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Let ϕ be an analytic function defined on the unit disk$D$, with ϕ($D$)⊂$D$, ϕ(0)=0, and ϕ′(0)=λ≠0. Then by a classical result of G. Kœnigs, the sequence of normalized iterates Φ_{$n$}/λ^{$n$}converges uniformly on compact subsets of$D$to a function σ analytic in$D$which satisfies$σ$°φ=λ$σ$. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy space$H$_{$p$}, the sequence Φ_{$n$}/λ^{$n$}converges to σ in the norm of$H$_{$p$}. We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ϕ is univalent.

Cwikel's bound is extended to an operator-valued setting. One application of this result is a semi-classical bound for the number of negative bound states for Schrödinger operators with operator-valued potentials. We recover Cwikel's bound for the Lieb-Thirring constant$L$_{0,3}which is far worse than the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimension$d$≥3) for the quotient$L$_{0,d}/L^{cl}_{0,d}is the so-called classical constant. This gives some improvement in large dimensions.

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No abstract uploaded!