## Communications in Mathematical Analysis

- Commun. Math. Anal.
- Volume 14, Number 2 (2013), 77-94.

### Radial Toeplitz Operators on the Unit Ball and Slowly Oscillating Sequences

Sergei M. Grudsky, Egor A. Maximenko, and Nikolai L. Vasilevski

#### Abstract

In the paper we deal with Toeplitz operators acting on the Bergman space
$\mathcal{A}^2(\mathbb{B}^n)$ of square integrable analytic functions on the unit ball
$\mathbb{B}^n$ in $\mathbb{C}^n$. A bounded linear operator acting on the space
$\mathcal{A}^2(\mathbb{B}^n)$ is called *radial* if it commutes with unitary changes
of variables. Zhou, Chen, and Dong [9] showed that every radial operator $S$ is diagonal
with respect to the standard orthonormal monomial basis $(e_\alpha)_{\alpha\in\mathbb{N}^n}$.
Extending their result we prove that the corresponding eigenvalues depend only on the
length of multi-index $\alpha$, i.e. there exists a bounded sequence
$(\lambda_k)_{k=0}^\infty$ of complex numbers such that
$Se_\alpha=\lambda_{|\alpha|}e_\alpha$. Toeplitz operator is known to be radial if and
only if its generating symbol $g$ is a radial function, i.e., there exists a function $a$,
defined on $[0,1]$, such that $g(z)=a(|z|)$ for almost all $z\in\mathbb{B}^n$. In this
case $T_g e_\alpha = \gamma_{n,a}(|\alpha|)e_\alpha$, where the eigenvalue sequence
$\bigl(\gamma_{n,a}(k)\bigr)_{k=0}^\infty$ is given by \[ \gamma_{n,a}(k) =2(k+n)\int_0^1
a(r)\,r^{2k+2n-1}\,dr =(k+n)\int_0^1 a(\sqrt{r})\,r^{k+n-1}\,dr. \] Denote by $\Gamma_n$
the set $\{\gamma_{n,a}\colon a\in L^\infty([0,1])\}$. By a result of Suárez [8], the
$C^\ast$-algebra generated by $\Gamma_1$ coincides with the closure of $\Gamma_1$ in
$\ell^\infty$ and is equal to the closure of $d_1$ in $\ell^\infty$, where $d_1$ consists
of all bounded sequences $x=(x_k)_{k=0}^\infty$ such that \[
\sup_{k\ge0}\,\Bigl((k+1)\,|x_{k+1}-x_k|\Bigr) \lt +\infty. \] We show that the
$C^\ast$-algebra generated by $\Gamma_n$ does not actually depend on $n$, and coincides
with the set of all bounded sequences $(x_k)_{k=0}^\infty$ that are *slowly
oscillating* in the following sense: $|x_j-x_k|$ tends to $0$ uniformly as
$\frac{j+1}{k+1}\to1$ or, in other words, the function
$x\colon\{0,1,2,\ldots\}\to\mathbb{C}$ is uniformly continuous with respect to the
distance $\rho(j,k)=|\ln(j+1)-\ln(k+1)|$. At the same time we give an example of a
complex-valued function $a\in L^1([0,1],r\,dr)$ such that its eigenvalue sequence
$\gamma_{n,a}$ is bounded but is not slowly oscillating in the indicated sense. This, in
particular, implies that a bounded Toeplitz operator having *unbounded* defining
symbol does not necessarily belong to the $C^*$-algebra generated by Toeplitz operators
with *bounded* defining symbols.

#### Article information

**Source**

Commun. Math. Anal., Volume 14, Number 2 (2013), 77-94.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.cma/1356039033

**Mathematical Reviews number (MathSciNet)**

MR3011521

**Zentralblatt MATH identifier**

1275.47058

**Subjects**

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]

Secondary: 32A36, 44A60

**Keywords**

Radial Toeplitz operator Bergman space unit ball slowly oscillating sequence

#### Citation

Grudsky, Sergei M.; Maximenko, Egor A.; Vasilevski, Nikolai L. Radial Toeplitz Operators on the Unit Ball and Slowly Oscillating Sequences. Commun. Math. Anal. 14 (2013), no. 2, 77--94. https://projecteuclid.org/euclid.cma/1356039033