A new approach to $γ$-bounded representations
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910334165450752 |
|---|---|
| author | Merdy, Christian Le |
| author_facet | Merdy, Christian Le |
| contents | Let $X$ be a Banach space, let $(Ω,μ)$ be a $σ$-finite measure space and let $A,B\colonΩ\to B(X)$ be strongly measurable $γ$-bounded functions. We show that for all $x\in X$ and all $x^*\in X^*$, there exist a Hilbert space $K$ and two measurable functions $a_1\in L^\infty(Ω;K)$ and $a_2\in L^\infty(Ω;K)$ such that $\langle B(t)A(s)x,x^*\rangle = (a_2(t)\,\vert\, a_1(s))_{K}$ for a.e. $(s,t)$ in $Ω^2$, with $\Vert a_1\Vert_\infty \Vert a_2\Vert_\infty\leq γ(A)γ(B)\Vert x\vert\vert x^*\Vert$. This factorization property allows us to improve or simplify some results concerning $γ$-bounded representations of groups or semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10736 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new approach to $γ$-bounded representations Merdy, Christian Le Functional Analysis 47A99, 22D12, 47D06, 47A60 Let $X$ be a Banach space, let $(Ω,μ)$ be a $σ$-finite measure space and let $A,B\colonΩ\to B(X)$ be strongly measurable $γ$-bounded functions. We show that for all $x\in X$ and all $x^*\in X^*$, there exist a Hilbert space $K$ and two measurable functions $a_1\in L^\infty(Ω;K)$ and $a_2\in L^\infty(Ω;K)$ such that $\langle B(t)A(s)x,x^*\rangle = (a_2(t)\,\vert\, a_1(s))_{K}$ for a.e. $(s,t)$ in $Ω^2$, with $\Vert a_1\Vert_\infty \Vert a_2\Vert_\infty\leq γ(A)γ(B)\Vert x\vert\vert x^*\Vert$. This factorization property allows us to improve or simplify some results concerning $γ$-bounded representations of groups or semigroups. |
| title | A new approach to $γ$-bounded representations |
| topic | Functional Analysis 47A99, 22D12, 47D06, 47A60 |
| url | https://arxiv.org/abs/2402.10736 |