We establish a Spectral Exclusion Principle for unbounded sub normals. As an application, we obtain some polynomial approximation results in the functional model spaces. 1. PRELIMINARIES For a subset A of the complex plane C, let A*, int(A), A and Ac respectively denote the conjugate, the interior, the closure and the complement of A in C. We use 7Z to denote the real line, and Rez and Imz respectively denote the real and imaginary parts of a complex number z. Let NH be a complex infinite-dimensional separable Hilbert space with the inner product (, )x and the corresponding norm 11 11-. If S is a densely defined linear operator in H with domain D(S), then we use u(S), up(S), Cap(S) to respectively denote the spectrum, the point spectrum and the approximate point spectrum of S. It may be recalled that op(S) is the set of eigenvalues of S, that Cap(S) is the set of those A in C for which S A is not bounded below, and that v(S) is the complement of the set of those A in C for which (T A)-1 exists as a bounded linear operator on N. For a normal operator N in H, u(N) = (ap(N). For a non-negative measure ,u on the complex plane C, we will use supp(,u) to denote the support of u. In the present section, we record a few requisites pertaining to the unbounded subnormals and the m-E-accretive operators. In Section 2, we prove a Spectral Exclusion Principle for unbounded subnormals. We use this principle to obtain H' functional calculi for unbounded subnormals. As an application, we present some polynomial approximation results in the functional model spaces. These results rely heavily on the results of Crouzeix and Delyon regarding the m-Z-accretive operators ([4], Chapter 7). 1.1. Unbounded subnormals. A densely defined linear operator S in NH with domain D(S) is said to be cyclic if there is a vector fo E vD (s) Fn=0 D(5f) (referred to as a cyclic vector of S) such that 1(S) is the linear span lin{Snfo n > O} of the set {Snfo: n > O}. If S is a densely defined linear operator in NH with domain D(S), then S is said to be subnormal if there exist a Hilbert space IC containing H and a densely defined Received by the editors April 26, 2007, and, in revised form, December 23, 2007. 2000 Mathematics Subject Classification. Primary 47A60, 47B20; Secondary 41A10.