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

Contents Full-Text PDF