  Taking matrix as a synonym for a numerical function on the Cartesian product
of two (in general, infinite) sets, a simple purely algebraic "reciprocity
property" says that the set of rows spans a finite-dim space iff the set of
columns does so. Similar topological reciprocity properties serve to define
strongly compact and weakly compact matrices, featured in the well-known basic
facts about almost periodic functions and about compact operators. Some
properties, especially for the weak compact case, are investigated, such as the
connection with the matrix having a Fubini-like property for general means.
These are applied to prove possibility of extension to the entire semigroup of
bounded densely defined semigroups of operators in a Banach space with weak
continuity properties.
