Spectral order for unbounded operators

The spectral order, a notion originated by Olson for bounded operators, is investigated here in the context of unbounded operators. Dissimilarities between bounded and unbounded cases are pointed out. New criteria for two operators to be comparable are supplied. A way of reducing the study of the spectral order to the case of bounded operators is proposed. Connections with essential selfadjointness are established. Integral inequalities for monotonically increasing functions are characterized in terms of distribution functions. Some illustrative examples are furnished.