Continuing the author’s previous works [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3, 263–295 (2000; Zbl 1037.47505); J. Funct. Anal. 194, 288–310 (2002; Zbl 1021.58007)], some generalization of Poincaré-Sobolev type inequalities is used for the study of the essential spectrum and the semigroup property for selfadjoint operators on abstract Hilbert spaces. Non-symmetric semigroups are also studied.