PROPER CONTRACTIONS AND INVARIANT SUBSPACES

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PII. S0161171201006287 http://ijmms.hindawi.com

PROPER CONTRACTIONS AND INVARIANT SUBSPACES

C. S. KUBRUSLY and N. LEVAN (Received 4 December 2000)

Abstract.LetT be a contraction andAthe strong limit of{T^∗ⁿTⁿ}n≥1. We prove the following theorem: if a hyponormal contractionT does not have a nontrivial invariant subspace, thenTis either a proper contraction of classᏯ00or a nonstrict proper contraction of classᏯ10for whichAis a completely nonprojective nonstrict proper contraction.

Moreover, its self-commutator[T^∗, T ]is a strict contraction.

2000 Mathematics Subject Classiﬁcation. 47A15, 47B20.

1. Introduction. LetᏴbe an inﬁnite-dimensional complex Hilbert space. By an operator onᏴwe mean a bounded linear transformation ofᏴinto itself. The null operator and the identity onᏴwill be denoted byOandI, respectively. IfT is an operator, thenT^∗is its adjoint, andT^∗ = T. The null space (kernel) ofT, which is the subspace ofᏴ, will be denoted byᏺ(T ). A contraction is an operatorTsuch thatT ≤1 (i.e.,T x ≤ xfor everyx inᏴor, equivalently,T^∗T ≤I). A strict contraction is an operatorT such thatT<1 (i.e., sup0=x(T x/x) <1; equivalently,T^∗T≺I, which means thatT^∗T ≤γI for some γ∈(0,1)). An isometry is a contraction for whichT x = xfor everyxinᏴ(i.e.,T^∗T=Iso thatT =1).

We summarize below some well-known results on contractions that will be applied throughout (cf. [16, page 40], [5,9,10,11,13], and [8, Chapter 3]). IfT is a contraction, then T^∗nTⁿ →^s A. That is, the sequence{T^∗nTⁿ}n≥1 of operators onᏴ converges strongly to an operatorAonᏴ, which means that(T^∗nTⁿ−A)x →0 for everyx inᏴ. Moreover,Ais a nonnegative contraction (i.e.,O≤A≤I),A =1 whenever A=O, T^∗nATⁿ=Afor every integern≥1 (so thatT is an isometry if and only if A=I),Tⁿx → A^1/2x for everyx inᏴ, and the null spaces ofAandI−A, viz.

ᏺ^(A)= {x∈Ᏼ^:^Ax=0}andᏺ^(I−A)= {x∈Ᏼ^:^Ax=x}, are given by ᏺ(A)=

x∈Ᏼ:Tⁿx →0 , ᏺ(I−A)=

x∈Ᏼ:Tⁿx= x ∀n≥1

=

x∈Ᏼ^:Ax = x .

(1.1)

Recall thatTis a contraction if and only ifT^∗is. ThusTⁿT^{∗n s}→A_∗, whereO≤A_∗≤I, A_∗=1 wheneverA_∗=O,TⁿA_∗T^∗n=A_∗for everyn≥1 (so thatTis a co-isometry—

i.e.,T^∗is an isometry—if and only ifA_∗=I),T^∗ⁿx → A^1/2_∗ xfor everyxinᏴ, and ᏺA_∗

=

x∈Ᏼ:T^∗nx →0 , ᏺ^I−A_∗

=

x∈Ᏼ^:^T^∗n^x= x ∀n≥1

=

x∈Ᏼ:A_∗x= x .

