Rank-1 representers (Q1407373)

This article gives a comprehensive review of the rank-1 and plenary subsets of continuous operators between Banach spaces. The main emphasis is placed on the separation-like properties stemming from convexity and particularly on some complicated and challenging examples. One of the main results offers a characterization of symmetric rank-1 supports stating that a two-place function is a rank-1 support of a nonempty set of bounded linear operators if and only if it is positively bihomogeneous, subadditive (in a certain sense) and lower semicontinuous. Analogous characterization of rank-1 supports of sets of continuous quadratic forms is also obtained, however, the lack of a subadditivity type condition is the most striking feature here. Thus, a wide class of degree-two-homogeneous functions can be considered as a dual set to continuous operators in a certain sense.