On positive differential operators (deficiency indices, factorization, perturbations)

Abstract

SynopsisA short survey is given of some recent results. The perturbations and stability of discrete spectrum and the problems of resolvent convergence of densely or non-densely defined elliptic differential operators are considered. The Courant theorem on variations of the domain is generalized. In connection with Berezanskiy's theorem on essential self-adjointness, the test for the finite velocity of propagation is extended. The Frobenius and the Krein-Heinz-Rellich factorization theorems and the Etgen Pawlowski oscillation criterion are generalized for equations of any order with operator-valued coefficients. Brusentsev's recent example of a two term fourth order differential operator with deficiency index 4 is discussed.