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In this paper, we introduce a polynomially normal operator on a complex Hilbert space, extending the notation of n-normal and normal operators. Several basic properties of polynomially normal operator are firstly presented. We show some spectral properties of polynomially normal operators under new assumption in literature. Precisely, we prove that σ (T) = σ a (T) , ker (T - z) ⊥ ker (T - w) if z and w are distinct eigen-values of T and others results. Thus, we generalize some results for n-normal and normal operators. [ABSTRACT FROM AUTHOR]

Property (UWΠ), introduced in Berkani and Kachad (BullKorean Math Soc 49:1027-1040, 2015) and studied more recently in Aiena and Kachad (Acta Sci Math (Szeged) 84:555-571, 2018) may be thought as a variant of Browder's theorem, or Weyl's theorem, for bounded linear operators acting on Banach spaces. In this article we study the stability of this property under some commuting perturbations, as quasinilpotent perturbation and, more in general, under Riesz commuting perturbations. We also study the transmission of property (UWΠ) from T to f (T), where f is an analytic function defined on a neighborhood of the spectrum of T . Furthermore, it is shown that this property is transferred from a Drazin invertible operator T to its Drazin inverse S. [ABSTRACT FROM AUTHOR]