Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Hyperinvariant Subspaces for Some 2 × 2 Operator Matrices, II

  • Jung, Il Bong (Department of Mathematics, Kyungpook National University) ;
  • Ko, Eungil (Department of Mathematics, Ewha Womans University) ;
  • Pearcy, Carl (Department of Mathematics, Texas A&M University)
  • Received : 2019.02.01
  • Accepted : 2019.06.14
  • Published : 2019.06.23
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Abstract

In a previous paper, the authors of this paper studied $2{\times}2$ matrices in upper triangular form, whose entries are operators on Hilbert spaces, and in which the the (1, 1) entry has a nontrivial hyperinvariant subspace. We were able to show, in certain cases, that the $2{\times}2$ matrix itself has a nontrivial hyperinvariant subspace. This generalized two earlier nice theorems of H. J. Kim from 2011 and 2012, and made some progress toward a solution of a problem that has been open for 45 years. In this paper we continue our investigation of such $2{\times}2$ operator matrices, and we improve our earlier results, perhaps bringing us closer to the resolution of the long-standing open problem, as mentioned above.

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References

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