Given a smooth family ${\hbox{\ac L}}$ of real or complex variable taking values within the class of Fredholm operators of index zero in a Banach space, there are some available definitions in the literature of the concept of algebraic multiplicity of the family ${\hbox{\ac L}}$ at a point $x_0$ of the parameter at which the operator ${\hbox{\ac L}}(x_0)$ becomes non-invertible. The purpose of the paper is to show that the algebraic multiplicity is uniquely determined by a few of its properties, independently of its construction. The main technical tools to obtain this uniqueness result are a Lyapunov–Schmidt reduction, the local Smith form and a new factorization result for general families at non-algebraic eigenvalues obtained in the paper.

Jensen's operator inequality and Jensen's trace inequality for real functions defined on an interval are established in what might be called their definitive versions. This is accomplished by the introduction of genuine non-commutative convex combinations of operators, as opposed to the contractions considered in earlier versions of the theory by the authors, and by Brown and Kosaki. As a consequence, one no longer needs to impose conditions on the interval of definition. It is shown how this relates to the pinching inequality of Davis, and how Jensen's trace inequality generalizes to C*-algebras.