On the Hyperinvariant Subspace Problem. II
S AMI H AMID , C ONSTANTIN O NICA & C ARL P EARCY
A BSTRACT. Recently in [6] the question of whether every nonscalar operator on a complex Hilbert space H of dimension ℵ0
has a nontrivial hyperinvariant subspace was reduced to a special
case; namely, the question whether every (BCP)-operator in C00
whose left essential spectrum is equal to some annulus centered at
the origin has a nontrivial hyperinvariant subspace. In this note,
we make additional contributions to this circle of ideas by showing that every (BCP)-operator in C00 is ampliation quasisimilar
to a quasidiagonal (BCP)-operator in C00 . Moreover, we show
that there exists a fixed block diagonal (BCP)-operator Bu with
the property that if every compact perturbation Bu + K of Bu
in (BCP) and C00 with kKk < ε has a nontrivial hyperinvariant
subspace, then every nonscalar operator on H has a nontrivial
hyperinvariant subspace.
1. I NTRODUCTION
In this note H will always be a fixed separable, infinite dimensional, complex
Hilbert space, and, as usual, L(H ) will denote the algebra of all bounded linear operators on H . If T ∈ L(H ) we write as usual, σ (T ), σe (T ), σle (T ),
and σre (T ) for the spectrum, essential (Calkin) spectrum, left essential spectrum and right essential spectrum of T , respectively. The closed ideal of compact operators in L(H ) will be denoted by K(H ) or K, and the quotient map
L(H ) → L(H )/K by π . The set of all scalar multiples of 1H will be written
consistently as C1H , and for any subset S ⊂ L(H ) we write S0 for the commutant of S, i.e., S0 = {T ∈ L(H ) : ST = T S for every S ∈ S}. Recall that
a subspace M ⊂ H is called a nontrivial hyperinvariant subspace (n.h.s.) for T if
743
c , Vol. 54, No. 3 (2005)
Indiana University Mathematics Journal 744
S AMI H AMID , C ONSTANTIN O NICA & C ARL P EARCY
(0) ≠ M ≠ H and SM ⊂ M for every S in {T }0 . The question whether every
operator T in L(H ) \ C1H has a nontrivial hyperinvariant subspace is a long-
standing and difficult open problem, but recently in [6] this problem was reduced
to a special subcase of itself, which we describe below.
Recall first that a completely nonunitary (c.n.u.) contraction T in L(H ) is
called a (BCP)-operator if D (= {ξ ∈ C : |ξ| < 1}) ∩ σe (T ) is a dominating
set for the unit circle T = ∂D and that the class C00 (H ) = C00 consists, by
definition, of the set of all (c.n.u.) contractions T such that the sequence {T n }∞
n=1
n ∗−SOT
satisfies T --------------------------→
--- 0 (see Section 2 for definitions). The class of (BCP)-operators,
introduced in [5], has been much studied and plays an important role in the theory
of dual algebras of operators. It is well known, for example, that the (BCP)operators are in a sense “universal dilations,” meaning that every direct sum of
strict contractions can be realized, up to unitary equivalence, as a compression
to some semi-invariant subspace of an arbitrary (BCP)-operator [3]. The lattice
Lat(T ) of invariant subspaces of any (BCP)-operator T is also known to be quite
large. In fact, it contains a sublattice isomorphic to the lattice of all subspaces
of H [3, Theorem 4.8] and also contains a countably infinite family of cyclic
invariant subspaces with the property that all pairwise intersections of members
of the family are trivial (i.e., (0)) [1]. Moreover, (BCP)-operators are reflexive [2].
Thus, having in mind the idea that it may be easier to find a n.h.s. for an operator
T in L(H ) \ C1H if one knows, at least, that T has a large family of nontrivial
invariant subspaces, the authors of [6] made the following reduction, alluded to
above.
Theorem 1.1. If either
(a) every (BCP)-operator T in C00 such that σle (T ) = D− := {ξ ∈ C : |ξ| ≤ 1} has
a n.h.s., or
(b) there exists 0 < θ < 1 such that every (BCP)-operator T satisfying σ (T ) =
σle (T ) = Aθ := {ξ ∈ C : θ ≤ |ξ| ≤ 1} and kT −1 k = 1/θ has a n.h.s.,
then every operator in L(H ) \ C1H has a n.h.s.
In this paper we contribute to this circle of ideas by establishing, in Section 3,
a new structure theorem for (BCP)-operators (Theorem 3.1), and we obtain as a
consequence of this result that every (BCP)-operator T is ampliation quasisimilar
(definition below) to a quasidiagonal (BCP)-operator. This shows, in conjunction with Theorem 1.1, that to solve the hyperinvariant subspace problem (for
operators on Hilbert space) it would suffice to show, for example, that every quasidiagonal (BCP)-operator T in C00 with σle (T ) = D− has a n.h.s. (Theorem 5.1).
We also obtain (Theorem 5.2) the striking result that there exists one quasidiagonal (BCP)-operator B0 with the property that if every small (in norm) compact
