On a Friedrichs Extension Related to Unbounded Subnormal Operators

Abstract

Abstract. We combine the theory of sectorial sesquilinear forms with the theory of unbounded subnormal operators in Hilbert spaces to characterize the Friedrichs extensions of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Such characterizations lead to abstract Galerkin approximations and generalized wave equations.2000 Mathematics Subject Classification. Primary 41A65, 47B20. Secondary 35K90, 41A10, 47A07, 47B32. Preliminaries.The theory of sectorial sesquilinear forms is a powerful tool in the study of differential operators (refer to [4] and [6], for example). The present paper is an illustration of the utility of that theory in the study of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Our methods make an essential use of the fact that the multiplication operators under consideration are unbounded cyclic subnormal operators (see definitions below), and rely crucially on the theory of such operators as expounded in [9]. (See also [7], [8].) The interaction of the theory of sectorial sesquilinear forms with that of unbounded subnormals was explored by the authors in [2] with a special attention paid to those subnormal operators that admit 'analytic models'. While the investigations here are in the same spirit as those carried out in [2], they are in no way dependent on the results obtained in [2]. The main result of the paper (Theorem 3 in Section 2) is the identification of the Friedrichs extension of a certain multiplication operator with a 'maximal' multiplication operator in the relevant functional Hilbert space. Such an identification has natural advantages as demonstrated by a couple of applications in Section 3 (Proposition 3 and Proposition 4). In the present section we record some requisites pertaining to sectorial sesquilinear forms and 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 Re z and Im z 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 at https://www.cambridge.org/core/terms. https://doi

Unbounded subnormal weighted shifts on directed trees

“…A systematic study of this class of operators was undertaken in the trilogy [57,58,59]. The theory of unbounded subnormal operators has intimate connections with other branches of mathematics and quantum physics (see [67,7,3] and [32,56,66,33]). It has been developed in two main directions, the first is purely theoretical (cf.…”