A new generalization of the class of \(m-\)symmetric operators

In this paper, we define and study a new class of bounded linear operators which is a generalization of the class of \(m-\)symmetric operators. Let m be a strictly positive integer number and \(U\in {\mathcal {B}}({\mathcal {H}})\) is a unitary operator, an operator \(T\in {\mathcal {B}}({\mathcal {H}})\) is said to be a \((U,m)-\)symmetry if it commutes with U such that

$$\begin{aligned} \sum _{k=0}^m (-1)^{k}\left( \begin{array}{l} m \\ k \end{array}\right) T^{*(m-k)}T^{k}U^{k}=0. \end{aligned}$$

It is shown that if T is a \((U,m)-\)symmetry, then \(T^{p}\) is a \((U^{p},m)-\)symmetry. We study the product and the sum of such a class. Moreover, if T is a \((U,m)-\)symmetry and m is even, we obtain that T is a \((U,m-1)-\)symmetry. We prove that if Q is a nilpotent operator of order n which commutes with both T and U, then \(T+Q\) is a \((U,m+2n-2)-\)symmetry. Also, we give some spectral properties of \((U,m)-\)symmetric operators. Finally, we show further results concerning this class of operators on a finite dimensional Hilbert space.