perturbation of B0 has a n.h.s., then every operator in L(H ) \ C1H has a n.h.s.
On the Hyperinvariant Subspace Problem. II
745
2. P RELIMINARIES
In this section we briefly review some definitions and notation that will be needed
in what follows. As usual, we write N for the set of positive integers. If {Tn }∞
n=1
SOT
is a sequence in L(H ), we will write Tn ---------------→
--- T0 to indicate that the sequence
∗−SOT
converges to T0 in the strong operator topology (SOT) and Tn --------------------------→
--- T0 to mean
SOT
∗
∗ SOT
that Tn ---------------→
--- T0 and Tn ---------------→
--- T0 . The normed ideal of trace-class operators in
L(H ) will be written as C1 (H ) and the corresponding trace-norm denoted by
k · k1 . As is well known, operators S and T in L(H ) are called quasisimilar
(notation: S ∼ T ) if there exist X and Y in L(H ) with ker X = ker X ∗ = ker Y =
ker Y ∗ = (0) such that SX = XT and Y S = T Y .
For any cardinal number n satisfying 1L≤ n ≤ ℵ0 , we denote by H (n) the
direct sum of n copies of H (i.e., H (n) = 0≤k<n Hk with Hk = H for every
k). For T ∈ L(H ) and 1 ≤ n ≤ ℵ0 , T (n) denotes the direct sum (ampliation)
of n copies of T acting on H (n) . For T1 and T2 in L(H ), we say, following
a
[6], that T1 is ampliation quasisimilar to T2 (notation: T1 ∼
T2 ) if there exist
(n1 )
(n2 )
∼ T2 , and it was shown
cardinal numbers 1 ≤ n1 , n2 ≤ ℵ0 such that T1
a
[6, Proposition 2.4] that if T1 ∼
T2 , then T1 has n.h.s. if and only if T2 does.
Recall next from [8] that an operator T in L(H ) is quasidiagonal (T ∈
(QD)(H )) if there exists an increasing sequence {Pn }∞
n=1 of finite rank projecSOT
tions such that Pn ---------------→
--- 1H and kT Pn − Pn T k → 0, and T is block diagonal
(notation: T ∈ (BD)(H )) if T is unitarily equivalent to a countably infinite (orthogonal) direct sum of operators each of which acts on a (nonzero) finite dimensional space. If, in addition, each of the direct summands Tn satisfies kTn k < 1,
then T will be called a strictly norm decreasing block diagonal operator. Recall
also that it is known from [8] that (QD)(H ) = (BD)(H ) + K(H ) and that
if T ∈ (QD)(H ) and ε > 0 are given, then there exist Bε ∈ (BD)(H ) and
Kε ∈ K(H ) such that T = Bε + Kε and kKε k < ε.
3. S TRUCTURE OF (BCP)- OPERATORS
In this section, we will establish the following new structure theorem for the class
of (BCP)-operators.
Theorem 3.1. Suppose T ∈ (BCP)(H ) and B is an arbitrary strictly norm
decreasing block diagonal operator. Then for every ε > 0 there exist c.n.u. contractions
T0 = T0 (ε) and Ki = Ki (ε), i = 1, 2, satisfying:
(a) Ki ∈ C1 (H ) and kKi k1 < ε for i = 1, 2,
a
(b) T ∼
T̂ , where T̂ ∈ (BCP) (H ⊕ H ) is the 2 × 2 operator matrix
(3.1)
T0 K1
K2 B
acting on H ⊕ H in the usual fashion,
!
S AMI H AMID , C ONSTANTIN O NICA & C ARL P EARCY
746
(c) σle (T̂ ) ⊃ σle (T ), σre (T̂ ) ⊃ σre (T ), and σ (T̂ ) ⊃ σ (T ), and
(d) if T ∈ C00 (H ), then also T̂ ∈ C00 (H ⊕ H ).
To prove Theorem 3.1, we must first establish the following lemmas.
Lemma 3.2. Suppose K1 and K2 are complex Hilbert spaces and T̂ ∈ L(K1 ⊕
K2 ) is a c.n.u. contraction defined matricially by
T̂ =
!
A B
0 C .
Then for every 0 < s < 1, the operator T̂s defined matricially by
T̂s =
A sB
0 C
!
is also a c.n.u. contraction.
Proof. Fix an arbitrary s such that 0 < s < 1, and let x ⊕ y be an arbitrary
vector in K1 ⊕ K2 . A short calculation gives that the inequality kT̂ (x ⊕ y)k ≤
kx ⊕ yk is equivalent to the inequality
(3.2)
h(1 − A∗ A)x, xi + h(1 − C ∗ C)y, yi ≥ kByk2 + 2<hB ∗ Ax, yi.
Now choose θ = θ(x, y) ∈ [0, 2π ) such that <e−iθ hB ∗ Ax, yi = |hB ∗ Ax, yi|.
Then we get from (3.2) and the fact that T̂ is a contraction that
(3.3)
h(1 − A∗ A)x, xi + h(1 − C ∗ C)y, yi
≥ kByk2 + 2|hB ∗ Ax, yi|
≥ s 2 kByk2 + 2s|hB ∗ Ax, yi|
≥ s 2 kByk2 + 2s<hB ∗ Ax, yi,
x ⊕ y ∈ K 1 ⊕ K2 ,
which proves that T̂s is a contraction. To show that T̂s is completely nonunitary,
suppose that M ⊂ K1 ⊕ K2 is an invariant (equivalently, reducing) subspace
for T̂s such that T̂s |M is a unitary operator. The argument will be completed
by showing that M = (0). Let PK2 ∈ L(K1 ⊕ K2 ) be the projection with the
subspace (0) ⊕ K2 as a range. If there exists x0 ⊕ y0 ∈ M with By0 ≠ 0, then the
inequality (3.3) becomes strict, and thus
kT̂s (x0 ⊕ y0 )k < kT̂ (x0 ⊕ y0 )k ≤ kx0 ⊕ y0 k,
❐
which contradicts the fact that T̂s |M is unitary. Thus PK2 M ⊂ (0) ⊕ ker B . But
this implies that M ∈ Lat(A⊕C) and that (A⊕C)|M = T̂s |M is a unitary operator.
Since T̂ is c.n.u., so are A, C , and A ⊕ C , so M = (0) as desired.
On the Hyperinvariant Subspace Problem. II
747
Lemma 3.3. Suppose that T̂ ∈ L(K1 ⊕ K2 ⊕ K3 ) is given matricially as

