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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.

##### MSC:
 47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX) 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
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