On commuting unbounded self-adjoint operators. II
The paper deals with couples of (unbounded) self-adjoint operators A and B of a Hilbert space which commute on a common core for A and B. As begun in part I (see the preceding review), these couples are studied via commutators of bounded operators (i.e., the resolvents of A and B). The main objects are couples \(\{\) A,B\(\}\) of the classes \({\mathfrak N}_{rs}\) which are defined as follows. Let \(r,s\in {\mathbb{N}}\cup \{+\infty \}\). A couple \(\{\) A,B\(\}\) of self-adjoint operators A and B is said to be in \({\mathfrak N}_{rs}\) if there exists a linear subspace \({\mathcal D}\) of the Hilbert space such that \({\mathcal D}\subseteq {\mathcal D}(A^ jB^ k)\cap {\mathcal D}(B^ kA^ j)\) and \(A^ jB^ k\phi = B^ kA^ j\phi\) for all \(\phi\in {\mathcal D}\) and j,k\(\in {\mathbb{N}}\), \(j\leq r\), \(k\leq s\) and such that \({\mathcal D}\) is a core for A and B. The classes \({\mathfrak N}_{rs}\) are characterized in terms of commutators of the resolvents of A and B. The examples from part I are considered again in this more general context. It is shown (for instance) that the classes \({\mathfrak N}_{rs}\) are different, i.e., \({\mathfrak N}_{rs}\neq {\mathfrak N}_{r's'}\) if (r,s)\(\neq (r',s')\) and \(r,s,r',s'\in {\mathbb{N}}\cup \{+\infty \}\).
scientific article; zbMATH DE number 3921311
scientific article