dc.contributor.advisor Misra, Gadadhar dc.contributor.author Hazra, Somnath dc.date.accessioned 2018-06-13T07:26:16Z dc.date.accessioned 2018-07-31T06:09:23Z dc.date.available 2018-06-13T07:26:16Z dc.date.available 2018-07-31T06:09:23Z dc.date.issued 2018-06-13 dc.date.submitted 2017 dc.identifier.uri http://etd.iisc.ac.in/handle/2005/3694 dc.identifier.abstract http://etd.iisc.ac.in/static/etd/abstracts/4564/G28564-Abs.pdf en_US dc.description.abstract A bounded operator T on a complex separable Hilbert space is said to be homogeneous if '(T ) is unitarily equivalent to T for all ' in M•ob, where M•ob is the M•obius group. A complete description of all homogeneous weighted shifts was obtained by Bagchi and Misra. The first examples of irreducible bi-lateral homogeneous 2-shifts were given by Koranyi. We describe all irreducible homogeneous 2-shifts up to unitary equivalence completing the list of homogeneous 2-shifts of Koranyi. After completing the list of all irreducible homogeneous 2-shifts, we show that every homogeneous operator whose associated representation is a direct sum of three copies of a Complementary series representation, is reducible. Moreover, we show that such an operator is either a direct sum of three bi-lateral weighted shifts, each of which is a homogeneous operator or a direct sum of a homogeneous bi-lateral weighted shift and an irreducible bi-lateral 2-shift. It is known that the characteristic function T of a homogeneous contraction T with an associated representation is of the form T (a) = L( a) T (0) R( a); where L and R are projective representations of the M•obius group M•ob with a common multiplier. We give another proof of the \product formula". We point out that the defect operators of a homogeneous contraction in B2(D) are not always quasi-invertible (recall that an operator T is said to be quasi-invertible if T is injective and ran(T ) is dense). We prove that when the defect operators of a homogeneous contraction in B2(D) are not quasi-invertible, the projective representations L and R are unitarily equivalent to the holomorphic Discrete series representations D+ 1 and D++3, respectively. Also, we prove that, when the defect operators of a homogeneous contraction in B2(D) are quasi-invertible, the two representations L and R are unitarily equivalent to certain known pairs of representations D 1; 2 and D +1; 1 ; respectively. These are described explicitly. Let G be either (i) the direct product of n-copies of the bi-holomorphic automorphism group of the disc or (ii) the bi-holomorphic automorphism group of the polydisc Dn: A commuting tuple of bounded operators T = (T1; T2; : : : ; Tn) is said to be homogeneous with respect to G if the joint spectrum of T lies in Dn and '(T); defined using the usual functional calculus, is unitarily equivalent to T for all ' 2 G: We show that a commuting tuple T in the Cowen-Douglas class of rank 1 is homogeneous with respect to G if and only if it is unitarily equivalent to the tuple of the multiplication operators on either the reproducing kernel Hilbert space with reproducing kernel n 1 i=1 (1 ziwi) i or Q n i i n; are positive real numbers, according asQG is as in (i) or 1 ; where ; i, 1 i i=1 (1 z w ) (ii). Finally, we show that a commuting tuple (T1; T2; : : : ; Tn) in the Cowen-Douglas class of rank 2 is homogeneous with respect to M•obn if and only if it is unitarily equivalent to the tuple of the multiplication operators on the reproducing kernel Hilbert space whose reproducing kernel is a product of n 1 rank one kernels and a rank two kernel. We also show that there is no irreducible tuple of operators in B2(Dn), which is homogeneous with respect to the group Aut(Dn): en_US dc.language.iso en_US en_US dc.relation.ispartofseries G28564 en_US dc.subject Homogeneous Operator en_US dc.subject Homogeneous 3-shifts en_US dc.subject Homogeneous Contractions en_US dc.subject Homogeneous Tuples en_US dc.subject Cowen-Douglas Class en_US dc.subject Irreducible Homogeneous 2-shifts en_US dc.subject Cowen-Douglas Class en_US dc.subject Polydisk en_US dc.subject.classification Mathematics en_US dc.title Homogeneous Operators en_US dc.type Thesis en_US dc.degree.name PhD en_US dc.degree.level Doctoral en_US dc.degree.discipline Faculty of Science en_US
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