Properties of the star supremum for arbitrary Hilbert space operators

We give a simple necessary and sufficient condition for the existence of star supremum for two arbitrary operators on a Hilbert space. It is shown that the results of Hartwig (1979) [15], and Janowitz (1983) [18], when applied to the ring of bounded operators on a Hilbert space, require much simpler conditions. We also give a complete answer to a question stated by Hartwig and Drazin (1982) in [16], regarding the extremal values of range and null-space of the star infimum.