The approximation numbers of Hardy--type operators on trees

Details The approximation numbers of Hardy--type operators on trees The approximation numbers of Hardy--type operators on trees

2000-03-30

arxiv.org/abs/math/0003215v1

Mathematics

Spectral Theory

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Scientific paper

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 \[ (*) \lim_{n\to \infty} na_n(T_a) = \alpha_p\int_{\G} |uv|dt \] for a specified constant $\alpha_p$ and $1

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