The theory of normal operators is so successful that much of the theory of non-normal operators is modeled after it.

Our aim is to consider weaker operators which satisfy some of the properties that normal operators fulfill or their weaker versions and find the interconnections, if any, between some of these classes of non-normal operators.

For normal operators T [member of] B(H), it is known that [sigma](T) and [bar.

Berberian, An extension of Weyl's theorem to a class of not necessarily normal operators, Michigan Math.

In the case of a normal operator T acting on a Hilbert space, Berkani [5, Theorem 4.

In this paper, the uncertainty principle is extended to symmetric operators and to normal operators.

uncertainty principle, self-adjoint operators, symmetric operators, normal operators, periodic functions, ultraspherical polynomials, sphere.

The present paper sheds some light behind the scene by extending the classical UP for self-adjoint operators to a wider class of operators, namely to symmetric operators and to normal operators.

Selig studies extensions of uncertainty principles to symmetric operators and to

normal operators and in various function spaces.

orthogonal polynomials, Hessenberg's matrix, normal operator.

Z] is empty, then the infinite Hessenberg matrix D is a normal operator in [l.

mu]], and also D = N, in consequence D is a normal operator.