dc.contributor.advisor Misra, Gadadhar dc.contributor.author Deb, Prahllad dc.date.accessioned 2014-03-03T09:38:06Z dc.date.accessioned 2018-07-31T06:08:58Z dc.date.available 2014-03-03T09:38:06Z dc.date.available 2018-07-31T06:08:58Z dc.date.issued 2014-03-03 dc.date.submitted 2012 dc.identifier.uri https://etd.iisc.ac.in/handle/2005/2284 dc.identifier.abstract http://etd.iisc.ac.in/static/etd/abstracts/2939/G25236-Abs.pdf en_US dc.description.abstract In a foundational paper “Operators Possesing an Open Set of Eigenvalues” written several decades ago, Cowen and Douglas showed that an operator T on a Hilbert space ‘H possessing an open set Ω C of eigenvalues determines a holomorphic Hermitian vector bundle ET . One of the basic theorems they prove states that the unitary equivalence class of the operator T and the equivalence class of the holomorphic Hermitian vector bundle ET are in one to one correspondence. This correspondence appears somewhat mysterious until one detects the invariants for the vector bundle ET in the operator T and vice-versa. Fortunately, this is possible in some cases. Thus they point out that if the operator T possesses the additional property that the dimension of the eigenspace at ω is 1 for all ω Ω then the map ω ker(T - ω) admits a non-zero holomorphic section, say γ, and therefore defines a line bundle on Ω. As is well known, the curvature defined by the formula is a complete invariant for the line bundle . On the other hand, define en_US and note that NT (ω)2 = 0. It follows that if T is unitarily equivalent to T˜, then the corresponding operators NT (ω) and NT˜(ω) are unitarily equivalent for all ω Ω. However, Cowen and Douglas prove the non-trivial converse, namely that if NT (ω) and NT˜(ω) are unitarily equivalent for all ω Ω then T and T˜ are unitarily equivalent. What does this have to do with the line bundles and .To answer this question, we must ask what is a complete invariant for the unitary equivalence class of the operator NT (ω). To find such a complete invariant we represent NT (ω) with respect to the orthonormal basis obtained from the two linearly independent vectors γ(ω),∂γ(ω) by Gram-Schmidt orthonormalization process. Then an easy computation shows that It then follows that is a complete invariant for NT (ω), ω Ω. This explains the relationship between the line bundle and the operator T in an explicit manner. Subsequently, in the paper ”Operators Possesing an Open Set of Eigenvalues”, Cowen and Douglas define a class of commuting operators possessing an open set of eigenvalues and attempt to provide similar computations as above. However, they give the details only for a pair of commuting operators. While the results of that paper remain true in the case of an arbitrary n tuple of commuting operators, it requires additional effort which we explain in this thesis. dc.language.iso en_US en_US dc.relation.ispartofseries G25236 en_US dc.subject Eigenvalues en_US dc.subject Curvature Calculations en_US dc.subject Mathematical Algebra en_US dc.subject Operator Algebras en_US dc.subject Cowen-Douglas Operator Class en_US dc.subject Operator Theory en_US dc.subject.classification Algebra en_US dc.title Curvature Calculations Of The Operators In Cowen-Douglas Class en_US dc.type Thesis en_US dc.degree.name MS en_US dc.degree.level Masters en_US dc.degree.discipline Faculty of Engineering en_US
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