A Note on n-Perinormal Operators

The study of operators satisfying σja(T)=σa(T)σja(T)=σa(T) is of significant interest. Does σja(T)=σa(T)σja(T)=σa(T) for n-perinormal operator T∈B(H)?T∈B(H)? This question was raised by Mecheri and Braha [Oper. Matrices 6 (2012), 725–734]. In the note we construct a counterexample to this question and obtain the following result: if T is a n-perinormal operator in B(H), then σja(T)\{0}=σa(T)\{0}.σja(T)\{0}=σa(T)\{0}. We also consider tensor product of n-perinormal operators.