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On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

100%
Michalak A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2014
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tom 62
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nr 1
61-76
EN
We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f:[0,1]^m → X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f:[0,1]^m → X$ with respect to any norming subset there exists a separately increasing function $g:[0,1]^m → ℝ$ such that the sets of points of discontinuity of f and g coincide.
2
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On some properties of quotients of homogeneous C(K) spaces

100%
Michalak A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2016
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tom 36
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nr 1
33-43
EN
We say that an infinite, zero dimensional, compact Hausdorff space K has property (*) if for every nonempty open subset U of K there exists an open and closed subset V of U which is homeomorphic to K. We show that if K is a compact Hausdorff space with property (*) and X is a Banach space which contains a subspace isomorphic to the space C(K) of all scalar (real or complex) continuous functions on K and Y is a closed linear subspace of X which does not contain any subspace isomorphic to the space C([0,1]), then the quotient space X/Y contains a subspace isomorphic to the space C(K).
3
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Notes on binary trees of elements in \(C(K)\) spaces with an application to a proof of a theorem of H. P. Rosenthal

100%
Michalak A.
Commentationes Mathematicae
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2006
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tom 46
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nr 2
EN
A Banach space \(X\) contains an isomorphic copy of \(C([0, 1])\), if it contains a binary tree \((e_n)\) with the following properties (1) \(e_n = e_{2n} + e_{2n+1}\) and (2) \(c \max_{2^n\leq k\tl 2^{n+1}} |a_k| \leq \|\sum_{k=2^n}^{2^{n+1}-1} a_k e_k \leq C\max_{2^n\leq k\lt 2^{n+1}} |a_k|\) for some constants \(0\lt c \leq C\) and every \(n\) and any scalars \(a_{2^n},\dots, a_{2^{n+1}-1}\). We present a proof of the following generalization of a Rosenthal result: if \(E\) is a closed subspace of a separable \(C(K)\) space with separable annihilator and\(S\colon E \to X\) is a continuous linear operator such that \(S^{∗}\) has nonseparable range, then there exists a subspace \(Y\) of \(E\) isomorphic to \(C([0, 1])\) such that \(S|_Y\) is an isomorphism, based on the fact.
4
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On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity

100%
Michalak A.
Studia Mathematica
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2003
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tom 155
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nr 2
171-182
EN
We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\overline{lin{f([0,1])}}$ contains an order isomorphic copy of D(0,1), (2) $\overline{lin{f(Q)}}$ contains an isomorphic copy of C([0,1]), (3) $\overline{lin{f([0,1])}}/\overline{lin{f(Q)}}$ contains an isomorphic copy of c₀(Γ) for some uncountable set Γ, (4) if I is an isomorphic embedding of $\overline{lin{f([0,1])}}$ into a Banach space Z, then no separable complemented subspace of Z contains $I(\overline{lin{f(Q)}})$.
5
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The Banach space \(D(0, 1)\) is primary

100%
Michalak A.
Commentationes Mathematicae
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2005
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tom 45
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nr 1
EN
We show that the Banach space \(D(0, 1)\) of all scalar (real or complex) functions on \([0, 1)\) that are right continuous at each point of \([0, 1)\) with left-hand limit at each point of \((0, 1]\) equipped with the uniform convergence topology is primary.
6
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On constructions of isometric embeddings of nonseparable \(L^p\) spaces, \(0 \lt p \leq 2\)

64%
Grala-Michalak J., Michalak A.
Commentationes Mathematicae
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2008
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tom 48
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nr 2
EN
Let \(J\) be an infinite set. Let \(X\) be a real or complex \(\sigma\)-order continuous rearrangement invariant quasi-Banach function space over \((\{0, 1\}^J,\ \mathcal{B}^J,\ \lambda_J)\), the product of \(J\) copies of the measure space \((\{0, 1\},\ 2^{0,1},\ \frac{1}{2} \delta_0 + \frac{1}{2}\delta_1)\). We show that if \(0 \lt p \lt 2\) and \(X\) contains a function \(f\) with the decreasing rearrangement \(f^∗\) such that \(f^∗(t) \gt t^{-\frac{1}{p}}\) for every \(t\in (0, 1)\), then it contains an isometric copy of the Lebesgue space \(L^p (\lambda_J)\). Moreover, if \(X\) contains a function \(f\) such that \(f^∗(t) \gt \sqrt{|\text{ln}(t)|}\) for every \(t\in (0, 1)\), then it contains an isometric copy of the Lebesgue space \(L^2(\lambda_J)\).
7
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Generalized Helly spaces, continuity of monotone functions, and metrizing maps

64%
Drewnowski L., Michalak A.
Fundamenta Mathematicae
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2008
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tom 200
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nr 2
161-184
EN
Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes from the fact that there is a natural one-to-one correspondence between increasing functions f: [0,1] → C(K,E) (with countably many discontinuities) and continuous maps F: K → H(E) (with metrizable ranges). It leads to the investigation of general continuous metrizing maps (those with metrizable ranges), and especially of the so called separately metrizing maps, and the results obtained are then used to derive some permanence properties of the class of spaces C(K,E) with property (λ). For instance, it is shown that if K is the product of compact spaces $K_{j}$ (j ∈ J) such that each of the spaces $C(K_{j},E)$ has property (λ), so does C(K,E); and, for any compact space K, if both C(K) and a Banach lattice E have property (λ), so does C(K,E).
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On constructions of isometric copies of \(L^p (0, 1)\) spaces \((0 \lt p \leq 2)\) by stochastic \(p\)-stable processes

64%
Grala-Michalak J., Michalak A.
Commentationes Mathematicae
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2008
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tom 48
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nr 1
EN
Let \(S^p = \{S_t^p : t = \frac{k}{2^n},\ 0 \leq k \leq 2^n,\ n \in\mathbb{N}\}\) be a stochastic process on a probability space \((\Omega, \Sigma, P)\) with independent and time homogeneous increments such that \(S_t^p - S_u^p\) is identically distributed as \((t- u)^{1/p} Z_p\) for each \(0 \leq u \lt t \leq 1\) where \(Z_p\) is a given symmetric \(p\)-stable distribution. We show that the closed linear hull of \(S^p\) forms an isometric copy of the real Lebesgue space \(L^p (0, 1)\) in any quasi-Banach space \(X\) consisting of \(P\)-a.e. equivalence classes of \(\Sigma\)-measurable real functions on \(\Omega\) equipped with a rearrangement invariant quasi-norm which contains \(S^p\) as a subset. It is possible to construct processes \(S^p\) for \(0 \lt p \leq 2\) on \([0, 1]\) with the Lebesgue measure. We show also a complex version of the result.
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