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1
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Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces

100%
Niemiec P.
Instytut Matematyczny Polskiej Akademii Nauk
EN
An ideal of N-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for N-tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class $CDD_{N}$ of all unitary equivalence classes of such N-tuples are established and certain ideals in $CDD_{N}$ are distinguished. It is proved that infinite operations in $CDD_{N}$ may be reconstructed from the direct sum operation of a pair. Prime decomposition in $CDD_{N}$ is proposed and its uniqueness (in a certain sense) is established. The issue of classification of ideals in $CDD_{N}$ (up to isomorphism) is discussed. A model for $CDD_{N}$ is described and its concrete realization is presented. A new partial order of N-tuples of operators is introduced and its fundamental properties are established. The importance of unitary disjointness of N-tuples and the way how it 'tidies up' the structure of $CDD_{N}$ are emphasized.
2
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Extending Maps in Hilbert Manifolds

95%
Niemiec P.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2012
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tom 60
|
nr 3
295-306
EN
Certain results on extending maps taking values in Hilbert manifolds by maps which are close to being embeddings are presented. Sufficient conditions on a map under which it is extendable by an embedding are given. In particular, it is shown that if X is a completely metrizable space of topological weight not greater than α ≥ ℵ₀, A is a closed set in X and f: X → M is a map into a manifold M modelled on a Hilbert space of dimension α such that $f(X∖A) ∩ \overline{f(∂A)} = ∅$, then for every open cover 𝒰 of M there is a map g: X → M which is 𝒰-close to f (on X), coincides with f on A and is an embedding of X∖A into M. If, in addition, X∖A is a connected manifold modelled on the same Hilbert space as M, and $\overline{f(∂A)}$ is a Z-set in M, then the above map g may be chosen so that $g|_{X∖A}$ be an open embedding.
3
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A problem with almost everywhere equality

95%
Niemiec P.
Annales Polonici Mathematici
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2012
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tom 104
|
nr 1
105-108
EN
A topological space Y is said to have (AEEP) if the following condition is satisfied: Whenever (X,𝔐) is a measurable space and f,g: X → Y are two measurable functions, then the set Δ(f,g) = {x ∈ X: f(x) = g(x)} is a member of 𝔐. It is shown that a metrizable space Y has (AEEP) iff the cardinality of Y is not greater than $2^{ℵ₀}$.
4
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Separate and joint similarity to families of normal operators

95%
Niemiec P.
Studia Mathematica
|
2002
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tom 149
|
nr 1
39-62
EN
Sets of bounded linear operators 𝓢,𝓣 ⊂ ℬ(H) (ℋ is a Hilbert space) are similar if there exists an invertible (in ℬ(H)) operator G such that $G^{-1}·𝓢·G = 𝓣$. A bounded operator is scalar if it is similar to a normal operator. 𝓢 is jointly scalar if there exists a set 𝓝 ⊂ ℬ(H) of normal operators such that 𝓢 and 𝓝 are similar. 𝓢 is separately scalar if all its elements are scalar. Some necessary and sufficient conditions for joint scalarity of a separately scalar abelian set of Hilbert space operators are presented (Theorems 3.7, 4.4 and 4.6). Continuous algebra homomorphisms between the algebra of all complex-valued continuous functions on a compact Hausdorff space and the algebra of all bounded operators in a Hilbert space are studied.
5
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Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps

95%
Niemiec P.
Studia Mathematica
|
2009
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tom 192
|
nr 2
97-110
EN
It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $CFL(𝕌_{r})$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space $𝕌_{r}$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $ LIP(𝕌_{r})$ of equivalence classes of all real-valued Lipschitz maps on $𝕌_{r}$. The space of all signed (real-valued) Borel measures on $𝕌_{r}$ is isometrically embedded in the dual space of $CFL(𝕌_{r})$ and it is shown that the image of the embedding is a proper weak* dense subspace of $CFL(𝕌_{r})*$. Some special properties of the space $CFL(𝕌_{r})$ are established.
6
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Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

95%
Niemiec P.
Studia Mathematica
|
2012
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tom 208
|
nr 1
77-85
EN
For a linear operator T in a Banach space let $σ_{p}(T)$ denote the point spectrum of T, let $σ_{p,n}(T)$ for finite n > 0 be the set of all $λ ∈ σ_{p}(T)$ such that dim ker(T - λ) = n and let $σ_{p,∞}(T)$ be the set of all $λ ∈ σ_{p}(T)$ for which ker(T - λ) is infinite-dimensional. It is shown that $σ_{p}(T)$ is $ℱ_{σ}$, $σ_{p,∞}(T)$ is $ℱ_{σδ}$ and for each finite n the set $σ_{p,n}(T)$ is the intersection of an $ℱ_{σ}$ set and a $𝒢_{δ}$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space 𝓗 a more detailed decomposition of the spectra is obtained and the algebra of all bounded linear operators on 𝓗 is decomposed into Borel parts. In particular, it is shown that the set of all closed range operators on 𝓗 is Borel.
7
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Spaces of measurable functions

95%
Niemiec P.
Open Mathematics
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2013
|
tom 11
|
nr 7
1304-1316
EN
For a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.
8
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Urysohn universal spaces as metric groups of exponent 2

95%
Niemiec P.
Fundamenta Mathematicae
|
2009
|
tom 204
|
nr 1
1-6
EN
The aim of the paper is to prove that the bounded and unbounded Urysohn universal spaces have unique (up to isometric isomorphism) structures of metric groups of exponent 2. An algebraic-geometric characterization of Boolean Urysohn spaces (i.e. metric groups of exponent 2 which are metrically Urysohn spaces) is given.
9
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Functor of extension in Hilbert cube and Hilbert space

95%
Niemiec P.
Open Mathematics
|
2014
|
tom 12
|
nr 6
887-895
EN
It is shown that if Ω = Q or Ω = ℓ 2, then there exists a functor of extension of maps between Z-sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365–383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity, Lipschitz property, nonexpansiveness of maps in appropriate metrics.
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