Speaker
Description
Let $\mathcal{H}$ be a complex Hilbert space, and let $\mathcal{B}(\mathcal{H})$ denote the algebra of all bounded linear operators on $\mathcal{H}$. For every $T\in \mathcal{B}(\mathcal{H})$, we denote by $T^{\ast}$ the adjoint of $T$. An operator $T\in \mathcal{B}(\mathcal{H})$ is said to be m-symmetric if
$$\sum_{k=0}^m (-1)^{k}\left(\begin{array}{l}
m \\
k
\end{array}\right)T^{\ast (m-k)}T^{k}=0,$$
where $\left(\begin{array}{l}
m \
k
\end{array}\right) $ is the binomial coefficient.
$T\in\mathcal{B}(\mathcal{H})$ is called skew m-symmetric if
$$\sum_{k=0}^m \left(\begin{array}{l}
m \\
k
\end{array}\right)T^{\ast (m-k)}T^{k}=0.$$
In this paper, we investigate in the products and the sums of m-symmetric and skew m-symmetric operators. More precisely, we show that if $T,S\in\mathcal{B}(\mathcal{H})$ are commuting operators such that T is m-symmetric and S is n-symmetric, then TS and T+S are (m+n-1)-symmetric. Furthermore, if $T,S\in\mathcal{B}(\mathcal{H})$ are commuting operators such that T is skew m-symmetric and S is skew n-symmetric, then TS is (m+n-1)-symmetric, and T+S is skew (m+n-1). Additionally, if $T,S\in\mathcal{B}(\mathcal{H})$ are commuting operators such that T is m-symmetric and S is skew n-symmetric, then TS is skew (m+n-1)-symmetric.