A11 A12 A13


T̂ =  0 A22 A23 
0 0 A33

and T̂ is a c.n.u. contraction. Then for every 0 < s < 1, the operator

A11 sA12 s 2 A13


T̂s =  0 A22 sA23 
0
0
A33

is also a c.n.u. contraction.
Proof. This follows immediately, after fixing 0 < s < 1, from Lemma 3.2
applied twice; first to give that

A11 sA12 sA13

 0

A23 
A33
A22
0

0
is a c.n.u. contraction, and then to give that


A11 sA12 s 2 A13
 0 A

22 sA23 

0
0
7A33
❐
is a c.n.u. contraction.
Proof of Theorem 3.1. Let ε > 0, let {r
P n }n∈N be a monotone decreasing sequence of positive real numbers such that rn < ε, and let B be any fixed strictly
norm decreasing operator in (BD)(H ). Then, by definition, there exist a sequence {Hn }n∈N of finite dimensional Hilbert spaces (with dim Hn := kn ∈ N),
and a Hilbert space isomorphism ϕ of H onto
aLsequence {Bn ∈ L(Hn )}n∈N , L
−1
= n∈N Bn and kBn k < 1 for every n. Fix now an
n∈N Hn , such that ϕBϕ
arbitrary T ∈ (BCP)(H ). For each n ∈ N, we may choose by [3, Theorem 4.8]
a Hilbert space isomorphism ϕn mapping H onto H ⊕ Hn ⊕ H such that

