An introduction to classical and \(p\)-adic theory of linear operators and applications.

*(English)*Zbl 1118.47323
Huntington, NY: Nova Science Publishers (ISBN 1-59454-424-7). x, 116 p. (2006).

The book consists of four chapters. Chapter 1, “Banach and Hilbert spaces”, contains a brief exposition of basic notions of functional analysis. In Chapter 2, “Bounded linear operators on classical and \(p\)-adic Hilbert spaces”, two kinds of material are included. The author gives basic notions of operator theory (normal and self-adjoint operators, compact operators, Hilbert-Schmidt operators), mostly without proofs; references to standard textbooks are provided. Then the author gives some notions regarding ultrametric Banach spaces and considers a “\(p\)-adic Hilbert space”, that is, a sequence space over a non-Archimedean field with a weighted supremum norm and a bilinear form satisfying the Cauchy-Schwarz inequality. Note that the norm of this space cannot be restored from the bilinear form. However, the availability of a form makes it possible to try to mimic some classical notions like that of a Hilbert-Schmidt operator. Some properties of the latter resemble the classical ones, not being non-Archimedean in spirit. For example, the Hilbert–Schmidt norm is of an \(\ell^2\) kind and does not satisfy the ultrametric inequality.

Chapter 3, “Unbounded linear operators on classical and \(p\)-adic Hilbert spaces”, contains some classical definitions and results (closed and closable operators, some information about semigroups of operators and spectral theory, etc.) and their counterparts, mostly on the level of simple examples, for operators on the above \(p\)-adic Hilbert space.

Chapter 4, “Almost automorphic and almost periodic solutions to differential equations”, is very different from the above ones. It is devoted to some generalizations of almost periodic functions, such as almost automorphic functions and pseudo-almost periodic functions. Existence of solutions from these classes for linear and semilinear operator-differential equations (in real or complex Banach spaces) is studied. For further results in this direction, see T. Diagana and E. Hernández Morales [J. Math. Anal. Appl. 327, No. 2, 776–791 (2007; Zbl 1123.34060); T. Diagana, Nonlinear Anal. 66, No. 1, 228–240 (2007; Zbl 1112.34034)].

Chapter 3, “Unbounded linear operators on classical and \(p\)-adic Hilbert spaces”, contains some classical definitions and results (closed and closable operators, some information about semigroups of operators and spectral theory, etc.) and their counterparts, mostly on the level of simple examples, for operators on the above \(p\)-adic Hilbert space.

Chapter 4, “Almost automorphic and almost periodic solutions to differential equations”, is very different from the above ones. It is devoted to some generalizations of almost periodic functions, such as almost automorphic functions and pseudo-almost periodic functions. Existence of solutions from these classes for linear and semilinear operator-differential equations (in real or complex Banach spaces) is studied. For further results in this direction, see T. Diagana and E. Hernández Morales [J. Math. Anal. Appl. 327, No. 2, 776–791 (2007; Zbl 1123.34060); T. Diagana, Nonlinear Anal. 66, No. 1, 228–240 (2007; Zbl 1112.34034)].

##### MSC:

47S10 | Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory |

47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

34D09 | Dichotomy, trichotomy of solutions to ordinary differential equations |

34G10 | Linear differential equations in abstract spaces |

47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |