Communications in Mathematical Analysis

Radial Toeplitz Operators on the Unit Ball and Slowly Oscillating Sequences

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
Secondary: 32A36, 44A60

Radial Toeplitz operator Bergman space unit ball slowly oscillating sequence


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.

Export citation