(n)
T11
(3.4)
0
Tn =
−1
ϕn T ϕn


=
 0

0
(n)
T12
Bn
0
(n)
T13



T23  ,

(n)
T33
(n) 
n ∈ N,
S AMI H AMID , C ONSTANTIN O NICA & C ARL P EARCY
748
(n)
(n)
and we note that the ranks of T12
and T23
are bounded above by kn . Define
sn = rn /kn for n ∈ N, and let Sn ∈ L(H ⊕ Hn ⊕ H ) be the invertible operator
defined by Sn = sn 1H ⊕ 1Hn ⊕ sn−1 1H . Then an easy calculation gives

(n)
(n)
sn T12
T11


00
0
Tn = Sn Tn Sn−1 = 
 0

(3.5)
Bn
0
0
(n)
2T
sn
13



(n)
sn T23 
,

(n)
T33
n ∈ N,
and by Lemma 3.3, Tn00 is a c.n.u. contraction. Moreover, for each n ∈ N, Tn00 is
obviously unitarily equivalent to the c.n.u. contraction

(n)
T11


Tn000 = 
 0

(3.6)
0
(n)
2T
sn
13
(n)
T33
(n)
sn T23
(n)
sn T12
0
Bn



.


Since, by construction, T is similar to each Tn000 , one knows that for each n ∈ N,
the spectrum and left and right essential spectra of Tn000 coincide with the corresponding parts of the spectrum of T , and if T ∈ C00 , then Tn000L∈ C00 also.
Moreover,
one knows (cf., e.g., [10, Proposition 9.7]) that T (ℵ0 ) ∼ n∈N Tn000 ∈
L
L( n∈N (H ⊕ H ⊕ Hn )). Furthermore, by reordering this direct sum of Hilbert
spaces as
M
M
M=
L
we see that
matricially as
(H ⊕ H ) ⊕
n∈N
n∈N
Hn ,
n∈N
Tn000 is unitarily equivalent to the operator M ∈ L(M) given
M
n∈N
M =
 K̂
(3.7)
Rn
2

K̂1

,
M
Bn 
n∈N
where
(A) Rn ∈ L(H ⊕ H ) is also defined matricially as
 (n)