(1.2)

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An operator T on Ᏼ is uniformly stable if the power sequence{Tⁿ}n≥1 converges uniformly to the null operator (i.e.,Tⁿ →0). It is strongly stable if{Tⁿ}n≥1converges strongly to the null operator (i.e.,Tⁿx →0 for everyxinᏴ), and weakly stable if {Tⁿ}n≥1converges weakly to the null operator (i.e.,Tⁿx;y →0 for everyx, y∈Ᏼ or, equivalently,Tⁿx;x →0 for everyx∈Ᏼ). It is clear that uniform stability implies strong stability, which implies weak stability. The converses fail (a unilateral shift is a weakly stable isometry and its adjoint is a strongly stable co-isometry) but hold for compact operators.T is uniformly stable if and only if T^∗ is uniformly stable, and T is weakly stable if and only if T^∗ is weakly stable. However, strong convergence is not preserved under the adjoint operation so that strong stability forT does not imply strong stability for T^∗ (and vice versa). IfT is a strongly stable contraction (i.e., ifᏺ^(A)=Ᏼ, which means that A=O), then it is usual to say thatT is aᏯ0·- contraction. IfT^∗ is a strongly stable contraction (i.e., ifᏺ(A_∗)=Ᏼ, which means thatA_∗=O), thenT is aᏯ·0-contraction. On the other extreme, if a contraction T is such thatTⁿx0 for every nonzero vectorx inᏴ^{(i.e., if}ᏺ^(A)= {0}), then it is said to be aᏯ1·-contraction. Dually, if a contractionTis such thatT^∗ⁿx0 for every nonzero vectorxinᏴ(i.e., ifᏺ(A_∗)= {0}), then it is aᏯ·1-contraction. These are the Nagy-Foia¸s classes of contractions (see [16, page 72]). All combinations are possible leading to classesᏯ00, Ꮿ01,Ꮿ10, andᏯ11. In particular,T and T^∗ are both strongly stable contractions if and only ifT is of classᏯ00. Generally,

T∈Ꮿ00⇐⇒A=A_∗=O,

T∈Ꮿ01⇐⇒A=O, ᏺ(A_∗)= {0}, T∈Ꮿ10⇐⇒ᏺ(A)= {0}, A_∗=O, T∈Ꮿ11⇐⇒ᏺ(A)=ᏺ(A_∗)= {0}.

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IfT is a strict contraction, then it is uniformly stable, and hence of classᏯ00. Thus, a contraction not inᏯ00 is necessarily nonstrict (i.e., if T ∉Ꮿ00, then T =1). In particular, contractions inᏯ1·or inᏯ·1are nonstrict.

2. Proper contractions. An operatorT is a proper contraction ifT x<xfor every nonzerox in Ᏼor, equivalently, if T^∗T < I. The terms “strict” and “proper”

contractions are sometimes interchanged in current literature. We adopt the termi- nology of [7, page 82] for strict contraction. Obviously, every strict contraction is a proper contraction, every proper contraction is a contraction, and the converses fail:

any isometry is a contraction but not a proper contraction, and the diagonal operator T =diag{(k+1)(k+2)⁻¹}^∞_k=0is a proper contraction on₊² but not a strict contraction. Thus, proper contractions comprise a class of operators that is properly included in the class of all contractions and properly includes the class of all strict contractions. IfT is a proper contraction, then so isT^∗T (reason:STis a proper contraction wheneverSis a contraction andT is a proper contraction). Thus, the point spectrum σP(T^∗T )lies in the open unit disc. If, in addition,T is compact, then so isT^∗T and hence its spectrumσ (T^∗T ), which is always closed, also lies in the open unit disc (for σ (K)\{0} =σP(K)\{0}wheneverKis compact). This implies that the spectral radius r (T^∗T )is less than one. Therefore,T²=r (T^∗T ) <1.

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Conclusion. The concepts of proper and strict contraction coincide for compact operators.

Proper contractions have been investigated in connection with unitary dilations (the minimal unitary dilation of a proper contraction is a bilateral shift whose multiplicity does not exceed the dimension ofᏴ—see [16, page 91]), and also with strong stability of contractive semigroups (cf. [1]). They were further investigated in [15] by considering diﬀerent topologies inᏴ. Here are three basic properties of proper contractions that will be needed in the sequel.

