Upper and lower Fredholm spectra II

Abstract

Joint upper and lower Fredholm spectra are defined for «-tuples of bounded linear operators, and the upper Fredholm spectrum is represen- ted both as the simultaneous eigenvalues and as the simultaneous approxi- mate eigenvalues of an /i-tuple of operators obtained by a Berberian-Quigley construction. Introduction. A bounded linear operator T: X -> Y between Banach spaces is said to be upper Fredholm if it has finite dimensional null space (T~x0) and closed range T(X), and is said to be lower Fredholm if its range T(X) is closed and has finite codimension. / is Fredholm iff it is upper and lower Fredholm. In this note we show that T is upper Fredholm iff a related operator P(T): <$(Y) is bounded below, where 9(X) is obtained as a certain quotient of the space lx(X) of all bounded A'-valued sequences. It is then shown that T is Fredholm iff P(T) is invertible. This construction is obtained in §1, the basic two theorems are established in §2, some immediate consequences in §3, and some consequences for joint upper and lower Fredholm spectra are obtained in §4. I. If X is a complex Banach space then let lx(X) denote the Banach space obtained from the space of all bounded sequences x = (xn) in X by imposing term-by-term linear combination and the supremum norm \\x\\ = supn||.xB||.