The commuting graph of bounded linear operators on a Hilbert space
The authors of the paper under review prove that there exists an operator \(T\) on a separable infinite-dimensional Hilbert space such that the commutant of every operator which is not a scalar multiple of the identity operator and commutes with \(T\) coincides with the commutant of \(T\). Furthermore, they study finite-rank operators, normal operators, partial isometries, \(\mathcal C_{0}\)-contractions, and show that for these classes of operators it is possible to construct a finite sequence of operators, starting with a given operator and ending with a rank-one projection, such that each operator in the sequence commutes with its predecessor. At the end, they prove that, for any given set of \textit{yes/no} conditions between points in some finite set, there always exist operators on a finite-dimensional Hilbert space such that their commutativity relations exactly satisfy those conditions.
scientific article; zbMATH DE number 6140893