Sobolev-Orlicz imbeddings, weak compactness, and spectrum
The aim of the paper under consideration is to show how weak compactness in the \(L^1(X,m)\) space may be used to relate the existence of a Sobolev-Orlicz embedding to the \(L^2(X,m)\)-spectral properties of an operator \(H\). First, it is shown that a Sobolev-Orlicz embedding implies that the bottom of the \(L^2\)-spectrum of \(H\) is an eigenvalue of finite multiplicity provided \(m\) is finite. Secondly, it is proved that for a large class of operators, namely those for which Persson's characterisation of the bottom of the essential spectrum is true, a Sobolev-Orlicz embedding always implies the discreteness of the \(L^2\)-spectrum of \(H\) provided \(m\) is finite. Finally, a certain converse of the last result is proved: the discreteness of the \(L^2\)-spectrum of \(H\) always implies the existence of an Orlicz space for which a Sobolev-Orlicz embedding holds true for \(H\). The case of logarithmic Sobolev inequalities is considered and provides the original motivation of the author's investigation.
scientific article; zbMATH DE number 1542415
scientific article