Analytic extension of a $N$th roots of $M$-hyponormal operator

Abstract

In this paper‎, ‎we study some properties of analytic extension of a $n$th roots of $M$-hyponormal operator‎. ‎We show that every analytic extension of a $n$th roots of $M$-hyponormal operator is subscalar of order $2k+2n$‎. ‎As a consequence‎, ‎we get that if the spectrum of such operator $T$ has a nonempty interior in $mathbb{C}$‎, ‎then $T$ has a nontrivial invariant subspace‎. ‎Finally‎, ‎we show that the sum of a $n$th roots of $M$-hyponormal operator and an algebraic operator of order $k$ which are commuting is subscalar of order $2kn+2$‎.