Compactness of operator integrators
Titarii Wootijirattikal, Sing-Cheong Ong, Yongwimon Lenbury · Operators and Matrices · 2019
A function f from a closed interval [a,b] to a Banach space X is a regulated function if one-sided limits of f exist at every point. A function from [a,b] to the space B(X,Y ) , of bounded linear transformations form X to a Banach space Y , is said to be an integrator if for each X -valued regulated function f , the Riemann-Stieltjes sums (with sampling points in the interior of subintervals) of f with respect to converge in Y . We use elementary methods to establish criteria for an integrator to induce a compact linear transformation from the space, Reg(X) , of X -valued regulated functions to Y . We give direct and elementary proofs for each result to be used, including, among other things, the fact that each integrator induces a bounded linear transformation, , from Reg(X) to Y , and other folklore or known results which required reading large amount of literature.