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non-Archimedean Hilbert space; bilinear form; continuous linear functionals; non-Archimedean Riesz theorem; bounded bilinear form; stable unbounded bilinear form; unstable unbounded bilinear form
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.
[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. MR 2092641 | Zbl 1077.47061
[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
[3] Diagana T.: Representation of bilinear forms in non-Archimedean Hilbert space by linear operators. Comment. Math. Univ. Carolin. 47 (2006), 4 695-705. MR 2337423 | Zbl 1150.47408
[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. MR 2126449 | Zbl 1087.47061
[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
[6] Diagana T.: Bilinear forms on non-Archimedean Hilbert spaces. preprint, 2005.
[7] Diagana T.: Fractional powers of the algebraic sum of normal operators. Proc. Amer. Math. Soc. 134 (2006), 6 1777-1782. MR 2207493 | Zbl 1092.47027
[8] Diagana T.: An Introduction to Classical and $p$-adic Theory of Linear Operators and Applications. Nova Science Publishers, New York, 2006. MR 2269328 | Zbl 1118.47323
[9] Diarra B.: An operator on some ultrametric Hilbert spaces. J. Analysis 6 (1998), 55-74. MR 1671148 | Zbl 0930.47049
[10] Diarra B.: Geometry of the $p$-adic Hilbert spaces. preprint, 1999.
[11] Johnson G.W., Lapidus M.L.: The Feynman Integral and Feynman Operational Calculus. Oxford Univ. Press, Oxford, 2000. MR 1771173
[12] Kato T.: Perturbation Theory for Linear Operators. Springer, New York, 1966. MR 0203473 | Zbl 0836.47009
[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
[14] van Rooij A.C.M.: Non-Archimedean Functional Analysis. Marcel Dekker, New York, 1978. MR 0512894 | Zbl 0396.46061
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