On a Friedrichs Extension Related to Unbounded Subnormals-Ii

Abstract

We study the Friedrichs extensions of unbounded cyclic subnormals. The main result of the present paper is the identification of the Friedrichs extensions of certain cyclic subnormals with their closures. This generalizes as well as complements the main result obtained in [5]. Such characterizations lead to abstract Galerkin approximations, generalized wave equations, and bounded H ∞ -functional calculi.2000 Mathematics Subject Classification. Primary 41A65, 47B20. Secondary 35K90, 41A10, 47A07, 47B32. Preliminaries.The present paper is a sequel to [2], [5], and continues the study of unbounded cyclic subnormals in the same spirit. The main result of the paper is the identification of the Friedrichs extensions of certain cyclic subnormals with their closures. As a corollary, we obtain a generalization of the main result of [5]. All the results in this paper rely heavily on the ideas developed in [5] and [2]. Also, in the present investigations, the notion of the minimal normal extension of spectral type ([9]) turns out to be an essential ingredient.The paper is organized as follows. In Section 2, we give a sufficient condition for unbounded subnormals to admit Friedrichs extensions, and characterize the Friedrichs extensions of certain cyclic subnormals. In Section 3, we discuss several applications of Theorem 2.3. These are the Galerkin approximation in the functional model space, existence and uniqueness of the Hilbert space valued solutions of a generalized wave equation, and an H ∞ -functional calculus for certain cyclic subnormals. In the last section, we obtain generalizations of some results obtained in [5]. In the present section, we fix the notation, and record a few requisites pertaining to unbounded subnormals and sectorial forms. Unbounded subnormals.For 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

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