Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals
Let <i>N</i> be the number operator in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>...
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MDPI AG
2022-07-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/15/2635 |
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| author | Jing Zhang Lixia Zhang Caishi Wang |
| author_facet | Jing Zhang Lixia Zhang Caishi Wang |
| author_sort | Jing Zhang |
| collection | DOAJ |
| description | Let <i>N</i> be the number operator in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of <i>N</i> from a probabilistic perspective. We first construct a nuclear space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>, which is also a dense linear subspace of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>, and then by taking its dual <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">G</mi><mo>*</mo></msup></semantics></math></inline-formula>, we obtain a real Gel’fand triple <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">G</mi><mo>⊂</mo><mi mathvariant="script">H</mi><mo>⊂</mo><msup><mi mathvariant="script">G</mi><mo>*</mo></msup></mrow></semantics></math></inline-formula>. Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>N</mi></msub></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">G</mi><mo>*</mo></msup></semantics></math></inline-formula> such that its covariance operator coincides with <i>N</i>. We examine the properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>N</mi></msub></semantics></math></inline-formula>, and, among others, we show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>N</mi></msub></semantics></math></inline-formula> can be represented as a convolution of a sequence of Borel probability measures on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">G</mi><mo>*</mo></msup></semantics></math></inline-formula>. Some other results are also obtained. |
| first_indexed | 2024-03-09T05:13:08Z |
| format | Article |
| id | doaj.art-89f1c0f3bb824d39a28d6ee6cfa5484b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T05:13:08Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-89f1c0f3bb824d39a28d6ee6cfa5484b2023-12-03T12:47:36ZengMDPI AGMathematics2227-73902022-07-011015263510.3390/math10152635Probabilistic Interpretation of Number Operator Acting on Bernoulli FunctionalsJing Zhang0Lixia Zhang1Caishi Wang2School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, ChinaSchool of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, ChinaSchool of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, ChinaLet <i>N</i> be the number operator in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of <i>N</i> from a probabilistic perspective. We first construct a nuclear space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>, which is also a dense linear subspace of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>, and then by taking its dual <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">G</mi><mo>*</mo></msup></semantics></math></inline-formula>, we obtain a real Gel’fand triple <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">G</mi><mo>⊂</mo><mi mathvariant="script">H</mi><mo>⊂</mo><msup><mi mathvariant="script">G</mi><mo>*</mo></msup></mrow></semantics></math></inline-formula>. Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>N</mi></msub></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">G</mi><mo>*</mo></msup></semantics></math></inline-formula> such that its covariance operator coincides with <i>N</i>. We examine the properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>N</mi></msub></semantics></math></inline-formula>, and, among others, we show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>N</mi></msub></semantics></math></inline-formula> can be represented as a convolution of a sequence of Borel probability measures on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">G</mi><mo>*</mo></msup></semantics></math></inline-formula>. Some other results are also obtained.https://www.mdpi.com/2227-7390/10/15/2635Bernoulli functionalsnumber operatorGel’fand tripleGauss measureconvolution of measures |
| spellingShingle | Jing Zhang Lixia Zhang Caishi Wang Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals Mathematics Bernoulli functionals number operator Gel’fand triple Gauss measure convolution of measures |
| title | Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals |
| title_full | Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals |
| title_fullStr | Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals |
| title_full_unstemmed | Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals |
| title_short | Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals |
| title_sort | probabilistic interpretation of number operator acting on bernoulli functionals |
| topic | Bernoulli functionals number operator Gel’fand triple Gauss measure convolution of measures |
| url | https://www.mdpi.com/2227-7390/10/15/2635 |
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