Addendum to the paper: “Some fixed point theorems for multivalued mappings”
This is a survey paper on additive combinations of certain special-type operators on a Hilbert space. We consider (finite) linear combinations, sums, convex combinations and/or averages of operators from the classes of diagonal operators, unitary operators, isometries, projections, symmetries, idempotents, square-zero operators, nilpotent operators, quasinilpotent operators, involutions, commutators, self-commutators, norm-attaining operators, numerical-radius-attaining operators, irreducible operators...
For a given bi-continuous semigroup on a Banach space we define its adjoint on an appropriate closed subspace of the norm dual . Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology . We give the following application: For a Polish space we consider operator semigroups on the space of bounded, continuous functions (endowed with the compact-open topology) and on the space of bounded Baire measures (endowed with the weak-topology)....
Characterisations are obtained for the following classes of unbounded linear operators between normed spaces: weakly compact, weakly completely continuous, and unconditionally converging operators. Examples of closed unbounded operators belonging to these classes are exhibited. A sufficient condition is obtained for the weak compactness of T' to imply that of T.
We obtain generalizations of the Fan's matching theorem for an open (or closed) covering related to an admissible map. Each of these is restated as a KKM theorem. Finally, applications concerning coincidence theorems and section results are given.