Artykuły naukowe (WMiI)
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Browsing Artykuły naukowe (WMiI) by Author "Śliwa, Wiesław"
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Item An explicit example concernig the invariant subspace problem for Banach spaces(Rocky Mountain Mathematics Consortium, 2010-04) Śliwa, WiesławWe simplify the negative solution to the invariant subspace problem for Banach spaces. Developing the ideas of Read we give an explicit example of a continuous linear operator on the Banach space of all absolute summable scalar sequences without nontrivial closed invariant subspacesItem Closed subspaces without Schauder bases in non-archimedean Frechet spaces(Royal Netherlands Academy of Arts and Sciences, 2001-06) Śliwa, WiesławLet E be an infinite-dimensional non-archimedean Frechet space which is not isomorphic to any of the following spaces: $c_0,c_0 x K^N,K^N$. It is proved that E contains a closed subspace without a Schauder basis (even without a strongly finite-dimensional Schauder decomposition). Conversely, it is shown that any closed subspace of $c_0 x K^N$ has a Schauder basis.Item Descriptive Topology in non-Archimedean Function Spaces Cp(X, K). Part I(2012) Śliwa, Wiesław; Kąkol, JerzyLet $K$ be a non-archimedean field and let $X$ be an ultraregular space. We study the non-archimedean locally convex space $C_p(X;K)$ of all $K$-valued continuous functions on $X$ endowed with the pointwise topology. We show that $K$ is spherically complete if and only if every polar metrizable locally convex space $E$ over $K$ is weakly angelic. This extends a result of Kiyosawa - Schikhof for polar Banach spaces. For any compact ultraregular space $X$ we prove that $C_p(X;K)$ is Frechet-Urysohn if and only if $X$ is scattered (a non-archimedean variant of Gerlits - Pytkeev's result). If $K$ is locally compact we show the following: (1) For any ultraregular space $X$ the space $C_p(X;K)$ is K-analytic if and only if it has a compact resolution (a non-archimedean variant of Tkachuk's theorem); (2) For any ultrametrizable space $X$ the space $C_p(X;K)$ is analytic if and only if $X$ is $\sigma$-compact (a non-archimedean variant of Christensen's theorem).Item Every infinite-dimensional non-archimedean Frechet space has an orthogonal basic sequence(Royal Netherlands Academy of Arts and Sciences, 2000-09) Śliwa, WiesławIt is proved that any infinite-dimensional non-archimedean metrizable locally convex space has an orthogonal basic sequence.Item Examples of non-archimedean nuclear Frechet spaces without a Schauder basis(Royal Netherlands Academy of Arts and Sciences, 2000-12) Śliwa, WiesławWe solve the problem of the existence of a Schauder basis in non-archimedean Frechet spaces of countable type. Using examples of real nuclear Frechet spaces without a Schauder basis we construct examples of non-archimedean nuclear Frechet spaces without a Schauder basis (even without the bounded approximation property).Item Frechet spaces of non-archimedean valued continuous functions(2012) Śliwa, WiesławLet $X$ be an ultraregular space and let $K$ be a complete non-archimedean non-trivially valued field. Assume that the locally convex space $E$ = $C_c(X;K)$ of all continuous functions from $X$ to $K$ with the topology of uniform convergence on compact subsets of $X$ is a Frechet space. We shall prove that $E$ has an orthogonal basis consisting of $K$-valued characteristic functions of clopen (i.e. closed and open) subsets of $X$ and that it is isomorphic to the product of a countable family of Banach spaces with an orthonormal basis.Item On closed subspaces with Schauder bases in non-archimedean Frechet spaces(Royal Netherlands Academy of Arts and Sciences, 2001-12) Śliwa, WiesławThe main purpose of this paper is to prove that a non-archimedean Frechet space of countable type is normable (respectively nuclear; reflexive; a Monte1 space) if and only if any its closed subspace with a Schauder basis is normable (respectively nuclear; reflexive; a Monte1 space). It is also shown that any Schauder basis in a non-normable non-archimedean Frechet space has a block basic sequence whose closed linear span is nuclear. It follows that any non-normable non-archimedean Frechet space contains an infinite-dimensional nuclear closed subspace with a Schauder basis. Moreover, it is proved that a non-archimedean Frechet space E with a Schauder basis contains an infinite-dimensional complemented nuclear closed subspace with a Schauder basis if and only if any Schauder basis in E has a subsequence whose closed linear span is nuclear.Item On linear