Turkish Journal of Mathematics
Article Title
DOI
10.3906/mat-1911-35
Abstract
Let $\mathcal{A}$ be a unital, complex normed $\ast$-algebra with the identity element $\textbf{e}$ such that the set of all algebraic elements of $\mathcal{A}$ is norm dense in the set of all self-adjoint elements of $\mathcal{A}$ and let $\{D_n\}_{n = 0}^{\infty}$ and $\{\Delta_n\}_{n = 0}^{\infty}$ be sequences of continuous linear mappings on $\mathcal{A}$ satisfying \[ \left\lbrace \begin{array}{c l} D_{n + 1}(p) = \sum_{k = 0}^{n}D_{n - k}(p)D_k(p),\\ \\ \Delta_{n + 1}(p) = \sum_{k = 0}^{n}\Delta_{n - k}(p)D_k(p), \end{array} \right. \] for all projections $p$ of $\mathcal{A}$ and all nonnegative integers $n$. Moreover, suppose that $D_0(p) = D_0(p)^2$ holds for all projections $p$ of $\mathcal{A}$. Then \begin{align*} \Delta_n = \frac{C_n}{2}\left(R_{D_0(\textbf{e})}\Delta_0 + L_{\Delta_0(\textbf{e})}D_0 \right) \end{align*} for all $n \in \mathbb{N}$, where $C_n$ denotes the $n$th Catalan number and $R_{D_0(\textbf{e})}(a) = a D_0(\textbf{e})$ and $L_{\Delta_0(\textbf{e})}(a) = \Delta_0(\textbf{e})a$ for all $a \in \mathcal{A}$. Using this result, we present a characterization of left $\tau$-centralizers satisfying a certain recursive relation. In addition, a characterization of generalized higher derivations is presented. Moreover, we show that higher derivations, prime higher derivations, left higher derivations, and $\sigma$-derivations are zero under certain conditions.
First Page
1578
Last Page
1594
Recommended Citation
HOSSEINI, AMIN and KARIZAKI, MEHDI MOHAMMADZADEH
(2020)
"Linear mappings satisfying some recursive sequences,"
Turkish Journal of Mathematics: Vol. 44:
No.
5, Article 5.
https://doi.org/10.3906/mat-1911-35
Available at:
https://journals.tubitak.gov.tr/math/vol44/iss5/5
