| Budget Amount *help |
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2002: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2001: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2000: ¥500,000 (Direct Cost: ¥500,000)
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| Research Abstract |
Motivated by recent developments in the theory of quantum groups, some classes of q-deformed operators (q-normal, q-quasinormal, q-deformed hyponormal operators) are investigated systematically, where q is a positive deformation parameter.
1. In the case when 0<q<1, a non-trivial q-deformed hyponormal closed operator must be unbounded. Its spectrum contains zero, the point spectrum is either {0} or empty set and its planar Lebesgue measure is always positive. If a q-deformed hyponormal closed operator T has dense range, then the inverse is also q-deformed hyponomal and the real and imaginary parts of its inverse are presented by those of T intertwined with some pure contraction.
2. On the other hand, if q>1 then there exists a q-deformed hyponormal operator with empty spectrum, which is constructed by an unbounded weighted shift, though the spectrum of every bounded q-qusinormal operator consists only of zero.
3. It is known that the Fuglede-Putnam theorem for usual normal operators is one of the most useful results in operator theory. Related to this, a q-analogue of the Fuglede-Putnam theorem for q-normal operators is proposed and is characterized in terms of the attached contraction.
4. The spectral analysis for q-deformed operators seems to be difficult. For example, the spectrum of all q-normal weighted shift is the whole complex plain. To analyze for such operators, a certain operator matrix representation induced by a closed operator is introduced and studied.
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