NAEEM AHMAD ATAULLAH

One of the current trends in operator theory is studying natural generalization of normal operators.
This project aims to introduce and study new concept of normality which we called polynomially quasi-(alpha,beta)-normal operators.
We prove a characterization of polynomially-quasi-totally-(α,β)-normal operators. Using this result we obtained several important spectral and algebraic properties of m-quasi-totally-(α,β)- normal operators.

In recent years, the study of tuple of operators was the subject of intensive study by many authors. In this project, we initiate the study of some classes of multivariable commuting operators. These classes of operators naturally generalize the notions of (\alpha, \beta)-class (Q) operators of a single operator and $(\alpha,\beta)$-normal for tuple of commuting operators done respectively in [8- 17]. We will study some properties of (\alpha, \beta)-class (Q) operators multioperators.