The commuting graph of bounded linear operators on a Hilbert space

An operator T on the separable infinite-dimensional Hilbert space is constructed so 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. On the other hand, it is shown that for several classes of operators it is possible to construct a finite sequence of operators, starting at a given operator from the class and ending in a rank-one projection such that each operator in the sequence commutes with its predecessor. The classes which we study are: finite-rank operators, normal operators, partial isometries, and C0 contractions. It is also shown that for any given set of 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.