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Katsuhisa Koshino

Publications and source records attributed to Katsuhisa Koshino.

15 recordsLinked to original sources
arXiv2026

Isometries between the unit spheres of spaces of metrics

Given a topological space $Z$, let $PM(Z)$ be the space of bounded continuous pseudometrics on $Z$, which is endowed with the sup-norm, and let $SPM(Z)$ be the unit sphere of $PM(Z)$. In this paper, we shall prove that for all non-degenerate compact metrizable spaces $X$ and $Y$, and for any surjective isometry $T : SPM(X) \to SPM(Y)$, there exists a homeomorphism $ϕ: Y \to X$ such that for any metric $d \in SPM(X)$ and for any pair of points $(x,y) \in Y^2$, $T(d)(x,y) = d(ϕ(x),ϕ(y))$. As a corollary, we can solve a variant of Tingley's problem on spaces of metics.

math.FA↗
arXiv2026

Compactness of averaging operators on non-reflexive Lebesgue spaces

Let $X$ be a Borel and Borel-regular metric measure space whose closed balls are of positive and finite measure. In this paper, we shall give equivalent conditions for averaging operators on non-reflexive Lebesgue spaces $L^1(X)$ and $L^\infty(X)$ on X to be compact, where X has some doubling property and satisfies certain uniform continuity between metric and measure.

math.FA↗
arXiv2025

Isometries between spaces of metrics

Given a metrizable space $Z$, denote by ${\rm PM}(Z)$ the space of continuous bounded pseudometrics on $Z$, and denote by ${\rm AM}(Z)$ the one of continuous bounded admissible metrics on $Z$, the both of which are equipped with the sup-norm $\|\cdot\|$. Let ${\rm Pc}(Z)$ be the subspace of ${\rm AM}(Z)$ satisfying the following: \begin{itemize} \item for every $d \in {\rm Pc}(Z)$, there exists a compact subset $K \subset Z$ such that if $d(x,y) = \|d\|$, then $x, y \in K$. \end{itemize} Moreover, set $${\rm Pp}(Z) = \{d \in {\rm AM}(Z) \mid \text{ there only exists } \{z,w\} \subset Z \text{ such that } d(z,w) = \|d\|\},$$ and let ${\rm M}(Z)$ be ${\rm Pc}(Z)$ or ${\rm Pp}(Z)$. In this paper, we shall prove the Banach-Stone type theorem on spaces of metrics, that is, for metrizable spaces $X$ and $Y$, the following are equivalent: \begin{enumerate} \item $X$ and $Y$ are homeomorphic; \item there exists a surjective isometry $T : {\rm PM}(X) \to {\rm PM}(Y)$ with $T({\rm M}(X)) = {\rm M}(Y)$; \item there exists a surjective isometry $T : {\rm AM}(X) \to {\rm AM}(Y)$ with $T({\rm M}(X)) = {\rm M}(Y)$; \item there exists a surjective isometry $T : {\rm M}(X) \to {\rm M}(Y)$. \end{enumerate} Then for each surjective isometry $T : {\rm PM}(X) \to {\rm PM}(Y)$ with $T({\rm M}(X)) = {\rm M}(Y)$, there is a homeomorphism $ϕ: Y \to X$ such that for any $d \in {\rm PM}(X)$ and for any $x, y \in Y$, $T(d)(x,y) = d(ϕ(x),ϕ(y))$. Except for the case where the cardinality of $X$ or $Y$ is equal to $2$, the homeomorphism $ϕ$ can be chosen uniquely.

math.FA↗
arXiv2024

On isometric universality of spaces of metrics

A metric space $(M, d)$ is said to be universal for a class of metric spaces if all metric spaces in the class can be isometrically embedded into $(M, d)$. In this paper, for a metrizable space $Z$ possessing abundant subspaces, we first prove that the space of bounded metrics on $Z$ is universal for all bounded metric spaces (with restricted cardinality). Next, in contrast, we show that if $Z$ is an infinite discrete space, then the space of metrics on $Z$ is universal for all separable metric spaces. As a corollary of our results, if $Z$ is non-compact, or uncountable and compact, then the space of metrics on $Z$ is universal for all compact metric spaces. In addition, if $Z$ is compact and countable, then there exists a compact metric space that can not be isometrically embedded into the space of metrics on $Z$.

