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The class of polynomially normal operators is a wider class than the class of all normal operators. Inspired by some interesting well known facts about normal operators and by some recent work, we present new properties of polynomially normal operators. Precisely, we prove that under certain conditions polynomially normal operators are Drazin or even group invertible and we also give necessary and sufficient conditions for a polynomially normal operator to have a closed range. In addition, we characterize polynomially normal operators with or without closed ranges applying the adequate operator matrix representations. Furthermore, we show that in some cases a polynomially normal operator can be written as a direct sum of a normal operator and a nilpotent operator. What is more, we state several examples to illustrate our results.

Drazin invertibility, characterizations and structure of polynomially normal operators