Approximately angle preserving mappings

In this paper, we present some characterizations of linear mappings, which preserve vectors at a specific angle. We introduce the concept of $(\epsilon ,c)$-angle preserving mappings for $\mathrm{\mid }c\mathrm{\mid }<1$ and $0\leq \varepsilon < 1 + |c|$\mathrm{.}Inaddition,wedefine$\widehat{\varepsilon},(T, c)$ as the ``smallest’’ number $\epsilon$ for which $T$ is $(\epsilon ,c)$-angle preserving mapping. We state some properties of the function $\widehat{\epsilon }(\mathrm{.},c)$, and then propose an exact formula for $\widehat{\epsilon }(T,c)$ in terms of the norm $\mathrm{\parallel }T\mathrm{\parallel }$ and the minimum modulus $[T]$ of $T$. Finally, we characterize the approximately angle preserving mappings.