math.MG↗
arXiv2024

On the Borel complexity and the complete metrizability of spaces of metrics

Given a metrizable space $X$, let $AM(X)$ be the space of continuous bounded admissible metrics on $X$, which is endowed with the sup-metric. In this paper, we shall investigate the Borel complexity and the complete metrizability of $AM(X)$ and show that a separable metrizable space $X$ is $σ$-compact if and only if $AM(X)$ is completely metrizable.

math.GN↗
arXiv2024

Compactness of averaging operators on Banach function spaces

Let $X$ be a Borel metric measure space such that each closed ball is of positive and finite measure. In this paper, we give a sufficient and necessary condition for averaging operators on a Banach function space $E(X)$ on $X$ to be compact. As a corollary, we show that the averaging operators on the Lorentz space $L^{p,q}(X)$ of $X$ is compact if and only if $X$ is bounded, in the case where $X$ is a doubling and Borel-regular metric measure space with some continuity between metric and measure.

math.FA↗
arXiv2022

Characterizing compact sets in $L^p$-spaces and its application

In this paper, we give a characterization of compact sets in $L^p$-spaces on metric measure spaces, which is a generalization of the Kolmogorov-Riesz theorem. Using the criterion, we investigate the topological type of the space consisting of lipschitz maps with bounded supports.

math.GN↗
arXiv2022

The topological classification of spaces of metrics with the uniform convergence topology

For a metrizable space $X$ of density $κ$, let $PM(X)$ be the space of continuous bounded pseudometrics on $X$ endowed with the uniform convergence topology. In this paper, its topology shall be classified as follows: (i) If $X$ is finite, then $PM(X)$ is homeomorphic to $\{0\}$ when $X$ is a singleton, and then $PM(X)$ is homeomorphic to $[0,1]^{κ(κ- 1)/2 - 1} \times [0,1)$ when $κ> 1$; (ii) If $X$ is infinite and generalized compact, then $PM(X)$ is homeomorphic to the Hilbert space $\ell_2(2^{< κ})$ of density $2^{< κ}$; (iii) If $X$ is not generalized compact, then $PM(X)$ is homeomorphic to the Hilbert space $\ell_2(2^κ)$ of density $2^κ$. Furthermore, letting $M(X)$ and $AM(X)$ be the spaces of continuous bounded metrics and bounded admissible metrics on $X$ with the subspace topology of $PM(X)$ respectively, we will recognize their topological types as follows: (iv) If $X$ is infinite and compact, then $M(X) (= AM(X))$ is homeomorphic to the separable Hilbert space $\ell_2$; (v) In the case that $X$ is not compact, $M(X)$ is homeomorphic to the Hilbert space $\ell_2(2^{\aleph_0})$ if $X$ is $σ$-compact, and moreover $AM(X)$ is also homeomorphic to the Hilbert space $\ell_2(2^{\aleph_0})$ if $X$ is separable locally compact.

math.GN↗
arXiv2022

The topological type of spaces consisting of certain metrics on locally compact metrizable spaces with the compact-open topology

For a separable locally compact but not compact metrizable space $X$, let $αX = X \cup \{x_\infty\}$ be the one-point compactification with the point at infinity $x_\infty$. We denote by $EM(X)$ the space consisting of admissible metrics on $X$, which can be extended to an admissible metric on $αX$, endowed with the compact-open topology. Let $ \mathbf{c}_0 \subset (0,1)^\mathbb{N}$ be the space of sequences converging to $0$. In this paper, we shall show that if $X$ is separable, locally connected and locally compact but not compact, and there exists a sequence $\{C_i\}$ of connected sets in $X$ such that for all positive integers $i, j \in \mathbb{N}$ with $|i - j| \leq 1$, $C_i \cap C_j \neq \emptyset$, and for each compact set $K \subset X$, there is a positive integer $i(K) \in \mathbb{N}$ such that for any $i \geq i(K)$, $C_i \subset X \setminus K$, then $EM(X)$ is homeomorphic to $ \mathbf{c}_0$.

math.GN↗
arXiv2015

On a hyperspace of compact subsets which is homeomorphic to a non-separable Hilbert space

Let $X$ be a metrizable space and ${\rm Comp}(X)$ be the hyperspace consisting of non-empty compact subsets of $X$ endowed with the Vietoris topology. In this paper, we give a necessary and sufficient condition on $X$ for ${\rm Comp}(X)$ to be homeomorphic to a non-separable Hilbert space. Moreover, we consider the topological structure of pair $({\rm Comp}(\overline{X}),{\rm Fin}(X))$ of hyperspaces of $X$ and its completion $\overline{X}$, where ${\rm Fin}(X)$ is the hyperspace of non-empty finite sets in $X$.

math.GN↗