On the closedness of the range of (fractional) powers of certain classes of possibly unbounded operators

Abstract

In this article, we establish certain identities about the closedness of the range of several classes of possibly unbounded operators defined on a Hilbert space, such as self-adjoint, positive, normal, hyponormal, and quasinormal operators as well as their powers, and fractional powers when the latter applies. In particular, we show that an unbounded self-adjoint positive operator has closed range if and only if one (and all) of its fractional powers has closed range. A substantial part of the paper is devoted to additivity properties of operator ranges, again in the context of non-necessarily bounded operators. The paper is accompanied by many counterexamples that demonstrate the boundaries of possible assertions.