c 2009 by Institut Mittag-Leffler. All rights reserved
Maximal invariant subspaces for a class of operators
Kunyu Guo, Wei He and Shengzhao Hou
Abstract. In this note, we characterize maximal invariant subspaces for a class of opera- tors. LetT be a Fredholm operator and 1−T T∗∈Spfor somep≥1. It is shown that ifM is an invariant subspace forT such that dimMT M <∞, then every maximal invariant subspace ofM is of codimension 1 inM. As an immediate consequence, we obtain that ifM is a shift invariant subspace of the Bergman space and dimMzM <∞, then every maximal invariant subspace of Mis of codimension 1 inM. We also apply the result to translation operators and their invariant subspaces.
1. Introduction
LetHbe a separable Hilbert space, and letT∈B(H), the set of bounded linear operators acting onH. A subspace M of His calledinvariant forT if it is closed andT M⊆M. The well-known invariant subspace problem is the following.
Question 1. Does every bounded linear operator on a separable Hilbert space have a nontrivial invariant subspace?
The invariant subspace problem has an equivalent form on the Bergman space.
Recall that the Bergman spaceL2a(D) over the unit diskDis the closed subspace of L2(D, dA) consisting of analytic functions, wheredAis the normalized area measure on D. Let Mz be the Bergman shift, i.e., the operator of multiplication by the coordinate functionz. The invariant subspace problem is equivalent to the following question (cf. [2] and [8]).
Question 2. For twoMz-invariant subspacesM andN of the Bergman space L2a(D) with N⊆M and dimMN≥2, is there another Mz-invariant subspace L such thatNLM?
Using methods from function theory and operator theory, Hedenmalm [6] got an affirmative answer to Question 2in the case M=L2a(D), by showing that every maximal invariant subspace ofL2a(D) is of codimension 1 and of the form
Mα={f∈L2a(D) :f(α) = 0}, α∈D.
Using completely different methods from classical operator theory, Trent [10] ob- tained the same result and generalized that result to theCn-valued Bergman space.
The result was also obtained in Atzmon’s paper [1].
In this note, enlightened by [6] and [10], we will study the invariant subspace problem in a much more general setting. Let T∈B(H) and let M be an invariant subspace forT. A T-invariant subspace N is called amaximal invariant subspace ofM, ifNM and there is noT-invariant subspaceLsuch that NLM.
One motivation for studying maximal invariant subspaces lies in the analogy between maximal invariant subspaces and maximal ideals of commutative Banach algebras in the Gelfand theory. Another motivation is that, if for any invariant subspace M for the Bergman shiftMz, the maximal invariant subspaces ofM are of codimension 1 inM, then Question2 will be answered affirmatively.
Our main result is as follows.
Theorem 1.1. Suppose T is a Fredholm operator and 1−T T∗∈Sp for some p≥1. If M is an invariant subspace for T such that dimMT M <∞, then every maximal invariant subspace of M is of codimension 1 inM.
Applying the above theorem to the Bergman shift Mz, we get an affirmative answer to Question2in the case dimMzM <∞. This covers the result of Heden- malm [6]. In a sense, our result provides weak support for the invariant subspace problem. Obviously, the case dimMzM=∞is the obstacle to the invariant sub- space problem.
In Section 2, we will give the proof of the main theorem. It is worth mentioning that the proof will make essential use of a classical result from operator theory (see Lemma2.2below), which was brought into our sight by Trent’s paper [10]. We also use some techniques in Trent’s paper [10].
In Section 3, we will apply the main theorem to concrete operators, which are closely related to the invariant subspace problem.
2. Proof of the main theorem
In this section, we will give the proof of the main theorem. To this end, we first establish some lemmas.
Forp>0, we use the notationSpto denote the set of Schatten-pclass operators.
Recall that an operatorT∈Sp if and only if the trace of (T∗T)p/2 is finite.
Lemma 2.1. If R is Fredholm and 1−R∗R∈Sp, then 1−RR∗∈Sp.
Proof. Since R is Fredholm, there is an operator Q∈B(H) and a finite-rank operatorF such thatRQ=1+F. Then
R(1−R∗R)Q= (1−RR∗)RQ
= (1−RR∗)(1+F)
= 1−RR∗+(1−RR∗)F.
