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Similarity of quotient Hilbert modules in the Cowen–Douglas class

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Abstract

We consider similarity and quasi-affinity problems for Hilbert modules in the Cowen–Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a “simple” rank one Cowen–Douglas Hilbert module \(\mathscr {M}\), we find necessary and sufficient conditions for a class of Cowen–Douglas Hilbert modules satisfying some positivity conditions to be similar to We also show that under certain uniform bound condition on the anti-holomorphic frame, a Cowen–Douglas Hilbert module is quasi-affinity to a submodule of the free module

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Authors and Affiliations

  1. Department of Mathematics, Hebei Normal University, Shijiazhuang, 050016, Hebei, China

    Kui Ji

  2. Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

    Jaydeb Sarkar

Authors
  1. Kui Ji
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  2. Jaydeb Sarkar
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Correspondence to Jaydeb Sarkar.

Additional information

Dedicated to the memory of Ron Douglas

The first author was supported by National Natural Science Foundation of China (Grant No. 11831006), the second author is supported in part by (1) National Board of Higher Mathematics (NBHM), India, Grant NBHM/R.P.64/2014, and (2) Mathematical Research Impact-Centric Support (MATRICS) Grant, File No: MTR/2017/000522, by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India. The second author would also like to thank the members of the Department of Mathematics of the Hebei Normal University, China, for their hospitality during a visit during which part of the work was carried out.

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Ji, K., Sarkar, J. Similarity of quotient Hilbert modules in the Cowen–Douglas class. European Journal of Mathematics 5, 1331–1351 (2019). https://doi.org/10.1007/s40879-018-0297-y

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