2 T (n)
T11 sn
13
,
Rn = 
(n)
0
T33
(3.8)
(B) K̂1 :
L
n∈N
L
xn in
Hn → (H ⊕ H )(ℵ0 ) is defined at an arbitrary vector
L
L
(n)
xn ) = (sn T12 xn ⊕ 0H ), and
n∈N Hn by K̂1 (
n∈N
L
On the Hyperinvariant Subspace Problem. II
749
L
L
(C) K̂2 : (H ⊕ H )(ℵ0 ) → n∈N Hn is defined at any n∈N (vn ⊕ yn ) in
L
L
(n)
(H ⊕ H )(ℵ0 ) by K̂2 ( n∈N (vn ⊕ yn )) = n∈N sn T23 yn .
L
Thus M , being unitarily equivalent to n∈N Tn000 , is a c.n.u. contraction satisfying
a
M ∼ T , σle (M) ⊃ σle (Tn000 ) = σle (T ), σre (M) ⊃ σre (T ), and σ (M) ⊃ σ (T ).
Moreover, if T ∈ C00 , then M ∈ C00 also.
Next, let ψ be some Hilbert space isomorphism of H onto H (ℵ0 ) and deL
fine K1 = ψ−1 K̂1 ϕ, K2 = ϕ−1 K̂2 ψ, T0 = ψ−1 ( n∈N Rn )ψ, and T̂ = (ψ ⊕
ϕ)−1 M(ψ ⊕ ϕ).
It is obvious that T̂ is given matricially by (3.1), and from above we know that
a
T ∼ T̂ and that T̂ has properties (b), (c), and (d) in the statement of the theorem.
Thus the proof can be completed by showing that K1 , K2 ∈ C1 (H ) and satisfy
kK1 k1 , kK2 k1 < ε.
(3.9)
Of course (3.9) is equivalent to k(Ki∗ Ki )1/2 k1 < ε, i = 1, 2, and since ϕ
and ψ are Hilbert space isomorphisms, it suffices to show that for i = 1, 2,
k(K̂i∗ K̂i )1/2 k1 < ε. Moreover, the above definitions together with a short calculation show that
(K̂1∗ K̂1 )1/2 =
∗
1/2
(K̂2 K̂2 )
M
n∈N
(n)
(n)
sn [(T12 )∗ T12 ]1/2 ,
= 0H ⊕
M
n∈N
(n)
(n)
sn [(T23 )∗ T23 ]1/2 .
Thus
k(K̂1∗ K̂1 )1/2 k1 =
=
X
(n)
n∈N
X
(n)
n∈N
(n)
sn k((T12 )∗ T12 )1/2 k1
(n)
sn tr((T12 )∗ T12 )1/2 ≤
X
sn kn < ε,
n∈N
(n) ∗ (n) 1/2
since for each n ∈ N, ((T12
) T12 )
∈ L(Hn ) and is a contraction, and
therefore must have trace at most dim Hn = kn . A similar calculation shows that
k(K̂2∗ K̂2 )1/2 k1 < ε,
❐
and the proof is complete.
4. R EPRESENTATIONS
In this section we continue with our game plan of eventually showing that every
(BCP)-operator is ampliation quasisimilar to a quasidiagonal operator (Theorem
5.1). A fundamental result that will be needed is Voiculescu’s representation theorem from [12]. If T ∈ L(H ), we will write C ∗ (T ) and C ∗ (π (T )) for the unital
C ∗ -algebras generated by T (and 1H ) and π (T ) (and 1L(H )/K ), respectively.
750
S AMI H AMID , C ONSTANTIN O NICA & C ARL P EARCY
Theorem 4.1 (Voiculescu). Let T ∈ L(H ) and let ρ̃ be a unital C ∗ -algebra
homomorphism from C ∗ (π (T )) into L(H ). Then there exists a sequence {Un }n∈N
of unitary operators from H to H ⊕ H such that
(I) Un AUn∗ − A ⊕ ρ̃(π (A)) ∈ K(H ⊕ H ), and
(II) kUn AUn∗ − A ⊕ ρ̃(π (A))k → 0, A ∈ C ∗ (T ), n ∈ N.
We will need several preparatory lemmas to enable us to apply Theorem 4.1
to obtain the desired conclusions. The following lemma is elementary and thus
needs no proof.
Lemma 4.2. Let A ⊂ L(H ) be a unital C ∗ -algebra, and let P ∈ L(H ) be
a projection in A0 . Then the map ϕ defined by ϕ(A) = P AP |range P is a unital
C ∗ -algebra homomorphism of A into P AP |range P (with ϕ(1H ) = 1range P ).
The following lemma is similar to lemmas used without proof in [7] and [9].
L
Lemma 4.3. Let T = n∈N Tn ∈ L(H (ℵ0 ) ), and suppose that {Tn } ⊂ L(H )
is a sequence of contractions that converges ∗-SOT to a nonzero contraction S . Then,
(a) there exists a unital C ∗ -algebra homomorphism ρ of C ∗ (T ) into C ∗ (S) (i.e.,
ρ(1H (ℵ0 ) ) = 1H ) such that ρ(T ) = S , and
(b) C ∗ (T ) ∩ K(H (ℵ0 ) ) ⊂ ker ρ . Hence
(c) there exists a unital C ∗ -algebra homomorphism ρ̃ of C ∗ (π (T )) such that ρ =
ρ̃ ◦ π , and therefore S = ρ̃(π (T )).
Proof. One knows that if {An } and {Bn } are sequences in L(H ) that converge in the SOT to A0 and B0 , respectively, then the sequence {An Bn } converges
in the SOT to A0 B0 . Using this fact and the hypothesis, we see easily that if
