Algebraic elements of a Banach algebra modulo an ideal
Let A be a Banach algebra. A classical result states that if \(f\in A\) and \(f^ 2-f\in rad(A)\), the Jacobson radical of A, then \(\exists e\in A\) such that \(e^ 2=e\) and e-f\(\in A\). A more recent result proved by C. Olsen states that when \(T\in B(H)\), the algebra of bounded linear operators on a Hilbert space H, and for some polynomial p, p(T)\(\in K(H)\), the ideal of compact operators, then \(\exists S\in B(H)\) such that \(p(S)=0\) and T-S\(\in K(H)\). In the general situation K is an ideal of A, p is a polynomial, and \(f\in A\) has the property p(f)\(\in K\). This paper is concerned with the question: under what conditions on K and f does there exist \(g\in A\) such that \(p(g)=0\) and f-g\(\in K\). Among the ideals K considered are rad(A), various ideals of B(H) in K(H), and the socle of A.
algebra of bounded linear operators on a Hilbert space
scientific article; zbMATH DE number 3923445