Moore-Penrose inverse and applications to linear equations

TL;DR

This paper explores the properties of the Moore-Penrose inverse for unbounded operators on Hilbert spaces and demonstrates their applications in solving linear and Fredholm integral equations.

Contribution

It extends the theory of Moore-Penrose inverse to unbounded operators and applies these results to variational regularization and integral equations.

Findings

Derived properties of Moore-Penrose inverse for unbounded operators.

Applied inverse theory to variational regularization problems.

Provided examples illustrating solutions to linear and Fredholm integral equations.

Abstract

In the present paper we investigate Moore-Penrose inverse and characteristic matrix of unbounded WCT operators on the Hilbert space $L^2(\mu)$. Also, we obtain some applications of the Moore-Penrose inverse of unbounded operators on the Hilbert space $\mathcal{H}$ to variational regularization problem. Moreover,some examples are provided to illustrate the applications of our results in linear equations and specially Fredholm integral equations.