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

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)}.