Effective Representations of the Space of Linear Bounded Operators
by
Vasco Brattka
Theoretische Informatik 1, FernUniversität Hagen

Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition.

In this talk a number of representations of the space of linear bounded operators in Banach spaces are discussed. Unfortunately, the operator norm topology of the operator space is non-separable in typical cases and hence the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs.

We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn.

The talk is based on results included in the following report.

Vasco Brattka, Computability of Banach Space Principles, Informatik-Bericht 286, FernUniversität Hagen, 2001

Date received: July 20, 2001