Weakly Compact "Matrices", Fubini-Like Property and Extension of Densely Defined Semigroups of Operators

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.