On higher order selfadjoint operators.

For a positive integer m and a Hilbert space H an operator T in B (H) , the space of all bounded linear operators on H , is called m -selfadjoint if ∑ k = 0 m (− 1) k ( m k ) T ⁎ k T m − k = 0. In this paper, we show that if T ∈ B (H) and the spectrum of T consists of a finite number of points then it is m -selfadjoint if and only if it is an n -Jordan operator for some integer n. Moreover, we prove that if T is m -selfadjoint then T is nilpotent when it is quasinilpotent. Then we characterize m -selfadjoint weighted shift operators. Also, we show that if T is m -selfadjoint then so is p (T) when p (z) is a polynomial with real coefficients. After that, we investigate an elementary operator τ and a generalized derivation operator δ on the Hilbert-Schmidt class C 2 (H) which are m -selfadjoint. Finally, we prove that no m -selfadjoint operator on an infinite-dimensional Hilbert space, can be N -supercyclic, for any N ≥ 1. [ABSTRACT FROM AUTHOR]