Nonselfadjoint Spectral Operators and Their Application to Control Theory

As is well known, there is no general spectral theory of nonselfadjoint operators in a Hubert space. There exists spectral theory only for some special classes of nonselfadjoint operators. One of these is the class of spectral operators. These operators were introduced by N. Dunford [1] (see, also, [2]), and the whole 3rd ed. volume of the N. Dunford and J. Schwartz monograph [1] is devoted to this class of operators. The problem is that it is quite difficult to find nontrivial examples of such operators.