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Spherically Quasinormal Pairs of Commuting Operators

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Analysis of Operators on Function Spaces

Part of the book series: Trends in Mathematics ((TM))

Abstract

We first discuss the spherical Aluthge and spherical Duggal transforms for commuting pairs of operators on Hilbert space. Second, we study the fixed points of these transforms, which are the spherically quasinormal commuting pairs. In the case of commuting 2-variable weighted shifts, we prove that spherically quasinormal pairs are intimately related to spherically isometric pairs. We show that each spherically quasinormal 2-variable weighted shift is completely determined by a subnormal unilateral weighted shift (either the 0-th row or the 0-th column in the weight diagram). We then focus our attention on the case when this unilateral weighted shift is recursively generated (which corresponds to a finitely atomic Berger measure). We show that in this case the 2-variable weighted shift is also recursively generated, with a finitely atomic Berger measure that can be computed from its 0-th row or 0-th column. We do this by invoking the relevant Riesz functionals and the functional calculus for the columns of the associated moment matrix.

The author “Raúl E. Curto” was partially supported by NSF Grant-1302666. The author “Jasang Yoon” was partially supported by a grant from the University of Texas System and the Consejo Nacional de Ciencia y Tecnología de México (CONACYT).

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Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Iowa, Iowa City, IA, USA

    Raúl E. Curto

  2. School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, TX, USA

    Jasang Yoon

Authors
  1. Raúl E. Curto
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  2. Jasang Yoon
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Correspondence to Raúl E. Curto .

Editor information

Editors and Affiliations

  1. Mathematics Department, Lund University, Lund, Sweden

    Alexandru Aleman

  2. Department of Mathematics, The Royal Institute of Technology, Stockholm, Sweden

    Haakan Hedenmalm

  3. Department of Mathematics, University of South California, Tampa, FL, USA

    Dmitry Khavinson

  4. Mathematics Department, University of California, Santa Barbara, CA, USA

    Mihai Putinar

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Curto, R.E., Yoon, J. (2019). Spherically Quasinormal Pairs of Commuting Operators. In: Aleman, A., Hedenmalm, H., Khavinson, D., Putinar, M. (eds) Analysis of Operators on Function Spaces. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-14640-5_7

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