arXiv subjects

Piotr Niemiec

Publications and source records attributed to Piotr Niemiec.

29 records · Page 2Linked to original sources

arXiv2011

A note on ANR's

It is shown that if for a complete metric space $(X,d)$ there is a constant $ε> 0$ such that the intersection $\bigcap_{j=1}^n B_d(x_j,r_j)$ of open balls is nonempty for every finite system $x_1,...,x_n \in X$ of centers and a corresponding system of radii $r_1,...,r_n > 0$ such that $d(x_j,x_k) \leqsl ε$ and $d(x_j,x_k) < r_j + r_k$ ($j,k = 1,...,n$), then $X$ is an ANR; and if in the above one may put $ε= \infty$, the space $X$ is an AR. A certain criterion for an incomplete metric space to be an A(N)R is presented.

math.GN↗

arXiv2011

Normal systems over ANR's, rigid embeddings and nonseparable absorbing sets

Most of results of Bestvina and Mogilski [\textit{Characterizing certain incomplete infinite-dimensional absolute retracts}, Michigan Math. J. \textbf{33} (1986), 291--313] on strong $Z$-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR $X$ is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and $w(U) = w(X)$ (where `$w$' is the topological weight) for each open nonempty subset $U$ of $X$,then $X$ itself is homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever $X$ is an AR, its weak product $W(X,*) = \{(x_n)_{n=1}^{\infty} \in X^ω:\ x_n = * \textup{for almost all} n\}$ is homeomorphic to a pre-Hilbert space $E$ with $E \cong ΣE$. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.

math.GN↗

arXiv2011

Spaces of measurable functions

For a metrizable space $X$ and a finite measure space $(Ω,\mathfrak{M},μ)$ let $M_μ(X)$ and $M^f_μ(X)$ be the spaces 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 and finite, respectively, equipped with the topology of convergence in measure. The main aim of the paper is to prove the following result: if $μ$ is (nonzero and) nonatomic and $X$ has more than one point, then the space $M_μ(X)$ is a noncompact absolute retract and $M^f_μ(A)$ is homotopy dense in $M_μ(X)$ for each dense subset $A$ of $X$. In particular, if $X$ is completely metrizable, then $M_μ(X)$ is homeomorphic to an infinite-dimensional Hilbert space.

math.GN↗

arXiv2011

Functor of continuation in Hilbert cube and Hilbert space

A $Z$-set in a metric space $X$ is a closed subset $K$ of $X$ such that each map of the Hilbert cube $Q$ into $X$ can uniformly be approximated by maps of $Q$ into $X \setminus K$. The aim of the paper is to show that there exists a functor of extension of maps between $Z$-sets of $Q$ [or $l_2$] to maps acting on the whole space $Q$ [resp. $l_2$]. Special properties of the functor are proved.

math.GN↗

arXiv2011

Problem with almost everywhere equality

A topological space $Y$ is said to have (AEEP) if the following condition is fulfilled. Whenever $(X,\mathfrak{M})$ is a measurable space and $f, g: X \to Y$ are two measurable functions, then the set $Δ(f,g) = \{x \in X:\ f(x) = g(x)\}$ is a member of $\mathfrak{M}$. It is shown that a metrizable space $Y$ has (AEEP) iff the cardinality of $Y$ is no greater than $2^{\aleph_0}$.

math.GN↗

arXiv2011

Extending maps by injective $σ$-$Z$-maps in Hilbert manifolds

The aim of the paper is to prove that if $M$ is a metrizable manifold modelled on a Hilbert space of dimension $α\geq \aleph_0$ and $F$ is its $σ$-$Z$-set, then for every completely metrizable space $X$ of weight no greater than $α$ and its closed subset $A$, for any map $f: X \to M$, each open cover $\mathcal{U}$ of $M$ and a sequnce $(A_n)_n$ of closed subsets of $X$ disjoint from $A$ there is a map $g: X \to M$ $\mathcal{U}$-homotopic to $f$ such that $g\bigr|_A = f\bigr|_A$, $g\bigr|_{A_n}$ is a closed embedding for each $n$ and $g(X \setminus A)$ is a $σ$-$Z$-set in $M$ disjoint from $F$. It is shown that if $f(\partial A)$ is contained in a locally closed $σ$-$Z$-set in $M$ or $f(X \setminus A) \cap \bar{f(\partial A)} = \empty$, the map $g$ may be taken so that $g\bigr|_{X \setminus A}$ be an embedding. If, in addition, $X \setminus A$ is a connected manifold modelled on the same Hilbert space as $M$ and $\bar{f(\partial A)}$ is a $Z$-set in $M$, then there is a $\mathcal{U}$-homotopic to $f$ map $h: X \to M$ such that $h\bigr|_A = f\bigr|_A$ and $h\bigr|_{X \setminus A}$ is an open embedding.

