KYUNGPOOK Math. J. 2019; 59(2): 225-231

Hyperinvariant Subspaces for Some 2×2 Operator Matrices, II

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

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.

© Kyungpook Mathematical Journal. All rights reserved.

- 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

_{ℋ}is denoted, as usual, by ℂ1_{ℋ}. ForT in ℬ(ℋ) we write$${\{T\}}^{\prime}=\{S\in \mathcal{B}(\mathcal{H}):ST=TS\},$$ for the

commutant ofT andσ (_{p}T ) for the point spectrum ofT . A subspace (i.e., a closed linear manifold) ℳ ⊂ ℋ is said to be anontrivial invariant subspace (notation: n.i.s.) for an operatorT in ℬ(ℋ) if (0) ≠ ℳ ≠ ℋ andT ℳ ⊂ ℳ. If ℳ is a n.i.s. forT and furthermore has the property thatT ′ℳ ⊂ ℳ for allT ′ ∈ {T }′, then ℳ is said to be anontrivial hyperinvariant subspace (notation: n.h.s.) forT . As is well-known, the problem of whether everyT in ℬ(ℋ) has a n.i.s. (called theinvariant 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_{ℋ}has a n.h.s., calledthe hyperinvariant subspace problem for operators on Hilbert space. The second (called sometimesthe hypertransitive operator problem for operators on Hilbert space) is the question of whether there exists an operatorT in ℬ(ℋ) such that forevery nonzero vectorx in ℋ, the orbit ofx underT , namely${\{{T}^{n}x\}}_{n=0}^{\infty}$ , 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 T ∈ ℬ(ℋ ⊕ ℋ)_{C}matricially as $${T}_{C}:=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right).$$ If there exists a pair (X , ℳ),where X ∈ ℬ(ℋ)with AX =XB ,and ℳis a n.h.s. for A such that X ℋ ⊄ ℳ,then for every D in ℬ(ℋ),T _{D}has a n.h.s. Observe now that every operator

S in ℬ(ℋ)ℂ1_{ℋ}that is known to have a n.i.s. but not known to have a n.h.s. is unitarily equivalent to some operatorT in ℬ(ℋ ⊕ ℋ) of the form (_{C}1.1 ) (but without the hypothesis thatA has a n.h.s.). This follows from the fact that if either the known n.i.s. forS or its orthocomplement is finite dimensional, thenS orS ^{*}has nonempty point spectrum, from which the existence of a n.h.s. forS follows trivially. Thus when studying operators likeS , no generality is lost by instead considering operators of the formT in (_{C}1.1 ). Moreover there are such operators for which the operatorA 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 {

e }_{n}_{n}_{∈ℤ}be an orthonormal basis for ℋ and letw = {w }_{n}_{n}_{∈ℤ}be a bounded sequence of positive numbers that is also bounded away from 0. DefineW ∈ ℬ(ℋ) by the equations_{w}$${W}_{w}{e}_{n}={w}_{n}{e}_{n-1},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}n\in \mathbb{Z}.$$ Obviously

W is an invertible bilateral weighted shift, and with ℳ defined as_{w}