## An Introduction to Classical and P-adic Theory of Linear Operators and ApplicationsThis book provides the reader with a self-contained treatment of the classical operator theory with significant applications to abstract differential equations, and an elegant introduction to basic concepts and methods of the rapidly growing theory of the so-called p-adic operator theory. |

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### Contents

1 | |

113 Examples of Banach Spaces | 3 |

12 Hilbert Spaces | 6 |

122 Hilbert Spaces | 7 |

123 Projections | 8 |

13 Bibliographical Notes | 10 |

Bounded Linear Operators on Classical and padic Hilbert Spaces | 11 |

212 The Adjoint of An Operator | 14 |

317 Maximal Linear Operators | 59 |

318 Algebraic Sum of Linear Operators | 60 |

323 Unbounded Linear Operators On Ew | 65 |

33 Closed Linear Operators on Ew | 67 |

34 The Diagonal Operator on E | 68 |

35 The Equation Ax yon Eo | 71 |

352 Application to the Perturbation of Bases on Ew | 72 |

353 The Unbounded Case | 73 |

213 The Inverse of An Operator | 15 |

214 Normal and Selfadjoint Operators | 17 |

215 Square Root of a Positive Selfadjoint Operator | 18 |

216 Compact Operators | 20 |

217 HilbertSchmidt Operators | 22 |

22 Bounded Linear Operators on padic Hilbert Spaces Ew | 24 |

223 Ultrametric Banach Spaces | 27 |

224 Free Banach Spaces | 28 |

225 The padic Hilbert Space Ew | 29 |

226 Bounded Linear Operators E | 31 |

23 padic HilbertSchmidt Operators | 35 |

233 Further Properties of HilbertSchmidt Operators on Ew | 39 |

234 Applications | 43 |

24 Bibliographical Notes | 46 |

Unbounded Linear Operators on Classical and padic Hilbert Spaces | 47 |

312 Closed and Closable Linear Operators | 49 |

313 Invariant Subspaces for Unbounded Linear Operators | 51 |

314 Semigroups of Linear Operators | 53 |

315 Spectral Theory for Unbounded Linear Operators | 55 |

316 Symmetric and Selfadjoint Linear Operators | 56 |

36 Bibliographical Notes | 75 |

Almost Automorphic and Almost Periodic Solutions to Differential Equations | 77 |

42 Basic Definitions and Notations | 78 |

422 Almost Periodic Functions | 82 |

43 The Equation ut Aut +But +ft | 83 |

432 Almost Periodic Mild Solutions | 85 |

44 The Method of Invariant Subspaces for Unbounded Linear Operators | 87 |

443 Applications to Some SecondOrder Differential Equations | 90 |

444 Almost Automorphic Solutions to Some SecondOrder Hyperbolic Equations | 91 |

45 Pseudo Almost Periodic Solutions to Some SecondOrder Differential Equations | 95 |

452 Applications | 98 |

46 Existence and Uniqueness of Almost Automorphic Mild Solutions to Semilinear Equations | 100 |

462 Existence and Uniqueness of Mild Solutions | 101 |

463 Case of Bounded Perturbations | 104 |

464 Applications | 105 |

47 Bibliographical Notes | 108 |

109 | |

115 | |

### Common terms and phrases

adjoint automorphic functions automorphic mild solution Automorphic Solutions basis for Ew bounded linear operator bounded operators bounded solution canonical orthogonal basis Cauchy sequence Cauchy-Schwarz inequality Classical and p-adic classical setting closable closed linear operator closed subspace co-semigroup complete continuous functions converges defined by D(A Definition denote densely defined closed differential equations Ew x Ew Example existence and uniqueness following assumptions free Banach space Furthermore G D(A G Ew G. M. N'Guerekata hence Hilbert-Schmidt operator infinitesimal inner product invariant subspaces Lemma Moreover non-Archimedean normed vector space operator on H operators on Ew orthogonal projections orthonormal basis p-adic Hilbert space periodic functions periodic solutions Proof Proposition pseudo almost periodic Psf(s PsX(s QsF(s Qsw(s QsX(s real numbers Remark satisfies self-adjoint operator semigroup sequence of nonzero solution to 4.1 space Ew space H subset Suppose Theorem ultrametric Banach space ultrametric field unique almost automorphic