math.GN↗

arXiv2011

Borel structure of the spectrum of a closed operator

For a linear operator $T$ in a Banach space let $σ_p(T)$ denote the point spectrum of $T$, $σ_{p[n]}(T)$ for finite $n > 0$ be the set of all $λ\in σ_p(T)$ such that $\dim \ker (T - λ) = n$ and let $σ_{p[\infty]}(T)$ be the set of all $λ\in σ_p(T)$ for which $\ker (T - λ)$ is infinite-dimensional. It is shown that $σ_p(T)$ is $\mathcal{F}_σ$, $σ_{p[\infty]}(T)$ is $\mathcal{F}_{σδ}$ and for each finite $n$ the set $σ_{p[n]}(T)$ is the intersection of an $\mathcal{F}_σ$ and a $\mathcal{G}_δ$ 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 $\mathcal{H}$ more detailed decomposition of the spectra is done and the algebra of all bounded linear operators on $\mathcal{H}$ is decomposed into Borel parts. In particular, it is shown that the set of all closed range operators on $\mathcal{H}$ is Borel.

math.FA↗

arXiv2011

Norm closures of orbits of bounded operators

To every bounded linear operator $A$ between Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$ three cardinals $ι_r(A)$, $ι_i(A)$ and $ι_f(A)$ and a binary number $ι_b(A)$ are assigned in terms of which the descriptions of the norm closures of the orbits $\{G A L^{-1}:\ L \in \mathcal{G}_1,\ G \in \mathcal{G}_2\}$ are given for $\mathcal{G}_1$ and $\mathcal{G}_2$ (chosen independently) being the trivial group, the unitary group or the group of all invertible operators on $\mathcal{H}$ and $\mathcal{K}$, respectively.

math.FA↗

arXiv2011

Central points and measures and dense subsets of compact metric spaces

For every nonempty compact convex subset $K$ of a normed linear space a (unique) point $c_K \in K$, called the generalized Chebyshev center, is distinguished. It is shown that $c_K$ is a common fixed point for the isometry group of the metric space $K$. With use of the generalized Chebyshev centers, the central measure $μ_X$ of an arbitrary compact metric space $X$ is defined. For a large class of compact metric spaces, including the interval $[0,1]$ and all compact metric groups, another `central' measure is distinguished, which turns out to coincide with the Lebesgue measure and the Haar one for the interval and a compact metric group, respectively. An idea of distinguishing infinitely many points forming a dense subset of an arbitrary compact metric space is also presented.

math.FA↗

arXiv2011

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

An \textit{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. \textit{Prime decomposition} in $CDD_N$ is proposed and its (in a sense) uniqueness 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. Extremal importance of unitary disjointness of $N$-tuples and the way how it `tidies up' the structure of $CDD_N$ are emphasized.

math.OA↗

arXiv2011

Universal valued Abelian groups

The counterparts of the Urysohn universal space in category of metric spaces and the Gurarii space in category of Banach spaces are constructed for separable valued Abelian groups of fixed (finite) exponents (and for valued groups of similar type) and their uniqueness is established. Geometry of these groups, denoted by G_r(N), is investigated and it is shown that each of G_r(N)'s is homeomorphic to the Hilbert space l^2. Those of G_r(N)'s which are Urysohn as metric spaces are recognized. `Linear-like' structures on G_r(N) are studied and it is proved that every separable metrizable topological vector space may be enlarged to G_r(0) with a `linear-like' structure which extends the linear structure of the given space.

math.GN↗