Journal of Operator Theory

Volume 91, Issue 2, Spring 2024 pp. 505-519.

Convergence and preservation of cyclicity

Authors: Alejandra Aguilera (1), Daniel Seco (2)
Author institution: (1) Departamento de Matematica, Universidad de Buenos Aires, Instituto de Matematica ``Luis Santalo'' (IMAS-CONICET-UBA), Buenos Aires, Argentina
(2) Universidad de la Laguna, Universidad Carlos III de Madrid and Instituto de Ciencias Matematicas (CSIC-UAM-UC3M-UCM) Avenida Astrofisico Francisco Sanchez, s/n. Facultad de Ciencias, seccion: Matematicas, apdo. 456. 38200 San Cristobal de La Laguna Santa Cruz de Tenerife, Spain

Summary: We prove that the set of cyclic (respectively, non-cyclic) functions in Dirichlet type spaces $D_{\alpha}$ is not closed in the topology induced by the norm. We also show that some additional conditions on a convergent sequence of cyclic functions $\{f_n\}$ force cyclicity of the limit $f$. We show counterexamples satisfying all but one of these conditions. Then we find precise estimates for the distance between the corresponding optimal polynomial approximants of each degree $d$ and control its blow up as the choice of $f$ moves within $D_\alpha$.

DOI: http://dx.doi.org/10.7900/jot.2022jun10.2403
Keywords: Optimal polynomial approximants, Dirichlet spaces, invariant subspaces

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