Localization of a simple eigenvalue and a corresponding eigenvector
Balmohan V. Limaye, M. Thamban Nair · 1988
ABSTRACT. Let A be a closed operator in a complex Banach space X with a dense domain D A. Let Q be a projection operator of rank one given by Q{-) = (-,x*)x, where x • D A and x * • X* are such that Ilxll = 1 = (x,x*). Sufficient conditions are given for the existence of a simple eigenvalue X of A near • = (Ax,x*) and for a corresponding eigenvector • of A near x. Bounds for the quantities 13,- •1 and 114- xll, and an isolation region for 3, from the rest of the spectrum of A are also obtained. The main theorem proved here improves upon a result of Lemordant (1980). The result is applied to a case of numerical approximation of operators to obtain computable error estimates. 1. Introduction. Let X be a complex Banach space with norm II'll and A be a closed operator in X with domain D A which is dense in X. Let x G D A and x * G X* be such that I',xl! = 1 = (x,x*), where (',-) denotes the "scalar product " on X X X* defined by (y,y*) = y*(y), the complex conjugate of y*(y) for y in X and y * in X*.