Perturbation of Normal Operators1

The perturbation theory of linear operators concerns itself with the eigenvalues and eigenelements of a variable operator. If T(e) is an operator depending on a real parameter e, the relation between the eigenvalues and eigenelements of the perturbed operator T(e) (e?^0) and those of the unperturbed operator 7\0) is of particular interest. This theory finds considerable application in the field of quantum mechanics. In such applications the eigenvalues of an operator represent certain quantities which, by their very nature, must be real. It is therefore not surprising to find that emphasis has been given to self-adjoint operators. The first person to handle systematically the regular perturbation of self-adjoint operators was Franz Rellich [7].2 By a different attack on the problem, Bela v. Sz. Nagy was able to improve on Rellich's theorem [S]. It was the conjecture of Frantisek Wolf that certain of these results would hold for normal operators—a class of operators which includes the self-adjoint. This paper substantiates that conjecture. Since the writing of this paper, Professor Wolf has rendered its main result a consequence of a general theory of analyticity for operators in a Banach space [9, section 6.4]. The proof given here exploits the relationship between a normal operator and a pair of self-adjoint operators. A bounded linear transformation of an abstract Hilbert space § into itself will be called an operator on §. The number A(e), the element/(e) of §, and the operator T(e) on § will be called regular on the interval | e| <p if they can be expressed as convergent power series in a real parameter e: