Multivalued Linear Operators

Ronald Cross, MA 1957

The simplest naturally occurring example of a multivalued linear operator (MVO) would be the inverse of a linear transformation (such as a matrix transformation for example) between two vector spaces.

Any linear subspace of the product of two vector spaces defines a MVO and vice versa. The Theory of Linear Operators, which embraces matrix algebra and differential and integral calculus, and is at the core of much of mathematical analysis, theoretical physics and engineering, can be viewed as a sub theory of the theory of MVOs embodying the single valued case.

The book sets up the basic theory, recovers all the canonical relationships between inverses, adjoints, closures and (normed space) completions which were not available in the single valued case and then goes on to study topics that include stability and perturbation theory, semi-Fredholm relations, spectral theory, compactness, and second adjoints (including Tauberian properties, weak compactness, etc.).

Linear relations were first introduced by John von Neumann in 1932 motivated by the need to consider adjoints of non-densely defined linear differential operators.

ISBN: 0-8247-0219-0