Öz
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$.
Anahtar Kelimeler
Pseudo-cyclic multipliers Hilbert function spaces invariant subspaces.
Kaynakça
- Agler, J., McCarthy, J.E., Pick Interpolation and Hilbert Function Spaces. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2002.
- 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.
- Aleman, A., Hartz, M., McCarthy, J.E., Richter, S., Factorizations induced by complete Nevanlinna-Pick factors, Adv. Math., 335(2018), 372–404.
- 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).
- 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).
- Beurling, A., On two problems concerning linear transformations in Hilbert space, Acta Math., 81(1949), 239–255.
- Duren, P., Schuster, A., Bergman Spaces. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2004.
- El-Fallah, O., Kellay, K., Mashreghi, J., Ransford, T., A Primer on the Dirichlet Space. Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2014.
- 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.
- Hedenmalm, H., Korenblum, B., Zhu, K., Theory of Bergman Spaces, Graduate Texts in Mathematics, Springer, New York, 2000.
- Luo, S., Corona theorem for the Dirichlet-type space, J. Geom. Anal., 32(3)(2022), 74–20.
- 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.
- Richter, S., Sundberg, C., Multipliers and invariant subspaces in the Dirichlet space, J. Operator Theory, 28(1) (1992), 167–186.
- Salas, H.N., A note on strictly cyclic weighted shifts, Proc. Amer. Math. Soc., 83(3)(1981), 555–556.
Öz
Kaynakça
- Agler, J., McCarthy, J.E., Pick Interpolation and Hilbert Function Spaces. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2002.
- 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.
- Aleman, A., Hartz, M., McCarthy, J.E., Richter, S., Factorizations induced by complete Nevanlinna-Pick factors, Adv. Math., 335(2018), 372–404.
- 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).
- 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).
- Beurling, A., On two problems concerning linear transformations in Hilbert space, Acta Math., 81(1949), 239–255.
- Duren, P., Schuster, A., Bergman Spaces. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2004.
- El-Fallah, O., Kellay, K., Mashreghi, J., Ransford, T., A Primer on the Dirichlet Space. Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2014.
- 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.
- Hedenmalm, H., Korenblum, B., Zhu, K., Theory of Bergman Spaces, Graduate Texts in Mathematics, Springer, New York, 2000.
- Luo, S., Corona theorem for the Dirichlet-type space, J. Geom. Anal., 32(3)(2022), 74–20.
- 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.
- Richter, S., Sundberg, C., Multipliers and invariant subspaces in the Dirichlet space, J. Operator Theory, 28(1) (1992), 167–186.
- Salas, H.N., A note on strictly cyclic weighted shifts, Proc. Amer. Math. Soc., 83(3)(1981), 555–556.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Operatör Cebirleri ve Fonksiyonel Analiz |
| Bölüm | Makaleler |
| Yazarlar | |
| Yayımlanma Tarihi | 31 Aralık 2024 |
| Gönderilme Tarihi | 29 Şubat 2024 |
| Kabul Tarihi | 24 Ekim 2024 |
| Yayımlandığı Sayı | Yıl 2024 Cilt: 16 Sayı: 2 |