On Equivalences of Some Classes of Operators in Hilbert Space

Abstract

The study explores various classes of operators introduced by different researchers, including n-normal, (n;m)-normal, k-quasi-(n;m)- normal, n-hyponormal, and (n;m)-hyponormal operators. Notably, the class of (n;m)-hyponormal operators, defined by specifc inequalities, is introduced, along with the concept of (n;m)-unitary quasiequivalence. The research also introduces the novel concept of (n;m)-binormal operators, characterized by specifc commutation conditions, and examines their properties, unitary equivalence, and closure under summations. Additionally, the class of skew (n;m)-binormal operators are introduced and investigated independently, highlighting their unique properties and unitary equivalence. The study concludes with a summary, conclusions, and suggestions for future research, providing a comprehensive overview of the diverse classes of operators studied and their implications.