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Compatibility and Schur complements of operators on Hilbert C*-module

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Abstract

Let E be a Hilbert C*-module, and S be an orthogonally complemented closed submodule of E. The authors generalize the definitions of S-complementability and S-compatibility for general (adjointable) operators from Hilbert space to Hilbert C*-module, and discuss the relationship between each other. Several equivalent statements about S-complementability and S-compatibility, and several representations of Schur complements of S-complementable operators (especially, of S-compatible operators and of positive S-compatible operators) on a Hilbert C*-module are obtained. In addition, the quotient property for Schur complements of matrices is generalized to the quotient property for Schur complements of S-complementable operators and S*-complementable operators on a Hilbert C*-module.

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Authors and Affiliations

  1. Department of Mathematics, Tongji University, Shanghai, 200092, China

    Xiaochun Fang

  2. Department of Mathematics, Qingdao University, Qingdao, 266071, Shandong, China

    Jing Yu

Authors
  1. Xiaochun Fang
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  2. Jing Yu
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Correspondence to Xiaochun Fang.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 10771161, 11071188).

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Fang, X., Yu, J. Compatibility and Schur complements of operators on Hilbert C*-module. Chin. Ann. Math. Ser. B 32, 69–88 (2011). https://doi.org/10.1007/s11401-010-0623-2

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  • DOI: https://doi.org/10.1007/s11401-010-0623-2

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