On symmetric functions and symmetric operators on Banach spaces

We study left symmetric and right symmetric elements in the space ${\mathrm{\ell }}_{\mathrm{\infty }}(K,\mathbb{X})$ of bounded functions from a non-empty set $K$ to a Banach space $\mathbb{X}\mathrm{.}$ We prove that a non-zero element $f\in {\mathrm{\ell }}_{\mathrm{\infty }}(K,\mathbb{X})$ is left symmetric if and only if $f$ is zero except for an element $k_0\in K$ and $f(k_0)$ is left symmetric in $\mathbb{X}\mathrm{.}$ We characterize left symmetric elements in the space $C_0(K,\mathbb{X}),$ where $K$ is a locally compact perfectly normal space. We also study the right symmetric elements in ${\mathrm{\ell }}_{\mathrm{\infty }}(K,\mathbb{X})\mathrm{.}$ Furthermore, we characterize right symmetric elements in $C_0(K,\mathbb{X}),$ where $K$ is a locally compact Hausdorff space and $\mathbb{X}$ is real Banach space. As an application of the results obtained in this article, we characterize the left symmetric and right symmetric operators on some special Banach spaces. These results improve and generalize the existing ones on the study of left and right symmetric elements in operator spaces.