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.

A solution must be found for the fact that all normal operators in separable Hilbert space possess granular eigenspaces.

It is possible to define a normal operator in separable Hilbert space whose eigenspace consists out of a set of chains that are put together from granules.

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.

Let N and M be two normal operators such that AN = MA.

Let M and N be two bounded normal operators and let A be a bounded operator all defined on a Hilbert space H.

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.

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.