Multi-interval linear ordinary boundary value problems and complex symplectic algebra. (English) Zbl 0982.47032
Mem. Am. Math. Soc. 715, 64 p. (2001).
The memoir begins with reviewing the role played by the Glazman-Krein-Naimark (GKN) theorem in relating the self-adjoint operators generated in the boundary value theory for quasi-differential systems. They are defined on a single interval of the real line \(\mathbb{R}\), to symplectic spaces. Similarly the same setup for the generalizations to quasi-differential systems, which are defined on \(\mathbb{R}\) for multi-interval problems.
Following Everitt-Zettl the authors define general multi-interval quasi-differential systems,
\(\{I_r, M_r,\omega_r:r\in \Omega\}\) where \(\Omega\) is general but non-empty index set that may be finite, denumerable or non-denumerable. They often term the problem a “multi-interval system”. Such a multi-interval system consists of a set of prescribed intervals \(I_r\subset\mathbb{R}\), each bearing a given positive weight \(\omega_r\), so as to define the usual Hilbert function space \(L^2_r(I_r; \omega_r)\equiv L^2_r\) of complex-valued square-integrable function on \(I_r\), and each supporting an assigned quasi-differential expression \(M_r\) which thus generates appropriate (unbounded) linear operators in our Hilbert function space for each \(r\). They show under suitable hypotheses, that a multi-interval system generates maximal and minimal operators, \(T_1\) and \(T_0\) with domains in the direct sum Hilbert space. Furthermore, the system generates self-adjoint operators in the direct sum Hilbert space, which are determined by kinds of generalized self-adjoint boundary conditions. Many of the results are illustrated through several kinds of examples. The examples include complete Lagrangians, for both finite- and infinite-dimensional complex symplectic spaces \(\mathbb{S}\) and illuminates new phenomena for the boundary value problems of multi-interval system. The book is very well organized and written in a clear concise manner. Highly recommended for graduate work.
Following Everitt-Zettl the authors define general multi-interval quasi-differential systems,
\(\{I_r, M_r,\omega_r:r\in \Omega\}\) where \(\Omega\) is general but non-empty index set that may be finite, denumerable or non-denumerable. They often term the problem a “multi-interval system”. Such a multi-interval system consists of a set of prescribed intervals \(I_r\subset\mathbb{R}\), each bearing a given positive weight \(\omega_r\), so as to define the usual Hilbert function space \(L^2_r(I_r; \omega_r)\equiv L^2_r\) of complex-valued square-integrable function on \(I_r\), and each supporting an assigned quasi-differential expression \(M_r\) which thus generates appropriate (unbounded) linear operators in our Hilbert function space for each \(r\). They show under suitable hypotheses, that a multi-interval system generates maximal and minimal operators, \(T_1\) and \(T_0\) with domains in the direct sum Hilbert space. Furthermore, the system generates self-adjoint operators in the direct sum Hilbert space, which are determined by kinds of generalized self-adjoint boundary conditions. Many of the results are illustrated through several kinds of examples. The examples include complete Lagrangians, for both finite- and infinite-dimensional complex symplectic spaces \(\mathbb{S}\) and illuminates new phenomena for the boundary value problems of multi-interval system. The book is very well organized and written in a clear concise manner. Highly recommended for graduate work.
Reviewer: J.Schmeelk (Richmond)
MSC:
47E05 | General theory of ordinary differential operators |
47B25 | Linear symmetric and selfadjoint operators (unbounded) |
34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
51A50 | Polar geometry, symplectic spaces, orthogonal spaces |
34L05 | General spectral theory of ordinary differential operators |
47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |
34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |