Commuting tuple of multiplication operators homogeneous under the unitary group

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dc.contributor.author Ghara, Soumitra
dc.contributor.author Kumar, Surjit
dc.contributor.author Misra, Gadadhar
dc.contributor.author Pramanick, Paramita
dc.date.accessioned 2022-02-11T08:02:49Z
dc.date.available 2022-02-11T08:02:49Z
dc.date.issued 2022-01
dc.identifier.citation Ghara, Soumitra; Kumar, Surjit; Misra, Gadadhar and Pramanick, Paramita, "Commuting tuple of multiplication operators homogeneous under the unitary group", arXiv, Cornell University Library, DOI: arXiv:2201.13228, Jan. 2022. en_US
dc.identifier.issn
dc.identifier.uri http://arxiv.org/abs/2201.13228
dc.identifier.uri https://repository.iitgn.ac.in/handle/123456789/7510
dc.description.abstract Let Bd be the open Euclidean ball in Cd and T:=(T1,…,Td) be a commuting tuple of bounded linear operators on a complex separable Hilbert space H. Let U(d) be the linear group of unitary transformations acting on Cd by the rule: z↦u⋅z, z∈Cd, and u⋅z is the usual matrix product. Consequently, u⋅z is a linear function taking values in Cd. Let u1(z),…,ud(z) be the coordinate functions of u⋅z. We define u⋅T to be the operator (u1(T),…,ud(T)) and say that T is U(d)-homogeneous if u⋅T is unitarily equivalent to T for all u∈U(d). We find conditions to ensure that a U(d)-homogeneous tuple T is unitarily equivalent to a tuple M of multiplication by coordinate functions acting on some reproducing kernel Hilbert space HK(Bd,Cn)⊆\rm Hol(Bd,Cn), where n is the dimension of the joint kernel of the d-tuple T. The U(d)-homogeneous operators in the case of n=1 have been classified under mild assumptions on the reproducing kernel K. In this paper, we study the class of U(d)-homogeneous tuples M in detail for n=d, or equivalently, kernels K quasi-invariant under the group U(d). Among other things, we describe a large class of U(d)-homogeneous operators and obtain explicit criterion for (i) boundedness, (ii) reducibility and (iii) mutual unitary equivalence of these operators. Finally, we classify the kernels K quasi-invariant under U(d), where these kernels transform under an irreducible unitary representation c of the group U(d).
dc.description.statementofresponsibility by Soumitra Ghara, Surjit Kumar, Gadadhar Misra and Paramita Pramanick
dc.format.extent
dc.language.iso en_US en_US
dc.publisher Cornell University Library en_US
dc.subject Functional Analysis en_US
dc.subject Operators en_US
dc.subject Unitary Group en_US
dc.subject kernel Hilbert space en_US
dc.subject Euclidean ball en_US
dc.title Commuting tuple of multiplication operators homogeneous under the unitary group en_US
dc.type Pre-Print en_US
dc.relation.journal arXiv


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