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A Wave Model of Metric Spaces. / Belishev, M. I.; Simonov, S. A.

In: Functional Analysis and its Applications, Vol. 53, No. 2, 01.04.2019, p. 79-85.

Research output: Contribution to journal › Article › peer-review

Harvard

Belishev, MI & Simonov, SA 2019, 'A Wave Model of Metric Spaces', Functional Analysis and its Applications, vol. 53, no. 2, pp. 79-85. https://doi.org/10.1134/S0016266319020011

APA

Belishev, M. I., & Simonov, S. A. (2019). A Wave Model of Metric Spaces. Functional Analysis and its Applications, 53(2), 79-85. https://doi.org/10.1134/S0016266319020011

Vancouver

Belishev MI, Simonov SA. A Wave Model of Metric Spaces. Functional Analysis and its Applications. 2019 Apr 1;53(2):79-85. https://doi.org/10.1134/S0016266319020011

Author

Belishev, M. I. ; Simonov, S. A. / A Wave Model of Metric Spaces. In: Functional Analysis and its Applications. 2019 ; Vol. 53, No. 2. pp. 79-85.

BibTeX

@article{9c8b6caeb4b94cdb8d82c3e22c8831da,
title = "A Wave Model of Metric Spaces",
abstract = "Let Ω be a metric space. By A t we denote the metric neighborhood of radius t of a set A ⊂ Ω and by D, the lattice of open sets in Ω with partial order ⊆ and order convergence. The lattice of D-valued functions of t ∈ (0, ∞) with pointwise partial order and convergence contains the family ID = {A(·)| A(t) = A t, A ∈ D}. Let{\~ }Ω be the set of atoms of the order closure ID¯. We describe a class of spaces for which the set{\~ }Ω equipped with an appropriate metric is isometric to the original space Ω. The space{\~ }Ω is the key element of the construction of the wave spectrum of a lower bounded symmetric operator, which was introduced in a work of one of the authors. In that work, a program for constructing a functional model of operators of the aforementioned class was laid down. The present paper is a step in the realization of this program. ",
keywords = "metric space, lattice of open subsets, isotony, lattice-valued function, atom, wave model",
author = "Belishev, {M. I.} and Simonov, {S. A.}",
year = "2019",
month = apr,
day = "1",
doi = "10.1134/S0016266319020011",
language = "Английский",
volume = "53",
pages = "79--85",
journal = "Functional Analysis and its Applications",
issn = "0016-2663",
publisher = "Springer Nature",
number = "2",

}

RIS

TY - JOUR

T1 - A Wave Model of Metric Spaces

AU - Belishev, M. I.

AU - Simonov, S. A.

PY - 2019/4/1

Y1 - 2019/4/1

N2 - Let Ω be a metric space. By A t we denote the metric neighborhood of radius t of a set A ⊂ Ω and by D, the lattice of open sets in Ω with partial order ⊆ and order convergence. The lattice of D-valued functions of t ∈ (0, ∞) with pointwise partial order and convergence contains the family ID = {A(·)| A(t) = A t, A ∈ D}. Let ̃Ω be the set of atoms of the order closure ID¯. We describe a class of spaces for which the set ̃Ω equipped with an appropriate metric is isometric to the original space Ω. The space ̃Ω is the key element of the construction of the wave spectrum of a lower bounded symmetric operator, which was introduced in a work of one of the authors. In that work, a program for constructing a functional model of operators of the aforementioned class was laid down. The present paper is a step in the realization of this program.

AB - Let Ω be a metric space. By A t we denote the metric neighborhood of radius t of a set A ⊂ Ω and by D, the lattice of open sets in Ω with partial order ⊆ and order convergence. The lattice of D-valued functions of t ∈ (0, ∞) with pointwise partial order and convergence contains the family ID = {A(·)| A(t) = A t, A ∈ D}. Let ̃Ω be the set of atoms of the order closure ID¯. We describe a class of spaces for which the set ̃Ω equipped with an appropriate metric is isometric to the original space Ω. The space ̃Ω is the key element of the construction of the wave spectrum of a lower bounded symmetric operator, which was introduced in a work of one of the authors. In that work, a program for constructing a functional model of operators of the aforementioned class was laid down. The present paper is a step in the realization of this program.

KW - metric space

KW - lattice of open subsets

KW - isotony

KW - lattice-valued function

KW - atom

KW - wave model

UR - http://www.scopus.com/inward/record.url?scp=85069466523&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/wave-model-metric-spaces

U2 - 10.1134/S0016266319020011

DO - 10.1134/S0016266319020011

M3 - статья

VL - 53

SP - 79

EP - 85

JO - Functional Analysis and its Applications

JF - Functional Analysis and its Applications

SN - 0016-2663

IS - 2

ER -

ID: 47875495