p(x, y) is any polynomial in the noncommuting variables x and y , we may define
(4.1) ρ(p(T , T ∗ )) = ρ
M
n∈N
p(Tn , Tn∗ ) := SOT- lim p(Tn , Tn∗ ) = p(S, S ∗ ),
n
❐
and it is obvious
Lthat ρ , so defined, is a contractive ∗-homomorphism. Moreover,
since 1H (ℵ0 ) = n∈N 1H , we clearly have ρ(1H (ℵ0 ) ) = 1H , so ρ is unital. Thus ρ
extends by continuity to a C ∗ -algebra homomorphism of C ∗ (T ) into C ∗ (S).
With respect to (b), let A ∈ C ∗ (T ) ∩LK(H (ℵ0 ) ) and set ρ(A) = B . It is
obvious that A must have the form A = n∈N An , and since A is compact it
follows easily that each An ∈ K(H ) and that kAn k → 0. Thus, if η > 0 and
p(x, y) is a polynomial such that kp(T , T ∗ ) − Ak < η, then for n sufficiently
large we have kp(Tn , Tn∗ )k ≤ η, so from (4.1) we get that kp(S, S ∗ )k ≤ η. Since
kp(S, S ∗ ) − Bk ≤ kp(T , T ∗ ) − Ak ≤ η, this shows that B = 0. That (c) is valid
is now just an application of the standard result about factoring through quotient
spaces.
We next construct a specific block diagonal operator, which, in the terminology
of [9], is called a universal block diagonal operator.
On the Hyperinvariant Subspace Problem. II
751
DefinitionW4.4. Fix an orthonormal basis {en }n∈N of HS and set, for each
j ∈ N, Mj = {e1 , e2 , . . . , ej }. Let N be partitioned
L as N = j∈N Pj , where for
each j ∈ N, Pj is an infinite set, and define K = m∈N Km , where Km = Mj
for each m ∈ Pj . Moreover, for each j ∈ N, let Dj be a countable set of strict
contractions norm-dense in the unit ball of L(Mj )), and enumerate the elements
of Dj as {Bk }k∈Pj . Define now
Bu :=
(4.2)
M
Bk ∈ L(K).
k∈N
It is clear that Bu is a strictly norm decreasing block diagonal operator in
L(K), and its universality is established by this next lemma.
Proposition 4.5. Let Bu be the operator in (BD)(K) constructed in Definition
4.4, and let S be any nonzero contraction in L(H ). Then there exist unital C ∗ algebra homomorphisms ρ : C ∗ (Bu ) → C ∗ (S) and ρ̃ : C ∗ (π (Bu )) → C ∗ (S) such
that ρ = ρ̃ ◦ π and ρ(Bu ) = ρ̃(π (Bu )) = S .
Proof. With the orthonormal basis {en }n∈N of H and the subspaces Mj ⊂
H as in Definition 4.4, let Pj be the projection in L(H ) with range Pj = Mj ,
SOT
SOT
(so Pj ---------------→
--- 1H ), and define Sj := Pj SPj ∈ L(H ). Clearly Sj ---------------→
--- S and since
SOT
Sj∗ = Pj S ∗ Pj , we get also Sj∗ ---------------→
--- S ∗ . Moreover, as a consequence of the way Bu
was constructed, for each j ∈ N there exists some mj ∈ Pj such that Bmj ∈ Dj
and
1
Bm − Sj j
Mj Mj < j ,
(4.3)
2
j ∈ N.
For each j ∈ N define now B̃mj ∈ L(H ) by B̃mj = Bmj ⊕ 0H Mj , and define
also
M
B̃ :=
B̃mj ∈ L(H (ℵ0 ) ).
j∈N
Clearly,
kB̃mj − Sj kH <
∗−SOT
1
,
2j
j ∈ N,
∗−SOT
❐
and since Sj --------------------------→
--- S , B̃mj --------------------------→
-- S also. Moreover, there exists a natural unital
∗
(surjective) C -algebra isomorphism ϕ : C ∗ (L
B̃) → C ∗ (QBu Q) where Q ∈ L(K)
is the projection of K onto the subspace j∈N Kmj . Thus, by Lemma 4.2,
to construct a unital C ∗ -algebra homomorphism ρ of C ∗ (Bu ) into C ∗ (S) such
that ρ(Bu ) = S , it suffices to construct a unital C ∗ -algebra homomorphism ψ of
∗−SOT
C ∗ (B̃) into C ∗ (S) such that ψ(B̃) = S , and since B̃mj --------------------------→
--- S , the existence of
ψ is immediate from Lemma 4.3.
752
S AMI H AMID , C ONSTANTIN O NICA & C ARL P EARCY
The following corollary of Theorem 4.1 and Proposition 4.5, which should probably be credited to Herrero [9], is quite interesting in itself.
Corollary 4.6. Let Bu be the operator in (BD)(H ) ∩ C00 constructed in Definition 4.4, let S ∈ L(H ) be any contraction, and let ε > 0 be given. Then there exist
