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Radial Toeplitz operators on the unit ball and slowly oscillating sequences. (English) Zbl 1275.47058

Authors’ 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. Z.-H. Zhou, W.-L. Chen and X.-T. Dong [Complex Anal. Oper. Theory 7, No. 1, 313–329 (2013; Zbl 1311.47038)] 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 \(Se_\alpha=\lambda_{|\alpha|}e_\alpha\).
A 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 \(\{\gamma_{n,a}: a\in L^\infty([0,1])\}\). By a result of D. Suárez [Bull. Lond. Math. Soc. 40, No. 4, 631–641 (2008; Zbl 1148.30033)], 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 \[ \sup_{k\geq0}\,\Bigl((k+1)\,|x_{k+1}-x_k|\Bigr)<+\infty. \] 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 \(x:\{0,1,2,\dots\}\to\mathbb{C}\) 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.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A36 Bergman spaces of functions in several complex variables
44A60 Moment problems
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Full Text: Euclid