Abstract. In this paper, we introduce the class of analytic extensionsof M-hyponormal operators and we study various properties of this class.We also use a special Sobolev space to show that every analytic extensionof an M-hyponormal operator T is subscalar of order 2k + 2. Finally weobtain that an analytic extension of an M-hyponormal operator satisfiesWeyl’s theorem. 1. IntroductionLet B(H) be the algebra of all bounded linear operators acting on infinitedimensional separable complex Hilbert space H. If T ∈ B(H), we shall writeN(T) and R(T) for the null space and range space of T. As an easy exten-sion of normal operators, hyponormal operators have been studied by manymathematicians. Though there are many unsolved interesting problems for hy-ponormal operators (e.g., the invariant subspace problem), one of recent trendsin operator theory is studying natural extensions of hyponormal operators. Sowe introduce some of these non-hyponormal operators. An operator T ∈ B(H)is said to be hyponormal if T