Compactness of operator integrators

• Main Authors: Titarii Wootijirattikal, Sing Cheong Ong, Yongwimon Lenbury
• Other Authors: Ubon Rajathanee University
• Format: Article
• Published: 2020
• Subjects: Mathematics
• Online Access: https://repository.li.mahidol.ac.th/handle/123456789/51229
• Institution: Mahidol University

© 2019, Element D.O.O.. All rights reserved. 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 α from [a,b] to the space B(X,Y), of bounded linear transformations form X to a Banach space Y,issaidtobeanintegrator 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 α converge in Y. We use elementary methods to establish criteria for an integrator α to induce a compact linear transformation from the space, Reg(X), ofX-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 α induces a bounded linear transformation, α, from Reg(X) to Y, and other folklore or known results which required reading large amount of literature.