operators U : H → H ⊕ H and K ∈ K(H ) with U unitary and kKk < ε such that
U(Bu + K)U ∗ = Bu ⊕ S .
5. T HE M AIN T HEOREMS
In this section we first show that every (BCP)-operator is ampliation quasisimilar to a quasidiagonal (BCP)-operator, and then we apply this result, together
with Corollary 4.6, to obtain a further reduction in the hyperinvariant subspace
problem for operators on Hilbert space.
Theorem 5.1. Suppose T ∈ (BCP)(H ), Bu is as in Definition 4.4, and ε > 0
is given. Then there exists T̂ ∈ (BCP) ∩(QD) satisfying
a
T̂ , so T has a n.h.s. if and only if T̂ does,
(I) T ∼
(II) T̂ = (T0 ⊕ Bu ) + J , where T0 is a c.n.u. contraction and J ∈ C1 (H ⊕ H )
satisfies kJk1 < ε,
(III) σle (T̂ ) ⊃ σle (T ), σre (T̂ ) ⊃ σre (T ), and σ (T̂ ) ⊃ σ (T ), and
(IV) if T ∈ C00 , then T̂ ∈ C00 also.
❐
Proof. Let Bu be the strictly norm decreasing operator in (BD)(H ) defined
in Definition 4.4 (with H replacing K ). Then, by Theorem 3.1 (with B = Bu ),
there exists an operator T̂ ∈ (BCP)(H ⊕ H ) such that conclusions (a)–(d) of
that theorem are valid. In particular, from (a) we have that T̂ = (T0 ⊕ Bu ) + J ,
where J ∈ C1 (H ⊕ H ) with kJk1 < ε, and since (III) and (IV) are immediate
from (c) and (d), it suffices to show that T0 ⊕ Bu is quasidiagonal. But this follows
immediately from Corollary 4.6 and the fact that (QD) = (BD) + K.
The following is our further reduction of the hyperinvariant subspace problem.
Theorem 5.2. Let Bu be the operator constructed in Definition 4.4, and let ε > 0
be given. Then
(A) Bu ∈ (BD) ∩ (BCP) ∩C00 (H ) and satisfies σ (Bu ) = σle (Bu ) = D− ,
(B) Bu has point spectrum dense in D, and thus has at least ℵ0 different (and “disjoint”) n.h.s., and
(C) if every C00 , quasidiagonal, (BCP)-operator of the form Bu + K has a n.h.s.,
where K ∈ K(H ) and satisfies kKk < ε, then every operator in L(H ) \ C1H
has a n.h.s.
Proof. Since every operator in the unit ball of operators on a finite dimensional Hilbert space is the limit of a sequence of direct summands of the operator
Bu , elementary spectral theory shows that σ (Bu ) = σle (Bu ) = D− which proves
(A), and (B) is obvious. To establish (C), we note first that it follows from (a)
On the Hyperinvariant Subspace Problem. II
753
of Theorem 1.1 that it suffices to fix an arbitrary (BCP)-operator T1 in C00 with
σle (T1 ) = D− and to show that T1 has a n.h.s. under the hypotheses in (C). With
a
T1 as indicated, we conclude from Theorem 5.1 that T1 ∼ T̂1 = (T0 ⊕ Bu ) + J ,
where T̂1 has properties (I)–(IV) of that theorem (with kJk1 < ε/2). Thus T1 has
a n.h.s. if and only if T̂1 does, by [6, Proposition 2.4], and moreover, by Corollary
4.6 (with S = T0 ), we know that there exist operators U and K ∈ K(H ) with U
unitary and kKk < ε/2 such that U(Bu + K)U ∗ = T0 ⊕ Bu . Thus
U(Bu + K + U ∗ JU)U ∗ = (T0 ⊕ Bu ) + J = T̂1 ,
❐
and K + U ∗ JU ∈ K(H ) and satisfies kK + U ∗ JUk < ε. Since U is unitary, T̂1
has a n.h.s. if and only if Bu + K + U ∗ JU does, and the proof is complete.
Remark 5.3 (Added in proof ). Two additional articles, entitled appropriately
. . . III and . . . IV, exist that make further contributions to the topic of the hyperinvariant subspace problem. The third article has appeared in the Journal of
Functional Analysis 222 (2005), 129–142.
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Department of Mathematics
Texas A&M University
College Station
TX 77843, U. S. A. .
E- MAIL: S AMI H AMID : [email protected]
E- MAIL: C ONSTANTIN O NICA : [email protected]
E- MAIL: C ARL P EARCY: [email protected]
K EY WORDS AND PHRASES: hyperinvariant subspace, quasisimilar, quasidiagonal.
2000 M ATHEMATICS S UBJECT C LASSIFICATION: Primary: 47A15, 47B.
Received : March 25th, 2004.