Associate Professor in Mathematics

Degrees: M.A. (St.Andrews), Ph.D. (London), D.Sc. (London)

Email: ronald.cross@maths.uct.ac.za

Personal Web Page: None

Teaching and research interests:

• Metamathematics. Applications of Non Standard Analysis to topological Vector Spaces.
• Classical and Functional Analysis. Vector Calculus.
• Vector Measures.
• Unbounded Linear Operators. Fredholm perturbation theory.
• Nonlinear Analysis. Set valued mappings on vector spaces. Multivalued Linear Operators (Linear Relations) in normed spaces and Banach spaces and their applications. A brief description is the following: Given a pair of normed spaces X and Y, any linear subspace of the Cartesian product defines a multivalued linear operator T whose graph is the given subspace (and conversely). Then the inverse, adjoint, closure and completion of T are all well defined MVLOs and together all follow simple canonical rules. Our investigations so far relate principally to index theory, perturbation properties of semi- Fredholm operators, measures of noncompactness and other operational quantities, partial continuity, weak compactness and Tauberian properties (for second adjoints), and stability properties for the essential spectrum. The book Multivalued Linear Operators (see below) develops the basic theory and techniques for this study. More recently (2011) applications to initial boundary value problems for degenerate (elliptic-hyperbolic) differential equations and integro-differential equations have been obtained as applications.

Representative publications:

Properties of some norm related functions of unbounded linear operators, Matematische Zeitschrift 199 (1988), 285-302.

On the perturbation of unbounded linear operators with topologically complemented ranges, Journal of Functional Analysis 92 (1990), 468-473.

Linear transformations of Tauberian type in normed spaces, Note di Matematica 10 (1990) (special `Köthe' volume), 193-203 (invited paper).

Adjoint characterisations of unbounded weakly compact, weakly continuous and unconditionally converging operators, (with T.Alvarez and A.I.Gouveia) Studia Math., 113(3) (1995), 283-298.

On a theorem of Kalton and Wilansky concerning Tauberian operators, Journal of Math.ematical Analysis and Applications 171, No.1 (1992), 156-170.

Operator ranges in Banach spaces I, (with M.I. Ostrovskii and V.V. Shevchik) Mathematische Nachrichten 173 (1995) no. 2, 91-114.

On certain densely invarient quantities of linear operators, Mathematische Nachrichten 178 (1996) no. 2, 91-114.

An index theorem for the product of linear relations , Linear Algebra and its Applications 277 (1998), 127-134.

Multivalued Linear operators, Marcel Dekker, New York, 1998.

Adjoint Characterisations of Quasi-weakly Compact Linear Relations, (with Alvarez and D.L.Wilcox), J.Math, Anal. Appl. 277 (2002) 257-271.

Perturbation Results for Multivalued Linear Operators (with Angelo Favini and Yakov Yakubov) Progress in Nonlinear Differential Equations and Their Applications 80 (2011), 111-130.