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, ‎w...

We construct a Banach space operator T which is not E(T)subscalar but ‖(T − z)−1‖ ≤ (|z| − 1)−1 for |z| > 1 and m(T − z) ≥ const · (1 − |z|)3 for |z| < 1 (here m denotes the minimum modulus). This gives a negative answer to a variant of a problem of Laursen and Neumann. We also give a sufficient condition (in terms of growth of resolvent and of an analytic left inverse of T − z) implying that T...

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, ‎w...

In 1984 M. Putinar proved that hyponormal operators are subscalar operators of order two. The proof provided a concrete structure of such operators. We will use this structure to give a sufficient condition for hyponormal operators T with trace-class commutator to admit a direct summand S so that T ⊕ S is the sum of a normal operator and a HilbertSchmidt operator. We investigate what this suffi...

We prove some further properties of the operator T ∈ nQN n-power quasinormal, defined in Sid Ahmed, 2011 . In particular we show that the operator T ∈ nQN satisfying the translation invariant property is normal and that the operator T ∈ nQN is not supercyclic provided that it is not invertible. Also, we study some cases in which an operator T ∈ 2QN is subscalar of order m; that is, it is simila...

In this paper we show that p-hyponormal operators with 0 / ∈ σ(|T | 1 2 r ) are subscalar. As a corollary, we get that such operators with rich spectra have non-trivial invariant subspaces.