2007 Fiscal Year Final Research Report Summary
UNBOUNDED REPRESENTATIONS OF QUANTUM ALGEBRAS AND THEIR ATTACHED
| Project/Area Number |
17540165
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Kyushu University |
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Principal Investigator |
OTA Schoichi Kyushu University, Faculty of Design, Professor (70107176)
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| Co-Investigator(Kenkyū-buntansha) |
INOUE Atushu Fukuoka University, Faculty of Science, Professor (50078557)
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| Project Period (FY) |
2005 – 2007
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| Keywords | UNBOUNDED OPERATOR / q-DEFORMED OPERATOR / q-NORMAL OPERATOR / q-HARMONIC OSCILLATOR / CIRCULAR OPERATOR |
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| Research Abstract |
The research is devoted to analyzing operator representations related to some q-deformed quantum harmonic oscillator and also to studying well-behaved unbounded representations of *-algebras. V. Bargmann showed in 1961 that the creation operator in the quantum harmonic oscillator is unitarily equivalent to the operator of multiplication by the independent variable on Segal-Bargmann space. Therefore, the creation operator is extended to a normal operator on a possibly larger Hilbert space. Recently, many variants of deformed oscillators with deformation parameter q occur in quantum groups theory. One possible variety is based on the relation xx*-qx*x=1, where q is positive real number and is not equal to 1. Motivated by the result of Bargmann, we naturally present a question whether a so-called q-creation operator x* is extended to some operator which is near to a kind of normal operator in a sense. We introduce a notion of 'q-positive definiteness' for a densely defined operator on Hilbert space and it is observed that the creation operator x* satisfies such a condition. We proved that, if a densely defined operator has invariant domain and satisfies 'q-positive definiteness' , then it is extended to a q-formally normal operator on a possibly larger space. Especially, the Bram-Halmos theorem for bounded subnormal operators is obtained by taking q=1 in the above. Moreover, it implies that the q-creation operator mentioned above is extended to a q-formally normal operator.
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