Berezin transform

Berezin transformation

The Berezin transform associates smooth functions with operators on Hilbert spaces of analytic functions. The usual setting involves an open set and a Hilbert space of analytic functions on (cf. also Analytic function). It is assumed that, for each , the point evaluation at is a continuous linear functional on . Thus, for each , there exists a such that for every . Because reproduces the value of functions in at , it is called the reproducing kernel. The normalized reproducing kernel is defined by .

For a bounded operator on , the Berezin transform of , denoted by , is the complex-valued function on defined by

For each bounded operator on , the Berezin transform is a bounded real-analytic function on . Properties of the operator are often reflected in properties of the Berezin transform .

The Berezin transform is named in honour of F. Berezin, who introduced this concept in [a4].

The Berezin transform has been useful in several contexts, ranging from the Hardy space (see, for example, [a8]) to the Bargmann–Segal space (see, for example, [a5]), with major connections to the Bloch space and functions of bounded mean oscillation (see, for example, [a9]). However, the Berezin transform has been most successful as a tool to study operators on the Bergman space. For concreteness and simplicity, attention below is restricted to the latter setting.

The Bergman space (cf. also Bergman spaces) consists of the analytic functions on the unit disc such that (here, denotes area measure, normalized so that the area of equals ). The normalized reproducing kernel is then given by the formula .

For , the Toeplitz operator with symbol is the operator on defined by , where is the orthogonal projection of onto (cf. also Toeplitz operator). The Berezin transform of the function , denoted by , is defined to be the Berezin transform of the Toeplitz operator . This definition easily leads to the formula

If is a bounded harmonic function on , then the mean-value property can be used to show that . The converse was proved by M. Engliš [a6]: if and , then is harmonic on . P. Ahern, M. Flores and W. Rudin [a1] extended this result to functions (the formula above for makes sense in this case) and showed that the higher-dimensional analogue is valid up to dimension but fails in dimensions and beyond.

The normalized reproducing kernel tends weakly to as . This implies that if is a compact operator on the Bergman space , then as . Unfortunately, the converse fails. For example, if is the operator on defined by , then . Thus, in this case as , but is not compact (in fact, this operator is unitary, cf. also Unitary operator).

However, the situation is much nicer for Toeplitz operators, and even, more generally, for finite sums of finite products of Toeplitz operators. S. Axler and D. Zheng [a2] proved that such an operator is compact if and only if its Berezin transform tends to at .

The Berezin transform also makes an appearance in the decomposition of the Toeplitz algebra generated by the Toeplitz operators with analytic symbol. Specifically, G. McDonald and C. Sundberg [a7] proved that if , then can be written in the form , where is in the closed algebra generated by the bounded harmonic functions on the unit disc and is in the commutator ideal of . The choice of is not unique, but taking to be the Berezin transform of always works (see [a3]).

References

[a1]	P. Ahern, M. Flores, W. Rudin, "An invariant volume-mean-value property" J. Funct. Anal. , 111 (1993) pp. 380–397
[a2]	S. Axler, D. Zheng, "Compact operators via the Berezin transform" Indiana Univ. Math. J. , 47 (1998) pp. 387–400
[a3]	S. Axler, D. Zheng, "The Berezin transform on the Toeplitz algebra" Studia Math. , 127 (1998) pp. 113–136
[a4]	F. Berezin, "Covariant and contravariant symbols of operators" Izv. Akad. Nauk. SSSR Ser. Mat. , 36 (1972) pp. 1134–1167 (In Russian)
[a5]	C. Berger, L. Coburn, "Toeplitz operators and quantum mechanics" J. Funct. Anal. , 68 (1986) pp. 273–299
[a6]	M. Engliš, "Functions invariant under the Berezin transform" J. Funct. Anal. , 121 (1994) pp. 233–254
[a7]	G. McDonald, C. Sundberg, "Toeplitz operators on the disc" Indiana Univ. Math. J. , 28 (1979) pp. 595–611
[a8]	K. Stroethoff, "Algebraic properties of Toeplitz operators on the Hardy space via the Berezin transform" , Function Spaces (Edwardsville, IL, 1998) , Contemp. Math. 232 , Amer. Math. Soc. (1999) pp. 313–319
[a9]	K. Zhu, "VMO, ESV, and Toeplitz operators on the Bergman space" Trans. Amer. Math. Soc. , 302 (1987) pp. 617–646