A spectral exclusion principle for unbounded subnormals

Abstract

Abstract. We establish a Spectral Exclusion Principle for unbounded subnormals. As an application, we obtain some polynomial approximation results in the functional model spaces. PreliminariesFor a subset A of the complex plane C, let A * , int(A), A and A c respectively denote the conjugate, the interior, the closure and the complement of A in C. We use R to denote the real line, and Rez and Imz respectively denote the real and imaginary parts of a complex number z. Let H be a complex infinite-dimensional separable Hilbert space with the inner product ·, · H and the corresponding norm · H . If S is a densely defined linear operator in H with domain D(S), then we use σ(S), σ p (S), σ ap (S) to respectively denote the spectrum, the point spectrum and the approximate point spectrum of S. It may be recalled that σ p (S) is the set of eigenvalues of S, that σ ap (S) is the set of those λ in C for which S − λ is not bounded below, and that σ(S) is the complement of the set of those λ in C for which (T − λ)−1 exists as a bounded linear operator on H. For a normal operator. For a non-negative measure µ on the complex plane C, we will use supp(µ) to denote the support of µ.In the present section, we record a few requisites pertaining to the unbounded subnormals and the m-Σ-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-Σ-accretive operators ([4], Chapter 7). Unbounded subnormals. A densely defined linear operator S in H with domain D(S) is said to be cyclic if there is a vector referred to as a cyclic vector of S) such that D(S) is the linear span lin{Sn f 0 : n ≥ 0} of the set {S n f 0 : n ≥ 0}. If S is a densely defined linear operator in H with domain D(S), then S is said to be subnormal if there exist a Hilbert space K containing H and a densely defined