On Pseudo-cyclic Multipliers in Hilbert Function Spaces

Abstract

Let $\mathcal{H}$ be a separable complete Pick space of continuous functions on a compact set $\Omega$ with multiplier algebra $\mathrm{M}(\mathcal{H})$. The notion of the pseudo-cyclicity is recently defined by Aleman et al. In this short paper, we first extend their definition of the pseudo-cyclic multipliers to all functions $f$ in $\mathcal{H}$. Then we show that whenever one-function corona theorem holds for $\mathrm{M}(\mathcal{H})$ then a function $f$ in $\mathcal{H}$ is in the pseudo-cyclic class $ \mathcal{C}_n(\mathcal{H})$ if and only if $1/f$ is in the corresponding Pick-Smirnov type class $N_n^+(\mathcal{H})$. Furthermore, we show that non-vanishing functions $f \in \mathcal{H}$ are in the class $\mathcal{C}_1(\mathcal{H})$. For functions $\varphi, \psi$ in $\mathrm{M}(\mathcal{H})$, with at least one being in $\mathcal{C}_1(\mathcal{H})$, we also show that the invariant subspace generated by $\varphi \psi$ is equal to the intersection of invariant subspaces generated by $\varphi$ and $ \psi$.

Keywords

• Pseudo-cyclic multipliers
• Hilbert function spaces
• invariant subspaces.

References

1. Agler, J., McCarthy, J.E., Pick Interpolation and Hilbert Function Spaces. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2002.
2. Aleman, A., Hartz, M., McCarthy, J.E., Richter, S., The Smirnov class for spaces with the complete Pick property, J. Lond. Math. Soc. (2), 96(1)(2017), 228–242.
3. Aleman, A., Hartz, M., McCarthy, J.E., Richter, S., Factorizations induced by complete Nevanlinna-Pick factors, Adv. Math., 335(2018), 372–404.
4. Aleman, A., Perfekt, K.-M., Richter, S., Sundberg, C., Sunkes, J., Cyclicity and iterated logarithms in the drury-arveson space, Preprint at https://arxiv.org/abs/2301.10091, (2023).
5. Aleman, A., Perfekt, K.-M., Richter, S., Sundberg, C., Sunkes, J., Cyclicity in the drury-arveson space and other weighted besov spaces,Preprint at https://arxiv.org/abs/2301.04994, (2023).
6. Beurling, A., On two problems concerning linear transformations in Hilbert space, Acta Math., 81(1949), 239–255.
7. Duren, P., Schuster, A., Bergman Spaces. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2004.
8. El-Fallah, O., Kellay, K., Mashreghi, J., Ransford, T., A Primer on the Dirichlet Space. Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2014.

9. Hartz, M., An Invitation to the Drury–arveson Space. In: Mashreghi, J. (ed.) Lectures on Analytic Function Spaces and Their Applications, Springer, Fields Institute Monographs, 2023.
10. Hedenmalm, H., Korenblum, B., Zhu, K., Theory of Bergman Spaces, Graduate Texts in Mathematics, Springer, New York, 2000.
11. Luo, S., Corona theorem for the Dirichlet-type space, J. Geom. Anal., 32(3)(2022), 74–20.
12. Paulsen, V.I., Raghupathi, M., An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2016.
13. Richter, S., Sundberg, C., Multipliers and invariant subspaces in the Dirichlet space, J. Operator Theory, 28(1) (1992), 167–186.
14. Salas, H.N., A note on strictly cyclic weighted shifts, Proc. Amer. Math. Soc., 83(3)(1981), 555–556.