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.
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}. For
for the
For the readers’ convenience we now restate [5, Theorem 2.1]:
Observe now that every operator
Let {
Obviously
one sees easily that ℳ is a n.i.s. for
Moreover
and if we define
then obviously
Moreover, it is well-known that all forward weighted unilateral shifts have nontrivial hyperinvariant subspaces (cf., e.g., [10]). Thus if all operators of the form
On the basis of Example 1.2 the question of
We next define a (perhaps new) class of operators to which our main theorem below (Theorem 2.4) applies.
An operator
(a) neither
(b)
(c) for every n.h.s. of
The name (RIH) comes from the phrase “restrictions inherit nontrivial hyperinvariant subspaces”.
The interest in the class (RIH) arises from the fact that operators
(I)
(II)
(III) ∩{ℳ ⊂ ℋ: ℳ ∈ Hlat(
(IV) ∨{ℳ ⊂ ℋ: ℳ ∈ Hlat(
All of I)–IV) follow easily from the definition of (RIH), the fact that ℳ is a n.h.s. for
The principal result of this note is the following, which is of interest because of the important classes of operators in ℬ(ℋ) that are subsets of (RIH), as we shall see below.
To apply Theorem 1.1, we observe that if
from which it follows trivially that we cannot have for every n.h.s. of
In this section we set forth some important classes of operators to which Theorem 2.4 applies, and thus we obtain new and better sufficient conditions on the operator
An operator
and
(CK) (ℋ) ⊂ (RIH) (ℋ).
Let
Then
By Proposition 3.2,
If
H.K. Kim in [6] raised the very interesting question of
We now turn to another important class of operators pertinent to the operator in (
An operator
and
It is well-known from the multiplicity theory of normal operators (cf., e.g., [2]) that every operator
We note in particular, that if
H.J. Kim also studied in [7] matrices
We close this note by posing some unsolved problems concerning hyperinvariant subspaces for certain operators
Let
Let
Let
The first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIT) (2018R1A2B6003660). The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03931937).