On higher order selfadjoint operators

Abstract

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=0m(−1)k(mk)T⁎kTm−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 C2(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.