Articles
KYUNGPOOK Math. J. 2019; 59(2): 225-231
Published online June 23, 2019
Copyright © Kyungpook Mathematical Journal.
Hyperinvariant Subspaces for Some 2 × 2 Operator Matrices, II
Department of Mathematics, Kyungpook National University, Daegu 41566, Korea, e-mail:
Received: February 1, 2019; Revised: June 12, 2019; Accepted: June 14, 2019
In a previous paper, the authors of this paper studied 2 × 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 × 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 × 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.
Keywords: invariant subspace, hyperinvariant subspace, compact operator.