Compact weighted composition operators on Sobolev related spaces (Q1103170)

If \(m\) is a positive integer and \(1\leq p<\infty\), then \(W_{m,p}\) denotes \(\{f\) on \([0,1]:\) \(f^{(r)}\) is absolutely continuous, \(r=1,2,...,m-1\), and \(f^{(m)}\in L^ p(0,1)\}\). If, for \(1\leq p<\infty\), \(\| f\|_{m,p}=(\sum^{m}_{s=0}\| f^{(s)}\|^ p_ p)^{1/p}\), then \(W_{m,p}\) is a Banach space. In this paper the authors show that if \(u\in W_{m,\infty}\), \(\phi: [0,1]\to [0,1]\), \(\phi \in W_{m,\infty}\cap C^ 1\), \(\phi^{- 1}([a,b])\) is expressible as a union of \(N\) intervals for all \(a,b\in [0,1]\), where \(N\) is a positive integer, and if \(uC_{\phi}(f)(x)= u(x)f(\phi(x))\), then \(uC_{\phi}\) is bounded on \(W_{m,p}\) which is compact if and only if \(u\phi '=0\). It is further shown that if \(uC_{\phi}\) is compact on \(W_{m,p}\), then the spectrum \(\sigma (uC_{\phi})\) is identifiable as \(\{0\}\cup \{\beta:\beta^ k= u(c)...u(\phi_{k-1}(c))\) for some integer \(k\) and fixed point \(c\) of \(\phi\) of order \(k\}\).