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 (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 |