KYUNGPOOK Math. J. 2019; 59(2): 225-231
Hyperinvariant Subspaces for Some 2×2 Operator Matrices, II
Il Bong Jung∗, Eungil Ko, Carl Pearcy
Department of Mathematics, Kyungpook National University, Daegu 41566, Korea
e-mail : ibjung@knu.ac.kr

Department of Mathematics, Ewha Womans University, Seoul 03760, Korea
e-mail : eiko@ewha.ac.kr

Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
e-mail : cpearcy@math.tamu.edu
* Corresponding Author.
Received: February 1, 2019; Revised: June 12, 2019; Accepted: June 14, 2019; Published online: June 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

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.
1. Introduction

The notation and terminology herein are completely standard and exactly the same as in [5]; nevertheless, we briefly review the main definitions. Throughout this note ℋ will always denote a separable, infinite dimensional, complex, Hilbert space, and ℬ(ℋ) the algebra of all bounded linear operators on ℋ. The space of scalar multiples of the identity operator 1 is denoted, as usual, by ℂ1. For T in ℬ(ℋ) we write

${T}′={S∈B(H):ST=TS},$

for the commutant of T and σp(T) for the point spectrum of T. A subspace (i.e., a closed linear manifold) ℳ ⊂ ℋ is said to be a nontrivial invariant subspace (notation: n.i.s.) for an operator T in ℬ(ℋ) if (0) ≠ ℳ ≠ ℋ and Tℳ ⊂ ℳ. If ℳ is a n.i.s. for T and furthermore has the property that T′ℳ ⊂ ℳ for all T′ ∈ {T}′, then ℳ is said to be a nontrivial hyperinvariant subspace (notation: n.h.s.) for T. As is well-known, the problem of whether every T in ℬ(ℋ) has a n.i.s. (called the invariant subspace problem for operators on Hilbert space) remains unsolved, although many partial results are known. (For more information about this topic, the reader may wish to consult the excellent book [1]). It is also the case that there are two related problems whose answers are not known. The first is the question of whether every operator in ℬ(ℋ)ℂ1 has a n.h.s., called the hyperinvariant subspace problem for operators on Hilbert space. The second (called sometimes the hypertransitive operator problem for operators on Hilbert space) is the question of whether there exists an operator T in ℬ(ℋ) such that for every nonzero vector x in ℋ, the orbit of x under T, namely ${Tnx}n=0∞$, is dense in ℋ.

For the readers’ convenience we now restate [5, Theorem 2.1]:

### Theorem 1.1

Let A, B, and C be arbitrary operators in ℬ(ℋ), and define TC ∈ ℬ(ℋ ⊕ ℋ) matricially as

$TC:=(AC0B).$

If there exists a pair (X, ℳ), where X ∈ ℬ(ℋ) with AX = XB, andis a n.h.s. for A such that Xℋ ⊄ ℳ, then for every D in ℬ(ℋ), TD has a n.h.s.

Observe now that every operator S in ℬ(ℋ)ℂ1 that is known to have a n.i.s. but not known to have a n.h.s. is unitarily equivalent to some operator TC in ℬ(ℋ ⊕ ℋ) of the form (1.1) (but without the hypothesis that A has a n.h.s.). This follows from the fact that if either the known n.i.s. for S or its orthocomplement is finite dimensional, then S or S* has nonempty point spectrum, from which the existence of a n.h.s. for S follows trivially. Thus when studying operators like S, no generality is lost by instead considering operators of the form TC in (1.1). Moreover there are such operators for which the operator A in (1.1) is known to have a n.h.s., and it is this class of operators to be studied herein.

### Example 1.2

Let {en}n∈ℤ be an orthonormal basis for ℋ and let w = {wn}n∈ℤ be a bounded sequence of positive numbers that is also bounded away from 0. Define Ww ∈ ℬ(ℋ) by the equations

$Wwen=wnen-1, n∈ℤ.$

Obviously Ww is an invertible bilateral weighted shift, and with ℳ defined as