Weak null maximum and integration of families of multiplication operators
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916935973732352 |
|---|---|
| author | Norrbo, David |
| author_facet | Norrbo, David |
| contents | Let $X$ be a reflexive Hardy space or weighted Bergman space on the unit disk in the complex plane. For a bounded linear operator $S$ on $X$, let $\textrm{wem}(S):= \sup_{(f_n)} \limsup_n \|Sf_n\|$, that is, the supremum of cluster points of $n\mapsto \|S f_n\|$, where $(f_n)$ is any unit norm weakly null sequence. This quantity coincides with the essential norm on the reflexive weighted Bergman spaces.
For a suitable family $\{ g_t : t\in]0,1[ \}$ of bounded analytic functions on the unit disk, we characterize when one can exchange $\textrm{wem}(\cdot)$ and integration over $t$ of the multiplication operators $M_{g_t}$, that is, when $\textrm{wem}( \int M_{g_t}\, dt ) = \int \textrm{wem}( M_{g_t} ) \, dt $; when the functions $g_t,t\in]0,1[$ can be continuously extended to the unit circle, we obtain a neat function-theoretic characterization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05173 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak null maximum and integration of families of multiplication operators Norrbo, David Functional Analysis Complex Variables 47B91, 47G10, (Primary) 30H10, 30H20 (Secondary) Let $X$ be a reflexive Hardy space or weighted Bergman space on the unit disk in the complex plane. For a bounded linear operator $S$ on $X$, let $\textrm{wem}(S):= \sup_{(f_n)} \limsup_n \|Sf_n\|$, that is, the supremum of cluster points of $n\mapsto \|S f_n\|$, where $(f_n)$ is any unit norm weakly null sequence. This quantity coincides with the essential norm on the reflexive weighted Bergman spaces. For a suitable family $\{ g_t : t\in]0,1[ \}$ of bounded analytic functions on the unit disk, we characterize when one can exchange $\textrm{wem}(\cdot)$ and integration over $t$ of the multiplication operators $M_{g_t}$, that is, when $\textrm{wem}( \int M_{g_t}\, dt ) = \int \textrm{wem}( M_{g_t} ) \, dt $; when the functions $g_t,t\in]0,1[$ can be continuously extended to the unit circle, we obtain a neat function-theoretic characterization. |
| title | Weak null maximum and integration of families of multiplication operators |
| topic | Functional Analysis Complex Variables 47B91, 47G10, (Primary) 30H10, 30H20 (Secondary) |
| url | https://arxiv.org/abs/2509.05173 |