# A Wave Model of Metric Spaces

M. I. Belishev, S. A. Simonov

Research output

### Abstract

Let Omega be a metric space. By A(t) we denote the metric neighborhood of radius t of a set A subset of Omega and by D, the lattice of open sets in Omega with partial order subset of and order convergence. The lattice of D-valued functions of t is an element of (0, infinity) with pointwise partial order and convergence contains the family ID = {A()| A(t) = A(t), A is an element of D}. Let Omega be the set of atoms of the order closure ID. We describe a class of spaces for which the set Omega equipped with an appropriate metric is isometric to the original space Omega.The space Omega 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.

Original language English 79-85 7 Functional Analysis and its Applications 53 2 https://doi.org/10.1134/S0016266319020011 Published - Apr 2019

### Cite this

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.
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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, 04.2019, p. 79-85.

Research output

TY - JOUR

T1 - A Wave Model of Metric Spaces

AU - Belishev, M. I.

AU - Simonov, S. A.

PY - 2019/4

Y1 - 2019/4

N2 - Let Omega be a metric space. By A(t) we denote the metric neighborhood of radius t of a set A subset of Omega and by D, the lattice of open sets in Omega with partial order subset of and order convergence. The lattice of D-valued functions of t is an element of (0, infinity) with pointwise partial order and convergence contains the family ID = {A()| A(t) = A(t), A is an element of D}. Let Omega be the set of atoms of the order closure ID. We describe a class of spaces for which the set Omega equipped with an appropriate metric is isometric to the original space Omega.The space Omega 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 Omega be a metric space. By A(t) we denote the metric neighborhood of radius t of a set A subset of Omega and by D, the lattice of open sets in Omega with partial order subset of and order convergence. The lattice of D-valued functions of t is an element of (0, infinity) with pointwise partial order and convergence contains the family ID = {A()| A(t) = A(t), A is an element of D}. Let Omega be the set of atoms of the order closure ID. We describe a class of spaces for which the set Omega equipped with an appropriate metric is isometric to the original space Omega.The space Omega 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

U2 - 10.1134/S0016266319020011

DO - 10.1134/S0016266319020011

M3 - статья

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SP - 79

EP - 85

JO - Functional Analysis and its Applications

JF - Functional Analysis and its Applications

SN - 0016-2663

IS - 2

ER -