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 T in ℬ(ℋ) we write

for the commutant of T and σ_p(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_ℋ 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 T_C ∈ ℬ(ℋ ⊕ ℋ) matricially as

If there exists a pair (X, ℳ), where X ∈ ℬ(ℋ) with AX = XB, and ℳ is a n.h.s. for A such that Xℋ ⊄ 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 operator T_C 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 T_C 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 {e_n}_n∈ℤ be an orthonormal basis for ℋ and let w = {w_n}_n∈ℤ be a bounded sequence of positive numbers that is also bounded away from 0. Define W_w ∈ ℬ(ℋ) by the equations

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

one sees easily that ℳ is a n.i.s. for W_w and W_w|_ℳ is unitarily equivalent to the forward weighted unilateral shift V_w− defined by

Moreover

and if we define Vw*+ by the equations

then obviously Vw*+ is a backward weighted unilateral shift and W_w is unitarily equivalent to the operator T_C in (1.1), where A is unitarily equivalent to V_w−, B is unitarily equivalent to Vw*+, and C is unitarily equivalent to the operator of rank one R: ℋ → ℋ defined by

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 T_C in (1.1), where A has a n.h.s., were known to have a n.h.s., then the longtime, still open problem of whether invertible weighted bilateral shifts have a n.h.s. would be solved.

On the basis of Example 1.2 the question of whether all operators of the form T_C in (1.1) have a n.h.s. when A does is of considerable interest, and in this note we continue to make progress on this problem, improving some results in [5].

We next define a (perhaps new) class of operators to which our main theorem below (Theorem 2.4) applies.

Definition 2.1

An operator T in ℬ(ℋ) will be said to belong to the class (RIH) (or (RIH)(ℋ) if necessary to avoid confusion) if T satisfies the following three conditions:

• (a) neither T nor T^* has nonempty point spectrum,

• (b) T has a n.h.s., and

• (c) for every n.h.s. of T, each of and has a n.h.s.

Remark 2.2

The name (RIH) comes from the phrase “restrictions inherit nontrivial hyperinvariant subspaces”.

The interest in the class (RIH) arises from the fact that operators T in (RIH) have particularly nice hyperinvariant subspace lattices (notation: Hlat(T)).

Proposition 2.3

Let T ∈ ℬ(ℋ) be an arbitrary operator in the class (RIH). Then Hlat(T) has the following properties.

• (I) T ∈ (RIH) if and only if T^* ∈ (RIH).

• (II) For every ℳ ≠ (0) in Hlat(T) and everyin Hlat(T^*), dimM=dimN=ℵ0.

• (III) ∩{ℳ ⊂ ℋ: ℳ ∈ Hlat(T) and ℳ ≠ (0)} = (0).

• (IV) ∨{ℳ ⊂ ℋ: ℳ ∈ Hlat(T) and ℳ ≠ ℋ} = ℋ.

Theorem 2.4

Let A ∈ (RIH)(ℋ) and let B be an arbitrary operator in ℬ(ℋ). If there exists a nonzero X ∈ ℬ(ℋ) such that AX = XB, then for every C in ℬ(ℋ), the operator T_C as in (1.1) has a n.h.s.

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 A in the matrix in (1.1) under which the operator T_C there has a n.h.s.

Definition 3.1

An operator A in ℬ(ℋ) will be said to belong to the class (CK)(or (CK) (ℋ)) if there exists a (nonzero) compact operator K in ℬ(ℋ) such that

and AK = KA. (The notation (CK) arises from the phrase “commutes with a compact operator”).

Proposition 3.2

(CK) (ℋ) ⊂ (RIH) (ℋ).

Corollary 3.3

Suppose A ∈ (CK) (ℋ) and B is an arbitrary operator in ℬ(ℋ). If there exists X ≠ 0 in ℬ(ℋ) such that AX = XB, then for every C ∈ ℬ(ℋ), the operator T_C as in (1.1) has a n.h.s.

Corollary 3.4

Suppose A is any nonzero compact operator in ℬ(ℋ) and B is an arbitrary operator there. If there exists a nonzero operator X such that AX = XB, then for every C in ℬ(ℋ), the operator T_C as in (1.1) has a n.h.s.

Remark 3.5

H.K. Kim in [6] raised the very interesting question of whether every operator T_C in (1.1) such that A is a nonzero compact operator has a n.h.s. This problem remains open still, and Corollary 3.4 above seems presently to be the best result in the direction of showing that the answer may be “yes”. Note that if the answer eventually turns out to be “yes”, then that theorem would be a beautiful generalization of V. Lomonosov’s first theorem in [9], namely that every nonzero compact operator in ℬ(ℋ) has a n.h.s.

We now turn to another important class of operators pertinent to the operator in (1.1), the treatment of which is parallel to that of the class (CK).

Definition 3.6

An operator A in ℬ(ℋ) will be said to belong to the class (CN) (or (CN)(ℋ)) if there exists a (nonzero) normal operator N in ℬ(ℋ) not of uniform multiplicity ℵ₀ such that

and AN = NA.

It is well-known from the multiplicity theory of normal operators (cf., e.g., [2]) that every operator A in the commutant of a normal operator N as in Definition 3.6 is an n-normal operator or a direct sum of operators at least one of which is an n-normal operator (for some n ∈ ℕ). And, via [4] and [3], all such operators are known to have a n.h.s. . Note that by Fuglede’s theorem, AN^* = N^*A, and therefore and . In other words, is a reducing subspace for N and is again a normal operator that commutes with and . But then, as above, has a n.h.s. These remarks are sufficient to constitute a proof of the following.

Corollary 3.7

Suppose A ∈ (CN) (ℋ) and B is an arbitrary operator in ℬ(ℋ). If there exists a nonzero operator X such that AX = XB, then for every C ∈ ℬ(ℋ) the operator T_C in (1.1) has a n.h.s.

We note in particular, that if A in Corollary 3.7 is a nonscalar normal operator, then the conclusions of that corollary remain true for all operators T_C as in (1.1).

Remark 3.8

H.J. Kim also studied in [7] matrices T_C as in (1.1), where A is a normal operator, and in some cases he obtained the existence of a n.h.s. for T_C. (This topic was also considered in the paper [8].)

We close this note by posing some unsolved problems concerning hyperinvariant subspaces for certain operators T_C as in (1.1).

Problem 3.9

Let T_C be as in (1.1), where A and C are compact and nonzero, and B is an arbitrary operator. Does T_C have a n.h.s.?

Problem 3.10

Let T_C be as in (1.1), where A has a n.h.s., B is an arbitrary operator, and C is nonzero and has finite rank. Does T_C have a n.h.s.?

Problem 3.11

Let T_C be as in (1.1), where A has a n.h.s., B is an arbitrary operator, and C = 1_ℋ. Does T_C have a n.h.s.?

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).