Proposition2.1. Tis a proper contraction if and only ifT^∗is a proper contraction.

Proof. Recall thatT^∗x²= T^∗x;T^∗x = T T^∗x;x ≤ T T^∗xxfor everyx inᏴ, for all operatorsTonᏴ. Take an arbitrary nonzero vectorxinᏴ. IfT^∗x=0, then T^∗x<xtrivially. On the other hand, ifT^∗x=0 andT is a proper contraction, thenT T^∗x<T^∗x =0 so thatT^∗x²<T^∗xx, and henceT^∗x<x.

That is,T^∗is a proper contraction. Dually, sinceT^∗∗=T, it follows thatT is a proper contraction wheneverT^∗is.

IfS is a contraction andT is a proper contraction, thenSTis a proper contraction (as we have already seen above) and so isS^∗T^∗byProposition 2.1. Another application ofProposition 2.1ensures thatT S=(S^∗T^∗)^∗ is still a proper contraction. Summing up:left or right product of a contraction and a proper contraction is again a proper contraction.

Proposition2.2. Every proper contraction is weakly stable.

Proof. IfT x<xfor every nonzeroxinᏴ, thenT is completely nonisomet- ric (i.e., there is no nonzero reducing subspaceᏹforT such thatTⁿx = xfor everyx∈ᏹ^{and every}ⁿ≥1), and therefore completely nonunitary. But a completely nonunitary contraction is weakly stable. In fact, the Foguel decomposition for contractions says that every contraction is the direct sum of a weakly stable contraction and a unitary operator (cf. [6, page 55] or [8, page 106]).

The converse ofProposition 2.2fails: shifts are weakly stable isometries. However, as it was raised in [1],a proper contraction is not necessarily strongly stable. Indeed, ifT is the weighted unilateral shiftT=shift{(k+1)^1/2(k+2)⁻¹(k+3)^1/2}^∞_k=0on²₊, which is a proper contraction because(k+1)(k+2)⁻²(k+3) <1 for everyk≥0, thenAis the diagonal operatorA=diag{(k+1)(k+2)⁻¹}^∞k=0=O(cf. [10] or [8, pages 51, 52]) so thatT is not strongly stable. As a matter of fact,ᏺ^(A)= {0}and (as it is read- ily veriﬁed)A_∗ =O. HenceT is a proper contraction of class Ꮿ10. The converse is much simpler:strongly stable contractions are not necessarily proper contractions. For instance, a backward unilateral shiftS₊^∗ is a strongly stable co-isometry (in fact, an operator is a strongly stable co-isometry if and only if it is a backward unilateral shift).

ThusS₊^∗is a strongly stable contraction but not a proper contraction (it is a nonproper contraction of classᏯ01). Actually, evenaᏯ00-contraction is not necessarily a proper contraction. For example, the weighted bilateral shift T =shift{(|k| +1)⁻¹}^∞_k=−∞ on ² is a contraction of class Ꮿ00 (reason: n

k=0(|k| +1)⁻¹ =(n!)⁻¹→ 0 asn → ∞, which means that both products _∞

k=0(|k| +1)⁻¹ and 0

k=−∞(|k| +1)⁻¹ diverge to

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0—see [3, page 181]) but not a proper contraction because(|k| +1)⁻¹=1 fork=0.

It is worth noticing that the weighted bilateral shiftT=shift{1−(|k|+2)⁻²}^∞k=−∞on ²is a proper contraction of classᏯ11. Indeed, 0<1−(|k|+2)⁻²<1 for each integer k, and both products_∞

k=0(1−(|k|+2)⁻²)and0

k=−∞(1−(|k|+2)⁻²)do not diverge to 0 (cf. [3, page 181] again)—these products converge once the series_∞

k=0(|k|+2)⁻² converges.

Proposition2.3. IfTis a proper contraction, thenAis a proper contraction.

