Representing Kernels of Perturbations of Toeplitz Operators by Backward Shift-Invariant Subspaces

Abstract

It is well known the kernel of a Toeplitz operator is nearly invariant under the backward shift \(S^*\). This paper shows that kernels of finite rank perturbations of Toeplitz operators are nearly \(S^*\)-invariant with finite defect. This enables us to apply a recent theorem by Chalendar–Gallardo–Partington to represent the kernel in terms of backward shift-invariant subspaces, which we identify in several important cases.

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Acknowledgements

The authors thank the referee for many useful comments which improve the presentation considerably. This work was done while the first author was visiting the University of Leeds. She is grateful to the School of Mathematics at the University of Leeds for its hospitality.

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Correspondence to Yuxia Liang.

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Yuxia Liang is supported by the National Natural Science Foundation of China (Grant No. 11701422).

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Liang, Y., Partington, J.R. Representing Kernels of Perturbations of Toeplitz Operators by Backward Shift-Invariant Subspaces. Integr. Equ. Oper. Theory 92, 35 (2020). https://doi.org/10.1007/s00020-020-02592-7

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Keywords

  • Shift-invariant subspace
  • Nearly \(S^*\)-invariant
  • Toeplitz operator

Mathematics Subject Classification

  • Primary 46E22
  • 47B38
  • Secondary 47A15