark 4.5. This result, includes the example mentioned in section 3, that show the limitation on the ability to compose Toeplitz operators [3]: If ? = e2(

1+2i 2 5 )| z |

, then by (3.2) γ? (n) = = = 1 n! e2(

R+

1+2i 5 )r

rn e?r dr

1?2

1 + 2i 5

?(n+1)

3 4 ? i 5 5

?(n+1)

4 ?(n+1) | = 1, therefore {γ? (n)} ∈ l∞ . Then, |γ? (n)| = |( 3 5 ? 5 i)

ABOUT COMPOSITION OF TOEPLITZ OPERATORS IN SEGAL-BARGMANN SPACE.

13

By ( 4.1) σ W (T? T? )(z, z ) = e?|z| = e?|z| =

2

|z |2n γ? (n)γ? (n) n! n=0 |z |2n n! n=0

?2 ∞

∞

2

3 4 ? i 5 5

32 24

?2(n+1)

3 4 ? i 5 5

32 24

e?( 25 + 25 i)

= ce?( 25 + 25 i)

Therefore, σ W (T? T? ) = e?K |z| with |K |2 = 2Re(K ) =

2

64 25 .

Taking into account,

the operator T? T? is bounded, by 1) in the Theorem 4.4, the operator T? T? can not be a Toeplitz operator in F 2 (C).

5. Appendix. Smooth Bochner Algebra. Consider ?(|z |) = ?(r), ψ (|z |) = ψ (r) radial functions in the Bochner algebra Ba (C). The functions in Ba (C) are bounded, uniformly continuous, with bounded derivatives of all orders. Therefore, by the hypothesis of ? and ψ we can apply the Theorem 4.1 and then, there exists τ such that Tτ = T? Tψ is a Toeplitz operator in the Fock space. As we have mentioned in section 3, we know that T? Tψ = T??ψ with ? ? ψ given by (3.1) and T??ψ is a Toeplitz operator. Then we have on F 2 (C) that Tτ = T??ψ . The question is if τ = ? ? ψ since for Toeplitz operators, is not always true that if Tη = 0 then η = 0. However, we can prove that if the anti Wick symbol η of Tη belongs to E ε (R+ , e?r )

?r (a subclass of L∞ )), then the operator Tη = 0 if and only if η = 0 almost 1 (R+ , e

2 2

everywhere (for details see [6]).

14

ROMINA RAMIREZ AND MARCELA SANMARTINO ?r The space E ε (R+ , e?r ) is the subclass of L∞ ) that consists of all 1 (R+ , e

2 2

functions η (r) satisfying at +∞ the following estimate: ?xed ε > 0, |η (r)|e?r

2

+εr

≤ C.

Proposition 5.1. Given ? and ψ two radial functions in the Bochner algebra Ba (C). Then (1) ? and ψ satis?es the hypothesis of Theorem 4.1 and (2) the function given by Theorem 4.1 satis?es τ = ? ? ψ . Proof : We must prove that there exists ε > 0 such that τ ???ψ ∈ E ε (R+ , e?r ). τ ∈ E ε (R+ , e?r ) :

∞ 1

2 2

|τ (r)|e?r

2

+εr

dr

≤

(

1

∞

|τ (r)|2 re?r dr)1/2 (

2

∞ 1

e?r e2εr dr)1/2 r

2

≤

(

1

∞

|τ (r)|2 re?r dr)1/2 (

2

∞ 1

e?r e2εr dr)1/2 .

2

Is trivial to see that the second integral converges for all ε. For another way, note that 1 ∈ F 2 (C) and Tτ is bounded, then 1 ∈ Dom (Tτ ), i.e., |τ.1|2 e?|z| = ω1

2

C

|τ (r)|2 re?r < ∞.

2

Then the ?rst integral converge.

Therefore +∞.

∞ 1

|τ (r)|e?r

2

+εr

dr converge and then |τ (r)|e?r

2

+εr

is bounded in

Taking into account that ? ? ψ is bounded, then η (r) := τ (r) ? ? ? ψ (r) ∈ E ε (R+ , e?r ).

2

ABOUT COMPOSITION OF TOEPLITZ OPERATORS IN SEGAL-BARGMANN SPACE.

15

References

[1] F. A. Berezin, Covariant and contravariant symbols of operators, Math. USSR Izv. 6 (1972). [2] F. A. Berezin, Quantization, Izv. Akad. Nauk SSSR Ser. Mat, 38 (1974), 1116-1174. Math. USSR Izv, 8 (1974), 1109-1165. [3] L.A Coburn On the Berezin -Toeplitz Calculus the Proceedings of the AMS 129 (2001) pp. 3331-3338. [4] G. Folland Harmonic analysis in phase space. Annals of Mathematics Studies number 122, Princenton University press 1989. [5] K. Grochenig Foundations of Time-Frequency Analysis. 360 pp., Birkh¨auser, Boston, 2001. [6] S. Grudsky, N. Vasilevski, Toeplitz operators on the Fock space: Radial component e?ects. Integral Equations and Operator Theory, v. 44, no. 1, 2002, p. 10-37 [7] A Berezin, M. A Shubin The Schrodinger equation Moscow State University, (1983), Mosc?u. [8] A. Brown, P.R Halmos, Algebraic properties of Toeplitz operators. J Rein ngew math 213: 89-102,1963. [9] M. A. Shubin, Pseudodi?erential Operators and Spectral Theory. Berl? ?n, Heidelberg, New York: Springer-Verlag (1987). [10] H. Ando and Y. Morimoto. Wick calculus and the Cauthy problem for some dispersive equations. Osaka J. Math., 39(1): 123-147, 2002. [11] E. Cordero and K. Grochenig. Symbolic calculus and Fredholm property for localization operators. Preprint, 2005. [12] E. Cordero and L. Rodino. Wick calculus: a time-frequency approach. Osaka J. Math., 42(1): 43-63, 2005. [13] N. Lerner. The Wick calculus of pseudo-di?erential operators and energy estimates. In New trends in microlocal analysis (Tokyo, 1995), pages 23-37. Springer, Tokyo, 1997. [14] D. Tataru. On the Fe?erman-Phong inequality and related problems. Comm. Partial Di?erential Equations, 27(11-12): 2101-2138, 2002. ? tica, Facultad de Ciencias Exactas, Universidad Nacional Departamento de Matema de La Plata, 1900 La Plata (Buenos Aires), Argentina E-mail address : romina@mate.unlp.edu.ar ? tica, Facultad de Ciencias Exactas, Universidad Nacional Departamento de Matema de La Plata, 1900 La Plata (Buenos Aires), Argentina E-mail address : tatu@mate.unlp.edu.ar

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