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 $\Omega \subset {\bf C} ^ { n }$ and a Hilbert space $H$ of analytic functions on $\Omega$ (cf. also Analytic function). It is assumed that, for each $z \in \Omega$, the point evaluation at $z$ is a continuous linear functional on $H$. Thus, for each $z \in \Omega$, there exists a $K _ { Z } \in H$ such that $f ( z ) = \langle f , K _ { z } \rangle$ for every $f \in H$. Because $K _ { z }$ reproduces the value of functions in $H$ at $z$, it is called the reproducing kernel. The normalized reproducing kernel $k _ { z }$ is defined by $k _ { z } = K _ { z } / \| K _ { z } \|$.

For $T$ a bounded operator on $H$, the Berezin transform of $T$, denoted by $\tilde{T}$, is the complex-valued function on $\Omega$ defined by

\begin{equation*} \widetilde{T} ( z ) = \langle T k _ { z } , k _ { z } \rangle. \end{equation*}

For each bounded operator $T$ on $H$, the Berezin transform $\tilde{T}$ is a bounded real-analytic function on $\Omega$. Properties of the operator $T$ are often reflected in properties of the Berezin transform $\tilde{T}$.

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 $L _ { a } ^ { 2 } ( D )$ (cf. also Bergman spaces) consists of the analytic functions $f$ on the unit disc $D \subset \mathbf{C}$ such that $\int _ { D } | f | ^ { 2 } d A < \infty$ (here, $d A$ denotes area measure, normalized so that the area of $D$ equals $1$). The normalized reproducing kernel is then given by the formula $k _ { \overline{z} } ( w ) = ( 1 - | z | ^ { 2 } ) / ( 1 - \overline{z} w ) ^ { 2 }$.

For $\varphi \in L ^ { \infty } ( D , d A )$, the Toeplitz operator with symbol $\varphi$ is the operator $T _ { \varphi }$ on $L _ { a } ^ { 2 } ( D )$ defined by $T _ { \varphi } f = P ( \varphi f )$, where $P$ is the orthogonal projection of $L ^ { 2 } ( D , d A )$ onto $L _ { a } ^ { 2 } ( D )$ (cf. also Toeplitz operator). The Berezin transform of the function $\varphi$, denoted by $\tilde { \varphi }$, is defined to be the Berezin transform of the Toeplitz operator $T _ { \varphi }$. This definition easily leads to the formula

\begin{equation*} \tilde { \varphi } ( z ) = ( 1 - | z | ^ { 2 } ) ^ { 2 } \int _ { D } \frac { \varphi ( w ) } { | 1 - z w | ^ { 4 } } d A ( w ). \end{equation*}

If $\varphi$ is a bounded harmonic function on $D$, then the mean-value property can be used to show that $\tilde { \varphi } = \varphi$. The converse was proved by M. Engliš [a6]: if $\varphi \in L ^ { \infty } ( D , d A )$ and $\tilde { \varphi } = \varphi$, then $\varphi$ is harmonic on $D$. P. Ahern, M. Flores and W. Rudin [a1] extended this result to functions $\varphi \in L ^ { 1 } ( D , d A )$ (the formula above for $\tilde { \varphi }$ makes sense in this case) and showed that the higher-dimensional analogue is valid up to dimension $11$ but fails in dimensions $12$ and beyond.

The normalized reproducing kernel $k _ { z }$ tends weakly to $0$ as $z \rightarrow \partial D$. This implies that if $T$ is a compact operator on the Bergman space $L_a^2$, then $\widetilde{T} ( z ) \rightarrow 0 $ as $z \rightarrow \partial D$. Unfortunately, the converse fails. For example, if $T$ is the operator on $L_a^2$ defined by $( T f ) ( z ) = f ( - z )$, then $\widetilde{T} ( z ) = ( 1 - | z | ^ { 2 } ) ^ { 2 } / ( 1 + | z | ^ { 2 } ) ^ { 2 }$. Thus, in this case $\widetilde{T} ( z ) \rightarrow 0 $ as $z \rightarrow \partial D$, but $T$ is not compact (in fact, this operator $T$ 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 $0$ at $\partial D$.

The Berezin transform also makes an appearance in the decomposition of the Toeplitz algebra $\mathcal{T}$ generated by the Toeplitz operators with analytic symbol. Specifically, G. McDonald and C. Sundberg [a7] proved that if $T \in \mathcal{T}$, then $T$ can be written in the form $T = T _ { \varphi } + C$, where $\varphi$ is in the closed algebra generated by the bounded harmonic functions on the unit disc and $C$ is in the commutator ideal of $\mathcal{T}$. The choice of $\varphi$ is not unique, but taking $\varphi$ to be the Berezin transform of $T$ 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