Abstract
We introduce a family of operators called the family of k-quasi-(n, m)-power normal operators. Such family includes normal, n-normal and (n, m)-power normal operators. An operator T is an element of B(H) is said to be k-quasi-(n, m)-power normal if it satisfies
T*(K) ((TT)-T-n*(m) - T*T-m(n))T-k = 0,
where k, n and in are natural numbers. Firstly, some basic structural properties of this family of operators are established with the help of special kind of operator matrix representation associated with such family of operators. Secondly, some properties of algebraically k-quasi-(n, m)-power normal operators are discussed. Thirdly, we consider the study of tensor products of k-quasi-(n, m)-power normal operators. A necessary and sufficient condition for T circle times S to be a k-quasi-(n, m)-power normal is given, when T not equal 0 and S not equal 0.