Commutativity of the Berezin transform of the pluriharmonic functions.

Abstract

Let B be the open unit ball in the n-space [C.sup.n] and [T.sub.u] be the Toeplitz operator on the Bergman space [L.sup.2.sub.a](B) with symbol u. For u [member of] [L.sup.[infinity]], the Berezin transform of u, is denoted by [??] and is defined as [??](z) =< [T.sub.u][k.sub.z]; [k.sub.z] >. In the present paper we discuss the multiplication and commutativity of the Berezin transform of the pluriharmonic functions.

AMS Subject Classification: Primary 47B35, 47B05.

Keywords and Phrases: Berezin transform, pluriharmonic functions, Bergman spaces.

1. Introduction

Let dv denote the normalized volume measure on the open unit ball B in the space [C.sup.n]. The Bergman space [L.sup.2.sub.a] is the set of all holomorphic functions on B which are in [L.sup.2](B; dv). We denote the set of all bounded linear operators on [L.sup.2.sub.a] by B([L.sup.2.sub.a]). Since [L.sup.2.sub.a] is a closed subspace of [L.sup.2] we can consider the orthogonal projection P from [L.sup.2] onto [L.sup.2.sub.a]. For [empty set] [member of] [L.sup.[infinity]](B; dv) the Toeplitz operator [T.sub.[empty set]] with symbol [empty set], is the bounded linear operator on [L.sup.2.sub.a] defined by [T.sub.[empty set]](f) = P([empty set]f). A function [empty set] [member of] [C.sup.2](B) is said to be pluriharmonic if it satisfies the [n.sup.2] differential equations [[bar.D.sub.j]] [[??].sub.k][empty set] = 0 for j; k = 1, ... , n. It is well known that every pluriharmonic function can be written as the sum of an analytic and a coanalytic function, for more details see [5].

Let [P.sub.a] be the orthogonal projection of [C.sup.n] onto the subspace [a] generated by a, i.e. [P.sub.a](z) = < z, a > / < a, a > a if a [not equal to] 0 and [P.sub.0] = 0, and [Q.sub.a] = I - [P.sub.a]. Define

[[empty set].sub.a](z) = a - [P.sub.a](z) - [(1 - [|a|.sup.2]).sup.1/2][Q.sub.a](z) / 1- < z, a >.

It is shown in [5] that [[empty set].sub.a] is a homeomorphism of [bar.B] onto [bar.B] and [[empty set].sub.a] [member of] Aut(B), the set of all automorphisms of B.

For f [member of] [C.sup.2](B) and z [member of] B, the invariant Laplacian of f, denoted [??]f, is defined by

([??]f)(z) = [DELTA](f [??] [[empty set].sub.z])(0)

where [DELTA] denotes the ordinary Laplacian. A function f [member of] [C.sup.2](B) is called M-harmonic on B if [??]f = 0 on B. Every pluriharmonic function on B is thus M-harmonic.

Since point evaluation at w [member of] B is bounded on [L.sup.2.sub.a], there exists a unique function [K.sub.w] in [L.sup.2.sub.a] such that < f, [K.sub.w] >= f(w) for all f [member of] [L.sup.2.sub.a]. The function K(z, w) defined on B x B by K(z, w) = [K.sub.w](z) is called the reproducing kernel or Bergman kernel. It is easy to see that K(z, w) = [(1- < z, w >).sup.-n-1] and by the reproducing property we have [[parallel]K(w,w)[parallel].sup.2.sub.2] = [K.sub.w](w) = [(1 - [|w|.sup.2]).sup.-n-1]. We will write [k.sub.w] for the normalized Bergman kernel, that is [k.sub.w] = [(1 - [|w|.sup.2]).sup.(n+1)/2] [K.sub.w]. If h [member of] [L.sup.2.sub.a], then it is easily verified that P([bar.h][k.sub.w]) = [bar.h](w)[k.sub.w] for every w [member of] B and specially P([bar.h]) = [bar.h](0). The real Jacobian of [[empty set].sub.w] at z [member of] B is ([J.sub.R][[empty set].sub.w])(z) = [(1 - [|w|.sup.2] / [|1- < z, w > |.sup.2]).sup.n+1] and so ([J.sub.R][[empty set].sub.w])(z) = [|kw(z)|.sup.2]. In the present paper we concern with the multiplication and commutativity properties of the Berezin transform in the case of pluriharmonic functions defined on the unit ball B in [C.sup.n]. Also we make use of the notion of M-harmonic function to deduce certain properties of the Berezin transform [??].

2. Berezin Transform

For T [member of] B([L.sup.2.sub.a](B)), the Berezin transform of T is the function [??] on B defined by [??](z) =< T[k.sub.z], [k.sub.z] > : For u [member of] [L.sup.[infinity]](B, dv) its Berezin transform [??] is defined to be the Berezin transform of the Toeplitz operator [T.sub.u]. Therefore we have,

< [T.sub.u][k.sub.z], [k.sub.z] >=< P(u[k.sub.z]), [k.sub.z] >=< u[k.sub.z], [k.sub.z] >

from which we obtain the formula,

[??](z) = [[integral].sub.B] u(w)[|[k.sub.z](w)|.sup.2]dv(w).

Also, since ([J.sub.R][[empty set].sub.w])(z) = [|k.sub.w](z)|.sup.2] we have the following change of variable formula

[[integral].sub.B] u[|[k.sub.w]|.sup.2]dv = [[integral].sub.B] u [??] [[empty set].sub.w]dv.

Good references for this topic are [7], [8], [5] and [9].

Kilic [3] considered the special case of multiplication of the Berezin transforms on certain functional Hilbert spaces, The author (joint with Seddighi) [8] considered the multiplication and commutativity of Berezin transform of the harmonic functions and in this paper we discuss these matters for the Berezin transform of the pluriharmonic functions.

