Journal of Operator Theory

Volume 95, Issue 1, Winter 2026 pp. 51-76.

Almost invariant subspaces of shift operators and products of Toeplitz and Hankel operators

Authors: Caixing Gu (1), In Sung Hwang (2), Hyoung Joon Kim (3), Woo Young Lee (4), Jaehui Park (5)
Author institution: (1) Department of Mathematics, California Polytechnic\break State University, San Luis Obispo, CA 93407, U.S.A.
(2) Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
(3) Department of Mathematics, Seoul National University, Seoul 08826, Korea
(4) HCMC, Korea Institute for Advanced Study (KIAS), Seoul, 02455, Korea
(5) Department of Mathematics Education, Chonnam National University, Gwangju, 61186, Korea

Summary: In this paper we formulate the almost invariant subspaces theorems of backward shift operators in terms of the ranges or kernels of product of Toeplitz and Hankel operators. This approach simplifies and gives more explicit forms of these almost invariant subspaces which are derived from related nearly backward shift invariant subspaces with finite defect. Furthermore, this approach also leads to the surprising result that the almost invariant subspaces of backward shift operators are the same as the almost invariant subspaces of forward shift operators which were treated only briefly in literature.

DOI: http://dx.doi.org/10.7900/jot.2024mar02.2476
Keywords: almost invariant subspaces, shift operators, Toeplitz operators, Hankel operators

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