# Article

 Title: Representation of bilinear forms in non-Archimedean Hilbert space by linear operators II (English) Author: Attimu, Dodzi Author: Diagana, Toka Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 48 Issue: 3 Year: 2007 Pages: 431-442 . Category: math . Summary: The paper considers the representation of non-degenerate bilinear forms on the non-Archimedean Hilbert space $\Bbb E_\omega \times \Bbb E_\omega$ by linear operators. More precisely, upon making some suitable assumptions we prove that if $\varphi$ is a non-degenerate bilinear form on $\Bbb E_\omega \times \Bbb E_\omega$, then $\varphi$ is representable by a unique linear operator $A$ whose adjoint operator $A^*$ exists. (English) Keyword: non-Archimedean Hilbert space Keyword: bilinear form Keyword: continuous linear functionals Keyword: non-Archimedean Riesz theorem Keyword: bounded bilinear form Keyword: stable unbounded bilinear form Keyword: unstable unbounded bilinear form MSC: 46S10 MSC: 47A07 MSC: 47S10 idZBL: Zbl 1199.47334 idMR: MR2374125 . Date available: 2009-05-05T17:03:56Z Last updated: 2012-05-01 Stable URL: http://hdl.handle.net/10338.dmlcz/119670 . Related article: http://dml.cz/handle/10338.dmlcz/119629 . Reference: [1] Basu S., Diagana T., Ramaroson F.: A $p$-adic version of Hilbert-Schmidt operators and applications.J. Anal. Appl. 2 (2004), 3 173-188. Zbl 1077.47061, MR 2092641 Reference: [2] de Bivar-Weinholtz A., Lapidus M.L.: Product formula for resolvents of normal operator and the modified Feynman integral.Proc. Amer. Math. Soc. 110 (1990), 2 449-460. MR 1013964 Reference: [3] Diagana T.: Representation of bilinear forms in non-Archimedean Hilbert space by linear operators.Comment. Math. Univ. Carolin. 47 (2006), 4 695-705. Zbl 1150.47408, MR 2337423 Reference: [4] Diagana T.: Towards a theory of some unbounded linear operators on $p$-adic Hilbert spaces and applications.Ann. Math. Blaise Pascal 12 (2005), 1 205-222. Zbl 1087.47061, MR 2126449 Reference: [5] Diagana T.: Erratum to: Towards a theory of some unbounded linear operators on $p$-adic Hilbert spaces and applications".Ann. Math. Blaise Pascal 13 (2006), 105-106. MR 2233015 Reference: [6] Diagana T.: Bilinear forms on non-Archimedean Hilbert spaces.preprint, 2005. Reference: [7] Diagana T.: Fractional powers of the algebraic sum of normal operators.Proc. Amer. Math. Soc. 134 (2006), 6 1777-1782. Zbl 1092.47027, MR 2207493 Reference: [8] Diagana T.: An Introduction to Classical and $p$-adic Theory of Linear Operators and Applications.Nova Science Publishers, New York, 2006. Zbl 1118.47323, MR 2269328 Reference: [9] Diarra B.: An operator on some ultrametric Hilbert spaces.J. Analysis 6 (1998), 55-74. Zbl 0930.47049, MR 1671148 Reference: [10] Diarra B.: Geometry of the $p$-adic Hilbert spaces.preprint, 1999. Reference: [11] Johnson G.W., Lapidus M.L.: The Feynman Integral and Feynman Operational Calculus.Oxford Univ. Press, Oxford, 2000. MR 1771173 Reference: [12] Kato T.: Perturbation Theory for Linear Operators.Springer, New York, 1966. Zbl 0836.47009, MR 0203473 Reference: [13] Ochsenius H., Schikhof W.H.: Banach Spaces Over Fields with an Infinite Rank Valuation, $p$-adic Functional Analysis.(Poznan, 1998), Marcel Dekker, New York, 1999, pp.233-293. MR 1703500 Reference: [14] van Rooij A.C.M.: Non-Archimedean Functional Analysis.Marcel Dekker, New York, 1978. Zbl 0396.46061, MR 0512894 .

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