isometries on non-archimedean powerseries spaces(2012-05) Śliwa, Wiesław; Ziemkowska, AgnieszkaThe non-archimedean power series spaces Ap(a; t) are the most known and important examples of non-archimedean nuclear Frechet spaces. We study when the spaces Ap(a; t) and Aq(b; s) are isometrically isomorphic. Next we determine all linear isometries on the space Ap(a; t) and show that all these maps are surjective.Item On non-archimedean Frechet spaces with nuclear Kothe spaces(The American Mathematical Society, 2010-06) Śliwa, WiesławAssume that K is a complete non-archimedean valued field. We prove that every Frechet-Montel space over K which is not of finite type has a nuclear Kothe quotient.Item On Quotients of Non-Archimedean Kothe Spaces(Canadian Mathematical Society, 2007) Śliwa, WiesławWe show that there exists a non-archimedean Frechet-Montel space W with a basis and with a continuous norm such that any non-archimedean Frechet space of countable type is isomorphic to a quotient of W. We also prove that any non-archimedean nuclear Frechet space is isomorphic to a quotient of some non-archimedean nuclear Frechet space with a basis and with a continuous norm.Item On tame operators between non-archimedean power seris spaces(2012) Śliwa, Wiesław; Ziemkowska, AgnieszkaLet p {1∞}. We show that any continuous linear operator T from A1(a) to Ap(b) is tame i.e. there exists a positive integer c such that supx IITxIIk=/IxIck < ∞ for every k N. Next we prove that a similar result holds for operators from A∞(a) to Ap(b) if and only if the set Mba of all finite limit points of the double sequence (bj/ai/I,j N is bounded. Finally we show that the range of every tame operator from A∞(a) to A∞(b) has a Schauder basis.Item On the quasi-equivalence of orthogonal bases in non-archimedean metrizable locally convex spaces(The Belgian Mathematical Society, 2002) Śliwa, WiesławWe prove that any non-archimedean metrizable locally convex space E with a regular orthogonal basis has the quasi-equivalence property, i.e. any two orthogonal bases in E are quasi-equivalent. In particular, the power series spaces, the most known and important examples of non-archimedean nuclear Frechet spaces, have the quasi-equivalence property. We also show that the Frechet spaces: K^N, c_0xK^N, c_0^N have the quasi-equivalence property.Item On the selection of basic orthogonal sequences in non-archimedean metrizable locally convex spaces(The Belgian Mathematical Society, 2002) Śliwa, WiesławOur main result follows that any infinite-dimensional sub-space F of a non-archimedean metrizable locally convex space E with an or- thogonal basis (e_n) contains a basic orthogonal sequence equivalent to a block basic orthogonal sequence relative to (e_n). Hence any infinite-dimensional non-archimedean metrizable locally convex space F has a basic orthogonal sequence equivalent to a block basic orthogonal sequence relative to an orthogonal basis in c_0^N.Item On the stability of orthogonal bases in non-archimedean metrizable locally convex spaces(The Belgian Mathematical Society, 2001) Śliwa, WiesławIt is proved that an orthogonal basis in a non-archimedean metrizable locally convex space E (in particular, a Schauder basis in a non-archimedean Frechet space E) is stable if and only if there is a continuous norm on E. Many others results are also obtained.Item On Universal Schauder Bases in Non-Archimedean Frechet Spaces(Canadian Mathematical Society, 2004) Śliwa, WiesławWe prove that there exists a non-archimedean Frechet space U with a basis $(u_n)$ such that any basis $(x_n)$ in a non-archimedean Frechet space X is equivalent to a subbasis $(u_k_n)$ of $(u_n)$. Then any non-archimedean Frechet space with a basis is isomorphic to a complemented subspace of U. Next we prove that there is no nuclear non-archimedean Frechet space H with a basis $(h_n)$ such that any basis $(y_n)$ in a nuclear non-archimedean Frechet space Y is equivalent to a subbasis $(h_k_n)$ of $(h_n)$.Item The Invariant Subspace Problem for Non-Archimedean Banach Spaces(Canadian Mathematical Society, 2008) Śliwa, WiesławIt is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A. C. M. van Rooij and W. H. Schikhof in 1992.Item The separable quotient problem and the strongly normal sequences(2012) Śliwa, WiesławWe study the notion of a strongly normal sequence in the dual $E^*$ of a Banach space $E$. In particular, we prove that the following three conditions are equivalent: (1) $E$ has a strongly normal sequence, (2) $(E^*;\sigma (E ^*;E))$ has a Schauder basic sequence, (3) $E$ has an infinite-dimensional separable quotient.