On Polynomial Operators and Equations

Abstract

This chapter provides an overview of polynomial operators and equations. It presents qualitative results such as a quadratic formula in Banach space, Weierstrass Theory in normed linear spaces, Eigenfunction expansions for quadratic operators, and existence theory for solutions of polynomial equations. It also explores a few computational techniques for solving polynomial equations in normed linear spaces. The simplest of all nonlinear operators on a normed linear space are the so-called polynomials operators. Equations in such operators are the linear space analog of ordinary polynomials in one or several variables over the fields of real or complex numbers. Such equations encompass a broad spectrum of applied problems including all linear equations.