Browsing by Author "Schikhof, W. H."

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    A new method for comparing two Norm Hilbert spaces and their operators
    (ELSEVIER SCIENCE BV, 2011) Ochsenius, H.; Schikhof, W. H.
    The paper deals with operators on Norm Hilbert spaces over a Krull valued field K. By using carefully selected equivalent norms (i) the perturbation theory of Fredholm operators (see Ochsenius and Schikhof (2010) [7]) is completed, and (ii) new matrix characterizations of operators are derived (compare Ochsenius and Schikhof (2007) [6]). For a prominent E, the first infinite-dimensional orthomodular space in history constructed by Keller (1980) [1], this leads to simple and elegant characterizations. (C) 2011 Royal Netherlands Academy of Arts and Sciences. Published by Elsevier B.V. All rights reserved.
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    Matrix characterizations of Lipschitz operators on Banach spaces over Krull valued fields
    (BELGIAN MATHEMATICAL SOC TRIOMPHE, 2007) Ochsenius, H.; Schikhof, W. H.
    Let K be a complete infinite rank valued field and E a K-Banach space with a countable orthogonal base. In [9] and [10] we have studied bounded (called Lipschitz) operators on E and introduced the notion of a strictly Lipschitz operator. Here we characterize them, as well as compact and nuclear operators, in terms of their (infinite) matrices. This results provide new insights and also useful criteria for constructing operators with given properties.
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    Norm Hilbert spaces over Krull valued fields
    (ELSEVIER SCIENCE BV, 2006) Ochsenius, H.; Schikhof, W. H.
    Norm Hilbert spaces (NHS) are defined as Banach spaces over valued fields (see 1.4) for which each closed subspace has a norm-orthogonal complement. For fields with a rank I valuation, these spaces were characterized already in [10, 5.13, 5.16], where it was proved that infinite-dimensional NHS exist only if the valuation of K is discrete. The first discussion of the case of (Krull) valued fields appeared in [1] and [3]. In this paper we continue and expand this work focussing on the most interesting cases, not covered before. If K is not metrizable then each NHS is finite-dimensional (Corollary 3.2.2), but otherwise there do exist infinite-dimensional NHS; they are completely described in 3.2.5. Our main result is Theorem 3.2.1, where various characterizations of NI-IS of different nature are presented. Typical results are that NHS arc of countable type, that they have orthogonal bases, and that no subspace is linearly homeomorphic to co.