Radial Toeplitz Operators on the Unit Ball and Slowly Oscillating Sequences

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.