As bothR(1−R∗R)Qand (1−RR∗)F are in Sp, we have 1−RR∗∈Sp. Lemma 2.2.([9], p. 107) SupposeT∈B(H) and 1−T T∗∈Sp for some p≥1, thenT has a nontrivial invariant subspace.
In the following, we use the notationPM to denote the orthogonal projection formHto the closed subspaceM.
Lemma 2.3. SupposeT is Fredholm and1−T T∗∈Spfor somep≥1. IfM and N are two invariant subspaces forT such thatdimMN≥2anddimMT M <∞, then there is an invariant subspaceL forT such that NLM.
Proof. Consider the operatorS=PMNT|MN. To prove the lemma, we first prove the following claim.
Claim 1. There is an invariant subspace L forT such that NLM if and only if there is an invariant subspaceL0 forS such that 0L0MN.
In fact, ifLis an invariant subspace forT such thatNLM, putL0=LN. Then
SL0=PMNT|MN(LN)⊆LN=L0.
Conversely, ifL0 is an invariant subspace forS such that 0L0MN, put L=
L0+N. SincePMNT|MNL0⊆L0, we havePN⊥T L0⊆L0. Hence T L0=PNT L0+PN⊥T L0⊆N+L0=L, and then
T L=T L0+T N⊆L.
By Claim1, it suffices to show that the operator S has a nontrivial invariant subspace. By Lemma2.2, the proof will be accomplished if the following claim is true.
Claim 2. 1−SS∗∈Sp.
LetR=T|M. To prove the claim, we first show that 1−RR∗∈Sp.
Since T is Fredholm and 1−T T∗∈Sp, we have 1−T∗T∈Sp by Lemma 2.1.
Thus
1−R∗R= 1−PMT∗T PM=PM(1−T∗T)PM∈ Sp. Since kerR⊆kerT, we have dim kerR<∞and
dim cokerR= dimMRM= dimMT M <∞. SoRis Fredholm. Using Lemma2.1again, we have 1−RR∗∈Sp.
SinceT∗N⊥⊆N⊥, we have
SS∗=PMNT PMNT∗PMN
=PMNT(PM−PN)T∗PMN
=PMNT PMT∗PMN−PMNT PNT∗PMN
=PMNT PMT∗PMN. Then
1−SS∗=PMN(1−T PMT∗)PMN
=PMN(1−(T PM)(T PM)∗)PMN
=PMN(1−RR∗)PMN.
As we have proved that 1−RR∗∈Sp, we obtain that 1−SS∗∈Sp. So Claim 2 is true and the proof is complete.
Now the main theorem is a direct consequence of the above lemma. For the convenience of the reader, we restate it here.
Theorem 2.4. Suppose T is a Fredholm operator and 1−T T∗∈Sp for some p≥1. If M is an invariant subspace for T such that dimMT M <∞, then every maximal invariant subspace of M is of codimension 1 inM.
Remark 2.5. In Theorem 2.4, if the conditions on the operator T is replaced by 1−T∗T∈Spfor somep≥1, andT has a finite generating set, then the conclusion still holds. In fact, the assumption that 1−T∗T∈Sp implies that the range of T∗ is closed and has finite codimension. Thus T is left semi-Fredholm. Moreover, the fact that T has a finite generating set implies that the range of T has finite codimension. Therefore,T is Fredholm. This reasoning is essentially due to [3]. By Lemma 2.1, 1−T T∗∈Sp. Thus the conditions on T in Theorem 2.4 are satisfied and the conclusion follows.
3. Applications
In this section, we will apply the main theorem to concrete operators on sepa- rable Hilbert spaces, which are closely related to the invariant subspace problem.
Example 3.1. Consider the Bergman shiftMz. As mentioned in the introduc- tion, the invariant subspace problem is equivalent to Question2about the invariant subspaces for the Bergman shiftMz. Now let us see what information we can get from the main theorem.
Obviously,Mzis Fredholm. Leten=zn/zn. Then{en}∞n=0is an orthonormal basis ofL2a(D). It is easy to verify that
1−MzMz∗=
∞
n=0
1
n+1en⊗en,
so 1−MzMz∗∈Sp for any p>1. Hence, if M is an invariant subspace for Mz such that dimMzM <∞, then by Theorem1.1, every maximal invariant subspace of M is of codimension 1 in M. This answers Question 2 affirmatively in the case dimMzM <∞.
