698 Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 2022, Volume 28, Number 4, 698–709 DOI: 10.7546/nntdm.2022.28.4.698-709 Linear mappings in paraletrix spaces and their application to fractional calculus R. U. Ndubuisi 1 , U. K. Nwajeri 2 , C. P. Onyenegecha 3 , K. M. Patil 4 , O. G. Udoaka 5 and W. I. Osuji 6 1 Department of Mathematics, Federal University of Technology, Owerri, Nigeria e-mails: u_ndubuisi@yahoo.com, rich.ndubuisi@futo.edu.ng 2 Department of Mathematics, Federal University of Technology, Owerri, Nigeria e-mail: ugochukwu.nwajeri@futo.edu.ng 3 Department of Physics, Federal University of Technology, Owerri, Nigeria e-mail: chibueze.onyenegecha@futo.edu.ng 4 Department of Mathematics, Dharmsinh Desai University, India e-mails: kailashpatil2111@gmail.com, kailashpatil2111.maths@ddu.ac.in 5 Department of Mathematics, Akwa Ibom State University, Ikot Akpaden, Nigeria e-mail: otobongawasi@yahoo.com 6 Department of Mathematics, Federal University of Technology, Owerri, Nigeria e-mail: osujiwilliams03@gmail.com Received: 3 June 2022 Revised: 25 October 2022 Accepted: 27 October 2022 Online First: 31 October 2022 Abstract: This paper considers linear mappings in paraletrix spaces as an extension of the one given for rhotrix vector spaces. Furthermore, the adjoints of these mappings are given with their application in fractional calculus. Keywords: Heart-oriented paraletrix, Linear mapping, Adjoints, Inner product, Fractional calculus, Rhotrix. 2020 Mathematics Subject Classification: 20M10. 1 Introduction In [1], the idea of rhotrix was introduced as an object whose elements are arranged in a rhomboidal nature which of course was an extension of matrix-tertions and matrix-noitrets given by Atanassov and Shannon [5]. Suppose  and  are two rhotrices such that