Frame properties induced by iteration of a multiplication operator on Hardy spaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915480413929472 |
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| author | Cheshmavar, Jahangir |
| author_facet | Cheshmavar, Jahangir |
| contents | Motivated by the study of frame properties arising from iterates of linear operators, it was previously established that the multiplication operator $T_ϕx(t) = ϕ(t)x(t)$ cannot generate a frame in $L^2(a,b)$ (Results Math, 2019). In this note, we examine the behavior of such operators on the Hardy space $H^2(D)$, the Hilbert space of holomorphic functions on the open unit disk $D$ with square integrable boundary values. We show that, in contrast to the $L^2$-setting, the iterates of $T_ϕ$, for $ϕ\in H^{\infty}(D)$, exhibit fundamentally different frame properties in $H^2(D)$, leading to new structural insights and results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04879 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Frame properties induced by iteration of a multiplication operator on Hardy spaces Cheshmavar, Jahangir Functional Analysis Motivated by the study of frame properties arising from iterates of linear operators, it was previously established that the multiplication operator $T_ϕx(t) = ϕ(t)x(t)$ cannot generate a frame in $L^2(a,b)$ (Results Math, 2019). In this note, we examine the behavior of such operators on the Hardy space $H^2(D)$, the Hilbert space of holomorphic functions on the open unit disk $D$ with square integrable boundary values. We show that, in contrast to the $L^2$-setting, the iterates of $T_ϕ$, for $ϕ\in H^{\infty}(D)$, exhibit fundamentally different frame properties in $H^2(D)$, leading to new structural insights and results. |
| title | Frame properties induced by iteration of a multiplication operator on Hardy spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2509.04879 |