On the generalized quadratic mappings in quasi-Banach modules over a $C^*$--algebra

TL;DR

This paper solves a specific functional equation related to quadratic mappings in quasi-Banach modules over a $C^*$-algebra and shows that approximately satisfying mappings can be closely approximated by exact solutions.

Contribution

It provides the general solution to a quadratic functional equation derived from the centroid of vectors and establishes stability results for mappings approximately satisfying this equation.

Findings

Explicit general solution to the functional equation.

Approximate solutions can be closely approximated by exact quadratic mappings.

Stability of quadratic mappings in quasi-Banach modules over $C^*$-algebras.

Abstract

Let $n>2$ be a positive integer. In this paper, we obtain the general solution of the following functional equation n \sum_{1 \le i<j \le n} Q(x_i-x_j)=\sum_{i=1}^{n}Q(\sum_{j =1}^n x_j -n x_i) which is derived from the centroid of the $n$ distinct vectors $x_1, ..., x_n$ in an inner product space. Furthermore, we prove that a mapping $f$ between quasi-Banach modules over a $C^*$-algebra satisfying approximately the equation can be approximated by a quadratic mapping $Q$ satisfying exactly the equation such that $|f(x)-Q(x)|$ is bounded.