Operator-valued
charges on finite sets

T.S.~Kudryk; V.E.~Lyantse

Investigations on applications of nonstandard methods to measure theory and operator theory (see [1] and [2]) are extended to operator-valued measures and charges. Spaces of operator-valued charges on the algebra $2^{\Bbb T}$ and corresponding charges on the algebra $2^{\text{\bf T}}$, where ${\text{\bf T}}$ is the standard filling of a finite (in the sense of IST) set ${\Bbb T}$, are introduced and examined. The notions of weak, strong and uniform nearstandardness are defined for such charges (including the technics of equipment of Hilbert space). It is proved that decomposition of the unity ${\frak N}$ of discrete differentiation operator $D$ is related to the decomposition of the unity ${\frak M}$ of usual differentiation operator ${\text{\bf D}}$ in the same way as the operators $D$ and ${\text{\bf D}}$: ${^{\circ}{\frak N}}={\frak M}$.

1. S. Albeverio, J. Fenstad, T. Hoegh-Krohn, T. Lindström, Nonstandard methods in stochastic analysis and mathematical physics. — New York: Academic Press, 1986.

2. N.J. Cutland, Nonstandard measure theory and its applications // Bull. Amer. Math. Soc. 1983, V.15, №6. P.529–589.

3. T.S. Kudryk, V.E. Lyantse, G.I. Chuiko, Nearstandardness on finite set // Математичні студії, 2, 1993, C.25–34.

4. T.S. Kudryk, V.E. Lyantse, G.I. Chuiko, Nearstandard operators // Математичні студії, 3, 1994, C.29–40.

5. Ю.М. Березанский, Г.Ф. Ус, З.Г. Шефтель, Функциональный анализ. — Киев.: Вища школа. 1990.

6. Т.С. Кудрик, В.Э. Лянце, Г.И. Чуйко. Нестандартные методы в теории операторов. 4,5. Околостандартные операторы / Львов. гос.ун-т. 1992. 19с. Деп. в ГНТБ Украины 28.06.93г., №1248-Ук93.

7. Т.С.Кудрик, В.Э.Лянце,Г.И.Чуйко. Нестандартные методы в теории операторов. 6. Дискретное преобразование Фурье / Львов. гос.ун-т. 1992. 15с. Деп. в ГНТБ Украины 28.10.93г., №2138–Ук93.

8. Т.С.Кудрик, В.Э.Лянце,Г.И.Чуйко. Нестандартные методы в теории операторов. 7. Околостандартность и оснащение / Львов. гос.ун-т. 1994. 11с. Деп. в ГНТБ Украины 25.08.94г., №1788-Ук94.

	Operator-valued charges on finite sets
Author	T.S.~Kudryk, V.E.~Lyantse Ukraine, 290602, Lviv, 1, Universytetska str, Lviv State University,Department of Mathematics
Abstract	Investigations on applications of nonstandard methods to measure theory and operator theory (see [1] and [2]) are extended to operator-valued measures and charges. Spaces of operator-valued charges on the algebra $2^{\Bbb T}$ and corresponding charges on the algebra $2^{\text{\bf T}}$, where ${\text{\bf T}}$ is the standard filling of a finite (in the sense of IST) set ${\Bbb T}$, are introduced and examined. The notions of weak, strong and uniform nearstandardness are defined for such charges (including the technics of equipment of Hilbert space). It is proved that decomposition of the unity ${\frak N}$ of discrete differentiation operator $D$ is related to the decomposition of the unity ${\frak M}$ of usual differentiation operator ${\text{\bf D}}$ in the same way as the operators $D$ and ${\text{\bf D}}$: ${^{\circ}{\frak N}}={\frak M}$.
Keywords	nonstandard methods, measure theory, operator theory, operator-valued measures, operator-valued charges, nearstandardness, Hilbert space
DOI	doi:10.30970/ms.7.2.145-156
Reference	1. S. Albeverio, J. Fenstad, T. Hoegh-Krohn, T. Lindström, Nonstandard methods in stochastic analysis and mathematical physics. — New York: Academic Press, 1986. 2. N.J. Cutland, Nonstandard measure theory and its applications // Bull. Amer. Math. Soc. 1983, V.15, №6. P.529–589. 3. T.S. Kudryk, V.E. Lyantse, G.I. Chuiko, Nearstandardness on finite set // Математичні студії, 2, 1993, C.25–34. 4. T.S. Kudryk, V.E. Lyantse, G.I. Chuiko, Nearstandard operators // Математичні студії, 3, 1994, C.29–40. 5. Ю.М. Березанский, Г.Ф. Ус, З.Г. Шефтель, Функциональный анализ. — Киев.: Вища школа. 1990. 6. Т.С. Кудрик, В.Э. Лянце, Г.И. Чуйко. Нестандартные методы в теории операторов. 4,5. Околостандартные операторы / Львов. гос.ун-т. 1992. 19с. Деп. в ГНТБ Украины 28.06.93г., №1248-Ук93. 7. Т.С.Кудрик, В.Э.Лянце,Г.И.Чуйко. Нестандартные методы в теории операторов. 6. Дискретное преобразование Фурье / Львов. гос.ун-т. 1992. 15с. Деп. в ГНТБ Украины 28.10.93г., №2138–Ук93. 8. Т.С.Кудрик, В.Э.Лянце,Г.И.Чуйко. Нестандартные методы в теории операторов. 7. Околостандартность и оснащение / Львов. гос.ун-т. 1994. 11с. Деп. в ГНТБ Украины 25.08.94г., №1788-Ук94.
Pages	145-156
Volume	7
Issue	2
Year	1997
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Operator-valued charges on finite sets