The approximation numbers of Hardy--type operators on trees
| dc.creator | Evans, W. D. | |
| dc.creator | Harris, D. J. | |
| dc.creator | Lang, J. | |
| dc.date | 2000-03-30 | |
| dc.date.accessioned | 2026-07-07T04:34:32Z | |
| dc.date.available | 2026-07-07T04:34:32Z | |
| dc.description | The Hardy operator $T_a$ on a tree $\G$ is defined by \[(T_af)(x):=v(x) \int^x_a u(t)f(t) dt \qquad {for} a, x\in \G. \] Properties of $T_a$ as a map from $L^p(\G)$ into itself are established for $1\le p \le \infty$. The main result is that, with appropriate assumptions on $u$ and $v$, the approximation numbers $a_n(T_a)$ of $T_a$ satisfy \[ (*) \lim_{n\to \infty} na_n(T_a) = α_p\int_{\G} |uv|dt \] for a specified constant $α_p$ and $1<p<\infty$. This extends results of Naimark, Newman and Solomyak for $p=2$. Hitherto, for $p\neq 2$, (*) was unknown even when $\G$ is an interval. Also, upper and lower estimates for the $l^q$ and weak-$l^q$ norms of $\{a_n(T_a)\}$ are determined. | |
| dc.description | LaTex file | |
| dc.identifier | https://arxiv.org/abs/math/0003215 | |
| dc.identifier | http://arxiv.org/abs/math/0003215 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/58940 | |
| dc.subject | Spectral Theory | |
| dc.subject | Functional Analysis | |
| dc.title | The approximation numbers of Hardy--type operators on trees | |
| dc.type | text |