A remark on limit operators
For the finite-dimensional Banach space \(E\), and the measurable subset \(\Omega\subset\mathbb{R}^n\), consider \(L_p(\Omega)\), \(1\leq p\leq\infty\), the space of equivalence classes of measurable functions \(u:\Omega\longrightarrow E\), and \(L_p=L_p(\mathbb{R}^n)\). Denoting for \(\xi\in\mathbb{R}^n\), \((\Psi_\xi u)(x)=e^{i\langle\xi,x\rangle}u(x)\), for the bounded linear operator \(T:L_p(\Omega)\longrightarrow L_p\) one defines \(|T|_+=\inf\{|Tu|:|u|=1\},|T|_{\lambda+}=\sup\{|T:M\longrightarrow L_p|_+:\)co\(\dim M<\infty,M\subset L_p(\Omega),M\) closed\}; the right limit operator will be \(\widetilde T\) such that there exists \(\xi_k\in\mathbb{R}^n,|\xi_k|\longrightarrow\infty\) and the sequence \(T\{\xi_k\}=\Psi_{\xi_k}^{-1}T\Psi_{\xi_k}\) strongly converges to \(\widetilde T\). A first theorem states that for \(p<\infty,|T|_{\lambda +}\leq|\widetilde T|_+\). If \(T:L_p\longrightarrow L_p\) is a bounded linear operator, \(\widetilde T\) will be a two-sided limit operator if there exists \(\xi_k\in\mathbb{R}^n,|\xi_k|\longrightarrow\infty\), \(T\{\xi_k\}\) strongly converges to \(\widetilde T\) and \(T\{\xi_k\}^*\) strongly converges to \(\widetilde T^*\). A corollary states that for \(1<p<\infty\) and \(T\) a Fredholm operator, all two-sided operators \(\widetilde T\) for \(T\) are invertible and \(|\widetilde T^{-1}|<1/|T|_{\lambda +}\). Let \(L_{p_\Omega}\subset L_p\) be the subspace of functions supported in \(\Omega\subset\mathbb{R}^n\) and \(|T|_{\Lambda +}=\inf\{|T:L_{p_\Omega}\longrightarrow L_p|_{\lambda +}\), \(\Omega\subset\mathbb{R}^n\) measurable bounded\}. A second theorem states that for \(p<\infty\) and \(T:L_p\longrightarrow L_p\) a bounded linear operator, a right limit operator \(\widetilde T\) for \(T\) satisfies \(|T|_{\Lambda +}\leq|\widetilde T|_+\). An example is given, involving the Hilbert operator and a locally compact operator.
scientific article; zbMATH DE number 1002986