Variations on a theme of Kaplansky

Abstract

We explore the relationship between Kaplansky's Lemma about locally algebraic operators, a dual to Kaplansky's lemma, and non singularity for pairs of operators in the sense of Joseph L. Taylor. Kaplansky's Lemma ([7] Lemma 14;[10] Theorem 4.8;[11] (3.5)) says that, for bounded linear operators on Banach spaces, 0.1 locally algebraic =⇒ algebraic .

Skew exactness perturbation

“…The first part of this is known as Kaplansky's Lemma; the proof [10], [12] is a combination of Baire's theorem and the Euclidean algorithm for polynomials. The Euclidean algorithm also gives equality…”

“…We cannot however remove the semi-Fredholm conditions: for example [12] if T = U ⊗ V is the tensor product of the shifts, then (14.3) holds but we never get T k (X) ∩ T −1 (0) = {0}. For operators which are both upper semi-Fredholm and of finite ascent, or lower semi-Fredholm of finite descent ("semi Browder" in the sense of [8], Definition 7.9.1) the conditions of Theorem 10 can be replaced by simple commutivity ( [8], Theorem 7.9.2).…”