Proof. LetT be a proper contraction and take an arbitrary nonzero vectorxin Ᏼ. IfT^mx=0 for some m≥1, thenTⁿx=0 for every integer n≥m. If Tⁿx=0 for every integern≥1, thenTⁿ⁺¹x = T Tⁿx<Tⁿx<xso that{Tⁿx}n≥1

is a strictly decreasing sequence of positive numbers. In the former case T is trivially strongly stable so thatA=O is a trivial proper contraction. In the latter case {Tⁿx}n≥1converges in the real line toA^1/2xso thatA^1/2x<x. ThusAx ≤ A^1/2x<x.

A backward unilateral shift shows that the converse ofProposition 2.3 does not hold true as well (i.e.,there exist nonproper contractions T for whichA is a proper contraction).

3. Invariant subspaces. A subspaceᏹofᏴis a closed linear manifold ofᏴ.ᏹis nontrivial if{0} =ᏹ=Ᏼ^{. If}^T is an operator onᏴând^{T (}ᏹ⁾⊆ᏹ^{, then}ᏹis invariant for T (orᏹ is T-invariant). If ᏹ is a nontrivial invariant subspace forT, then its orthogonal complementᏹ^⊥is a nontrivial invariant subspace forT^∗. Ifᏹis invariant for bothT andT^∗ (equivalently, if bothᏹândᏹ^⊥ âre^T-invariant), thenᏹ^reduces T. A classical open question in operator theory is:does a contraction not inᏯ00have a nontrivial invariant subspace? Although this is still an unsolved problem we know that the following result holds true.

Lemma3.1. If a contraction has no nontrivial invariant subspace, then it is either a Ꮿ00, aᏯ01, or aᏯ10-contraction.

Proof. See, for instance, [8, page 71].

The class of contractionsTfor whichAis a projection was investigated in [4,10]. It coincides with the class of all contractionsT that commute withA; that is,A=A²if and only ifAT=T A(cf. [4]). Equivalently,ᏺ(A−A²)=Ᏼif and only ifᏺ(AT−T A)=Ᏼ. The next proposition extends this equivalence.

Proposition3.2. ᏺ(A−A²)is the largest subspace ofᏴthat is included inᏺ(AT− T A)and isT-invariant.

Proof. See [10] (or [8, page 52]).

We will say thatAis completely nonprojective if Ax=A²x for every nonzero x in Ᏼ(i.e., if ᏺ(A−A²)= {0}). Sinceᏺ(A−A²) reduces the selfadjoint operatorA, this means that no nonzero direct summand ofAis a projection. IfAis completely nonprojective, thenT is aᏯ1·-contraction (forᏺ(A)⊆ᏺ(A−A²)).

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Lemma3.3. If a contractionThas no nontrivial invariant subspace, then eitherT is strongly stable orAis a completely nonprojective nonstrict proper contraction.

Proof. Suppose thatT is a contraction without a nontrivial invariant subspace.

Sinceᏺ(A−A²) is an invariant subspace forT (byProposition 3.2), it follows that eitherᏺ(A−A²)=Ᏼor ᏺ(A−A²)= {0}. In the former case Ais a projection (i.e., A=A²). However, as it was shown in [10], ifAis a projection thenT is the direct sum of a strongly stable contractionG, a unilateral shiftS₊, and a unitary operator U, where any of the direct summands of the decomposition

T=G⊕S₊⊕U (3.1)

may be missing (see also [8, page 83]). ButT has no nontrivial invariant subspace so thatT=G. That is,T is a strongly stable contraction, forS₊andUclearly have nontrivial invariant subspaces (isometries have nontrivial invariant subspaces). In the latter caseAis a completely nonprojective proper contraction. Indeed,{x∈Ᏼ:Ax = x} =ᏺ(I−A)⊆ᏺ(A−A²)= {0}. Finally, the contractionAis not strict (i.e.,A =1) wheneverT is not strongly stable (i.e., wheneverA=O).

