Abstract:

The property （ω′） and the generalized property（ω′） are variants of Weyl′s theorem. The sufficient and necessary conditions are established for which the property （ω′） and the generalized property（ω′）for T and perturbations by power finite rank operators commuting with T by means of a new spectrum defined on a bounded linear operator  T acting on a Hilbert space by the essential spectrum, singlevalued extension property and the operator of consistence in Fredholm and index.

Key words:  property (ω′); generalized property (ω′); perturbations