Ronald Cross

Emeritus Associate Professor in Mathematics

Degrees:

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

Email: cross@maths.uct.ac.za

Teaching and research interests:

Published papers on infinite matrices, Banach spaces, non-standard analysis, multivalued linear operators.

Linear operators and linear relations: Given a linear relation T : D (T) X -> Y, where X and Y are normed spaces, to investigate properties of T given various conditions satisfied by T and/or the underlying spaces X and Y. In the special case when T is single-valued, T is termed a `linear operator'. Most of the investigations so far have been concerned with operators and relate principally to a study of operational quantities of operators, partial continuity, compactness, Tauberian theory, perturbation theory, index theory, and stability properties of the essential spectra. The results have applications to the theory of closed operators in Banach spaces, and in particular to Fredholm theory and differential operators. Some new results in the (relatively new) theory of linear relations have also been established.

Representative publications:

1. Multivalued Linear operators , Dekker, New York, 1998
   
   
2. Properties of some norm related functions of unbounded linear operators, Matematische Zeitschrift 199 (1988), 285-302.

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

4. (with M.I. Ostrovskii and V.V. Shevchik) Operator ranges in Banach spaces I, Mathematische Nachrichten 173 (1995) no. 2, 91-114.
   
   
5. with T. Alvarez and A.I. Gooveia) Adjoint characterisations of unbounded weakly compact, weekly completley continuous and unconditionally converging operators, Studia Math., 113(3) (1995), 283-298.

    

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

    

7. An index theorem for the product of linear relations , Linear Algebra and its Applications, 277 (1998), 127-134
   
   
8. (with T.Alvarez and Diane Wilcox) Adjoint characterisations of quasi-weekly-compact linear relations, J. Math. Anal. Appl. 277 (2002), 257-271