Another classical open question in operator theory is:does a hyponormal operator have a nontrivial invariant subspace?Recall that an operatorT onᏴis hyponormal if T T^∗≤T^∗T (equivalently, ifT^∗x ≤ T xfor everyxinᏴ), andT is cohyponormal ifT^∗is hyponormal. Here is a consequence of Lemmas3.1and3.3for hyponormal contractions. It uses the fact that a cohyponormal contractionT is such thatAis a projection. This implies that a completely nonunitary cohyponormal contraction is strongly stable (cf. [9,12,14]).

Theorem3.4. If a hyponormal contractionT has no nontrivial invariant subspace, then it is either a Ꮿ00-contraction or aᏯ10-contraction for whichA is a completely nonprojective nonstrict proper contraction.

Proof. IfT has no nontrivial invariant subspace, thenT^∗has no nontrivial invariant subspace. IfT is a contraction, then Lemmas3.1and3.3ensure that eitherA= A_∗=O,A=OandA_∗is a completely nonprojective nonstrict proper contraction, orA is a completely nonprojective nonstrict proper contraction andA_∗=O. However, ifT is hyponormal, thenA_∗is a projection [9] so thatA_∗=O(see also [8, page 78]).

Can the conclusion inTheorem 3.4be sharpened toT∈Ꮿ00? In other words, does a hyponormal contraction not inᏯ00have a nontrivial invariant subspace?The question has an aﬃrmative answer if we replace “Ꮿ00-contraction” with “proper contraction.”

That is,if a hyponormal contraction is not a proper contraction, then it has a nontrivial invariant subspace. This will be proved inTheorem 3.6below, but ﬁrst we consider the following auxiliary result. LetDdenote the self-commutator ofT; that is,

D=

T^∗, T =T^∗T−T T^∗. (3.2)

Thus, a hyponormal is precisely an operatorTfor whichDis nonnegative (i.e.,D≥O).

Proposition3.5. IfT is a hyponormal contraction, thenDis a contraction whose power sequence converges strongly. IfPis the strong limit of{Dⁿ}n≥1, thenP T=O.

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Proof. Take an arbitraryxinᏴand an arbitrary nonnegative integern. Suppose thatT is hyponormal and letR=D^1/2≥Obe the unique nonnegative square root of D≥O. If, in addition,T is a contraction, then

Dⁿ⁺¹x;x

=Rⁿ⁺¹x²=

DRⁿx;Rⁿx

=T Rⁿx²−T^∗Rⁿx²

≤Rⁿx²−T^∗Rⁿx²≤Rⁿx²

= Dⁿx;x

.

(3.3)

This shows that R (and soD) is a contraction: setn=0 above. It also shows that {Dⁿ}n≥1is a decreasing sequence of nonnegative contractions. Since a bounded mono- tone sequence of selfadjoint operators converges strongly,

Dⁿ ^s→P≥O. (3.4)

Indeed, the strong limitP of{Dⁿ}n≥1is nonnegative, for the set of all nonnegative operators onᏴis weakly (thus strongly) closed. As a matter of fact,P=P²(the weak limit of any weakly convergent power sequence is idempotent) and so P ≥O is a projection. Moreover,

m n=0

T^∗Rⁿx²≤ m n=0

Rⁿx²−Rⁿ⁺¹x²

= x²−R^m+1x²≤ x² (3.5)

for allm≥0 so thatT^∗Rⁿx →0 asn→ ∞. Hence T^∗P x=T^∗lim

n Dⁿx=lim

n T^∗R²ⁿx=0 (3.6)

for everyxinᏴ, and thereforeP T=O(sincePis selfadjoint).

Theorem3.6. If a hyponormal contraction has no nontrivial invariant subspace, then it is a proper contraction and its self-commutator is a strict contraction.