It is clear that if u is M-harmonic then [??] = u. The converse is also true and we state it as the following theorem. One can find its proof in [1].

Theorem 2.1. If u [member of] [L.sup.[infinity]](B) and [[integral].sub.B] u [??] [[empty set].sub.w]dv = u(w), then [??]u = 0, or equivalently if [??] = u, then u is M-harmonic.

Now using the above result and the idea of the proof of [8] and [6] we can derive the following result.

Theorem 2.2. Let [empty set] and [psi] be bounded pluriharmonic functions on B, [empty set] = [[empty set].sub.1] + [[bar.[empty set].sub.2]] and [psi] = [[psi].sub.1] + [[bar.[psi].sub.2]] where [[empty set].sub.1], [[empty set].sub.2], [[psi].sub.1] and [[psi].sub.2] are holomorphic functions on B, and u = [[bar.[empty set].sub.2]][[psi].sub.1] - [[empty set].sub.1] [[bar.[psi].sub.2]], then, u is M-harmonic if and only if, [([T.sub.[empty set]][T.sub.[psi]]).sup.~] = [([T.sub.[psi]][T.sub.[empty set]]).sup.~].

Proof: At first we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (1)

On the other hand,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Now by Theorem 2.1 u is M-harmonic if and only if [??](w) = u(w) and the proof is completed.

Theorem 2.3. Let [empty set] and [psi] be bounded pluriharmonic functions on B, [empty set] = [[empty set].sub.1] + [[bar.[empty set].sub.2]] and [psi] = [[psi].sub.1] + [[bar.[psi].sub.2] where [empty set].sub.1], [[empty set].sub.2], [[psi].sub.1] and [[psi].sub.2] are holomorphic functions on B, then, [([empty set][psi]).sup.~] = ([empty set][psi]) if and only if, [[bar.[empty set]].sub.2][[psi].sub.1] + [[empty set].sub.1] [[bar.[psi].sub.2]] is M-harmonic.

Proof. By the similar computation as in the proof of last theorem we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

So we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and by using the Theorem 2.1 the proof is completed.

Theorem 2.4. Let [T.sub.[empty set]] and [T.sub.[psi]] be Toeplitz operators with bounded pluriharmonic symbols [empty set] = [[empty set].sub.1] + [[bar.[empty set].sub.2]] and [psi] = [[psi].sub.1] + [[bar.[psi].sub.2]] where [[empty set].sub.1], [[empty set].sub.2], [[psi].sub.1] and [[psi].sub.2] are holomorphic functions on B. Then [([T.sub.[empty set]][T.sub.[psi]]).sup.~] = [[??].sub.[empty set]] [[??].sub.[psi]] = [empty set][psi] if and only if [[bar.[empty set].sub.2]][[psi].sub.1] is M-harmonic.

Proof: At first by Theorem 2.1 [[??].sub.[empty set]][[??].sub.[psi]] =[??][??] = [empty set][psi]. Now by equation (1) in the proof of Theorem 2.2 we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

On the other hand we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

So we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Now by using Theorem 2.1 the proof is completed.

Theorem 2.5. Let A be an analytic function and A be a bounded pluriharmonic function on B, then [([T.sub.[empty set]][T.sub.[psi]]).sup.~](w) = [empty set][psi] = [[??].sub.[empty set]] [[??].sub.[psi]].

Proof: Suppose [[psi].sub.1] and [[psi].sub.2] are analytic functions on B such that [psi] = [[psi].sub.1] + [[bar.[psi].sub.2], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

On the other hand [[??].sub.[empty set]] [[??].sub.[psi]] = [??][??] = [empty set][psi], since [empty set] and [psi] are M-harmonic.

Corollary 2.6. Let A be a bounded pluriharmonic function and [bar.[psi]] be an analytic function on B, then [([T.sub.[empty set]][T.sub.[psi]]).sup.~](w) = [empty sey][psi] = [[??].sub.[empty set]] [[??].sub.[psi]].

The author would like to thank the referee for the helpful comments and a careful reading of the article.

References

[1] Ahern P., 1993, Flores M. and Rudin W., An Invariant Volume Mean Value Property, J. Funct. Anal., 111, pp. 380-397.

[2] Choe, Boo Rim, Lee, Young Joo, 1998, Pluriharmonic Symbols of Essentially Commuting Toeplitz Operators, Illinois J. Math., 42, pp. 280-293.

[3] Kilic S., 1995, The Berezin Symbol and Multipliers of Functional Hilbert Spaces, Proc. Amer. Math. Soc., 123, pp. 3687-3691.

[4] Lee, Young Joo, 1998, Essentialy Commuting Toeplitz Operators on theWeighted Bergman Spaces, Far East J. of Math. Sci., 1, pp. 280-293.

[5] Rudin W., Function Theory in the Unit Ball of [C.sup.n], Springer-Verlag, New York, 1980.

[6] Stroethoff K., 1993, Essentially Commuting Toeplitz Operators with Harmonic Symbols, Canad. J. Math., 45(5), pp. 1080-1093.

[7] Vaezpour S. M. and Seddighi K., 2000, Essentially Commuting Toeplitz Operators with Pluriharmonic Symbols, Iranian J. of Sci. and Tech., 24, pp. 413-417.

[8] Vaezpour S. M., 2004, A Comment on the Berezin Transform of Certain Operators, Italian J. of Pure and Appl. Math., 15, pp. 39-44.

[9] Zhu K., Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.

S. M. Vaezpour

Department of Mathematics,

Amirkabir University of Technology,

Tehran, Iran

[e-mail]