Strictly singular and strictly cosingular linear relations and their conjugates (original) (raw)

2005, Bulletin of the Australian Mathematical Society

In this paper various conditions are given under which the strict singularity (respectively, strict cosingularity) of a linear relation implies the strict singularity (respectively, strict cosingularity) of its conjugate. Serial-fee code: 0004-9727/05 SA2.00+0.00. 2 T. Alvarez [2] for all nonzero scalars a, ft € K and X\,x 2 € D(T). The class of such relations T is denoted by LR(X, Y). If T maps the points of its domain to singletons, then T is said to be a single valued or simple operator. The graph G{T) of T e LR(X, Y) is G(T) := {(*,!/) € * x r : a; €£>(r), j / e T i } , Let X denote the completion of the normed space X. The completion T of T £ LR(X, Y) is the linear relation in LR(X, Y) whose graph is G(T). Let M be a subspace of D(T). Then the restriction T \ M is defined by G(T \ M ) := {(m,y) :meM, ye Tm). For any subspace M of X such that M n D(T) ^ 0, we write T | M = T \ M nD(T)-The inverse of T is the linear relation X 1 " 1 defined by G t T -1 ) : = {(y,x) eYxX:(x,y)e G(T)}.