Proof. (a) Take an arbitrary operatorT onᏴand an arbitraryxinᏴ. Note that T^∗T x= T²xif and only ifT x = Tx. (3.7) Indeed, ifT^∗T x= T²x, thenT x²= T^∗T x;x = T²x². Conversely, ifT x = Tx, thenT^∗T x;T²x = T⁴x², and hence

T^∗T x−T²x²=T^∗T x²−2 Re

T^∗T x;T²x

+T⁴x²

=T^∗T x²−T⁴x²≤T^∗T²−T⁴

x²=0. (3.8) Putᏹ= {x∈Ᏼ:T x = Tx} =ᏺ(T²I−T^∗T ), which is a subspace ofᏴ. IfT is hyponormal, thenᏹisT-invariant. In fact, ifT is hyponormal andx∈ᏹ, then

T (T x)≤ TT x =T²x=T^∗T x≤T (T x) (3.9)

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and soT x∈ᏹ(see also [6, page 9]). Now letTbe a hyponormal contraction. IfT<1, then it is trivially a proper contraction. IfT =1 andT has no nontrivial invariant subspace, thenᏹ= {x∈Ᏼ:T x = x} = {0}(actually, ifᏹ=Ᏼ, then T is an isometry, and isometries have invariant subspaces). HenceT is a proper contraction.

(b) LetD≥O be the self-commutator of a hyponormal contraction T and letP be the strong limit of{Dⁿ}n≥1so thatP T =O (cf.Proposition 3.5). Suppose T has no nontrivial invariant subspace. Sinceᏺ(P )is a nonzero invariant subspace forT wheneverP T =O and T =O, it follows thatᏺ(P )=Ᏼ. HenceP =O and soD is strongly stable (D^{n s}→O). Moreover, since

{Tⁿx}n≥0is a nonzero invariant subspace forT wheneverx=0, it follows that

{Tⁿx}n≥0=Ᏼfor eachx=0 (every nonzero vector inᏴis a cyclic vector forT). Thus the Berger-Shaw theorem (see, for instance, [2, page 152]) ensures thatDis a trace-class operator so thatDis compact (i.e.,T is essentially normal). But for compact operators strong stability coincides with uniform stability, and uniform stability always means spectral radius less than one. Hence the nonnegativeDis a strict contraction because it is clearly normaloid (i.e.,D = r (D) < 1).

Remark 3.7. According to the Berger-Shaw theorem, a hyponormal contraction without a nontrivial invariant subspace has a trace-class self-commutatorDwith trace- normD1≤1. IfD=Ois not a rank-one operator, thenD<D1≤1. The above argument ensures the inequalityD<1 whenever a hyponormal contraction has no nontrivial invariant subspace, including the case of a hyponormal contraction with a rank-one self-commutator.

An operator is seminormal if it is hyponormal or cohyponormal. Recall thatT^∗has a nontrivial invariant subspace if and only ifT has,T^∗is a proper contraction if and only ifT is (Proposition 2.1), and[T , T^∗]= −[T^∗, T ]. Thus, the above theorem also holds for cohyponormal contractions.If a seminormal contraction has no nontrivial invariant subspace, then it is a proper contraction and its self-commutator is a strict contraction. This prompts the question: can we drop “hyponormal” from the theorem statement? In particular,is it true that every nonproper contraction has a nontrivial invariant subspace? Theorems3.4and3.6yield the following result.

Corollary3.8. If a hyponormal contractionThas no nontrivial invariant subspace, then it is either a proper contraction of class Ꮿ00 or a nonstrict proper contraction of classᏯ10 for which A is a completely nonprojective nonstrict proper contraction.

Moreover, its self-commutator[T^∗, T ]is a strict contraction.

Acknowledgement. This work was supported in part by CNPq-Brazilian National Research Council.

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C. S. Kubrusly: Catholic University of Rio de Janeiro,22453-900Rio de Janeiro, RJ, Brazil

E-mail address:[email protected]

N. Levan: University of California at Los Angeles, Los Angeles, CA90024-1594, USA E-mail address:[email protected]