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**Towards a theory of some unbounded linear operators on \(p\)-adic Hilbert spaces and applications.**
*(English)*
Zbl 1087.47061

The author presents some examples of (un-)bounded operators acting on a
“\(p\)-adic Hilbert space” \(E\). This is a free Banach-space endowed with a bilinear form satisfying the Cauchy-Schwarz inequality. It should be
mentioned that only little is known about such structures (cf., in
particular, the references to Bertin Diarra’s papers in the article under
review). The paper, therefore, is mainly dedicated to a study of
particular examples.

The question of the existence of adjoints of unbounded operators is addressed; a satisfying characterization, however, is not achieved. Motivated by the established characterization of bounded operators with adjoint, the author introduces the notion of “an operator \(A\) which is said to have an adjoint \(A^*\) if and only if it satisfies condition (2.2)” (cf. Definition 2.2 in the paper). It is, however, not proven in the sequel that this notion is necessary for the existence of an adjoint of \(A^*\) in the usual sense. This interesting question therefore remains open.

The paper contains the following misprints:

Proof of Proposition 2.4 (page 211, line 9): “what we have to show” is “there exists a positive integer \(i\) such that...” (quantifiers). Proof of Theorem 5.1 (last line of the proof): the conclusion “\(A\) has norm \(1\), hence \(A\) is an isometry” should be replaced by “\(A-I\) has norm strict less than \(1\), therefore \(A\) is an isometry.”

The question of the existence of adjoints of unbounded operators is addressed; a satisfying characterization, however, is not achieved. Motivated by the established characterization of bounded operators with adjoint, the author introduces the notion of “an operator \(A\) which is said to have an adjoint \(A^*\) if and only if it satisfies condition (2.2)” (cf. Definition 2.2 in the paper). It is, however, not proven in the sequel that this notion is necessary for the existence of an adjoint of \(A^*\) in the usual sense. This interesting question therefore remains open.

The paper contains the following misprints:

Proof of Proposition 2.4 (page 211, line 9): “what we have to show” is “there exists a positive integer \(i\) such that...” (quantifiers). Proof of Theorem 5.1 (last line of the proof): the conclusion “\(A\) has norm \(1\), hence \(A\) is an isometry” should be replaced by “\(A-I\) has norm strict less than \(1\), therefore \(A\) is an isometry.”

Reviewer: Eberhard Mayerhofer (Wien)

### MSC:

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

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |

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

47B25 | Linear symmetric and selfadjoint operators (unbounded) |

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\textit{T. Diagana}, Ann. Math. Blaise Pascal 12, No. 1, 205--222 (2005; Zbl 1087.47061)

### References:

[1] | Albeverio, S.; Bayod, J. M.; Perez-Gargia, C.; Cianci, R.; Khrennikov, A. Y., Non-Archimedean analogues of orthogonal and symmetric operators and \(p\)-adic quantization, Acta Appl. Math., 57, 3, 205-237, (1999) · Zbl 0943.46044 |

[2] | Basu, S.; Diagana, T.; Ramaroson, F., A \(p\)-adic version of Hilbert-Schmidt operators and applications, J. Anal. Appl., 2, 3, 173-188, (2004) · Zbl 1077.47061 |

[3] | Diarra, B.; Ludkovsky, S., Spectral integration and spectral theory for non-Archimedean Banach spaces, Int. J. Math. Math. Sci., 31, 7, 421-442, (2002) · Zbl 0999.47063 |

[4] | Diarra, B., An operator on some ultrametric Hilbert spaces, J. Analysis, 6, 55-74, (1998) · Zbl 0930.47049 |

[5] | Diarra, B., Geometry of the \(p\)-adic Hilbert spaces, Preprint, (1999) |

[6] | Keller, H. A.; Ochsenius, H., Algebras of bounded operators on nonclassical orthomodular spaces. Proceedings of the international quantum structures association, part III (Castiglioncello, 1992), Internat. J. Theoret. Phys., 33, 1, 1-11, (1994) · Zbl 0809.46094 |

[7] | Khrennikov, A. Y., Mathematical methods in non-Archimedean physics. (Russian)., Uspekhi Math. Nauk., 45, 79-110, (1990) · Zbl 0722.46040 |

[8] | Khrennikov, A. Y., Generalized functions on a non-Archimedean super space, (Russian) Izv. Akad. Nauk SSSR Ser. Math., 55, 6, 1257-1286, (1991) · Zbl 0755.46048 |

[9] | Khrennikov, A. Y.\(, p\)-adic quantum mechanics with \(p\)-adic valued functions, J. Math. Phys., 32, 4, 932-937, (1991) · Zbl 0746.46067 |

[10] | Ochsenius, H.; Schikhof, W. H., Banach spaces over fields with an infinite rank valuation \(, p\)-adic functional analysis (Poznań, 1998), Lectures Notes in Pure and Appl. Math., 207, 233-293, (1999) · Zbl 0938.46056 |

[11] | van Rooij, A. C. M., Non-Archimedean Functional Analysis, (1978), Marcel Dekker, Inc. · Zbl 0396.46061 |

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