Gen. Math. Notes, Vol. 26, No. 1, January 2015, pp.46-52 ISSN 2219-7184; Copyright c ICSRS Publication, 2015 www.i-csrs.org Available free online at http://www.geman.in Some Properties of M −Class Q and M −Class Q ∗ Operators Valdete Rexh¨ ebeqaj Hamiti Faculty of Electrical and Computer Engineering University of Prishtina, Prishtin¨ e, 10000, Kosov¨ e E-mail: valdete r@hotmail.com (Received: 21-8-14 / Accepted: 18-11-14) Abstract Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class of operators: M −class Q * operators. An op- erator T ∈L(H) is of M −class Q * , for a fixed real number M ≥ 1, if T satisfies ‖T * x‖ 2 ≤ 1 2 (M 2 ‖T 2 x‖ 2 + ‖x‖ 2 ), for all x ∈H. We prove the basic properties of this class of operators and the M −class Q operators. Keywords: M −paranormal operator, operator of M −class Q, M * −paranormal operator, operator of M −class Q * . 1 Introduction Throughout this paper, let H be a complex Hilbert space with inner product 〈·, ·〉. Let L(H) denote the C * algebra of all bounded operators on H. For T ∈L(H), we denote by kerT the null space, by T (H) the range of T and by σ(T ) the spectrum of T . The null operator and the identity on H will be denoted by O and I , respectively. If T is an operator, then T * is its adjoint, and ‖T ‖ = ‖T * ‖. We shall denote the set of all complex numbers by C, the set of all non- negative integers by N and the complex conjugate of a complex number λ by λ. The closure of a set M will be denoted by M. An operator T ∈L(H) is