Given an invariant subspaceM ofL2a(D) such that dimMzM <∞, what do the maximal invariant subspaces ofM look like? To discuss this, we first need a fact. Let ϕλ(z)=(z−λ)/(1−λz) be the M¨¯ obius transform. If M is an invariant subspace such that dimMzM <∞, then
dimMϕλM= dimMzM for any λ∈D.
In fact, since dimMzM <∞, σe(Mz|M)=∂D (cf. [11]). Hence for any λ∈D, λ−Mz is Fredholm. Thus
dimMϕλM=−Index(λ−Mz) =−IndexMz= dimMzM.
Now consider the case dimMzM=1. By the above fact, for anyλ∈D,ϕλM is an invariant subspace of codimension 1 inM, and thus maximal inM. In fact, there are no other forms of maximal invariant subspaces ofM, which is the following theorem.
Theorem 3.2. Let M be an invariant subspace of the Bergman space L2a(D) such that dimMzM=1. Then all the maximal invariant subspaces of M are of the formϕλM for someλ∈D.
Proof. Let N be a maximal invariant subspace of M. By Theorem 1.1,N is of codimension 1 inM. This implies that there is a unit vectore in M such that M=N⊕Ce. Define a linear functional ψon the disk algebraA(D) by
ψ(f) =f e, e forf∈A(D).
We claim that ψ is multiplicative. To see this, note that for g∈A(D), ge∈M. Thusgecan be decomposed asge=ge−ge, ee+ge, ee, wherege−ge, ee∈Nand ge, ee⊥N. SinceN is invariant for Mz, we havef(ge−ge, ee)∈N forf∈A(D).
Hence
ψ(f g) =f ge, e
=f(ge−ge, ee+ge, ee), e
=f(ge−ge, ee), e+f e, ege, e
=f e, ege, e
=ψ(f)ψ(g).
Asψis multiplicative, there is aλ0∈Dsuch that ψ(f) =f e, e=f(λ0).
We claim thatλ0∈D. To complete the proof, suppose that |λ0|=1. Put B={f∈A(D) :f(λ0) = 0}.
For anyf∈B, we have
f e, e=
D
f|e|2dA=f(λ0) = 0.
SinceB is weak-star dense inH∞(D), we have
D
f|e|2dA= 0 for anyf∈H∞(D), and hence
D|e|2dA=0, a contradiction. Therefore, we haveλ0∈Das desired.
As ϕλ0e, e=ϕλ0(λ0)=0, we have ϕλ0e⊥e, and thus ϕλ0e∈N. Hence ϕλ0M=ϕλ0(N⊕Ce)⊆N⊆M.
Noting that as dimMϕλ0M=1 and N is maximal in M, we have N=ϕλ0M, completing the proof.
IfM=L2a(D), the above theorem is Hedenmalm’s result [6].
Now consider the case dimMzM=n>1. By the fact preceding Theorem3.2, for anyλ∈D, dimMϕλM=n, and thusϕλM is not maximal inM. Givenλ∈D, for anyf∈M, there is a nonnegative integernf such thatf=ϕnλfgfor some analytic functiong withg(λ)=0. Let
n0= inf{nf:f∈M}. (3.1)
ThenM can be written as M=ϕnλ0M. Let
M={f∈M:f(λ) = 0}.
Then ϕnλ0M is of codimension 1 in M, and thus maximal inM. But unlike the case dimMzM=1, these are not all the maximal invariant subspaces ofM. Here is an example.
Example 3.3. For any positive integer n, there is a sampling sequence A for L2a(D) and a decomposition of it as a finite disjoint union A=n
j=1Aj, with the property that each A\Aj is an interpolating sequence for L2a(D) (cf. [5] or [7, Lemma 6.3]). This can be used to construct an invariant subspace I such that dimIzI=n. Assume without loss of generality that 0∈/A, for if 0∈A, we can replaceAwithϕλ(A) for someλ /∈A.
For 1≤j≤n, let Ij be the set of functions in L2a(D) vanishing onA\Aj, then Ij is a zero-based invariant subspace. Let Gj be the extremal function of Ij, by [7, Theorem 3.31 and Corollary 6.16], we haveGj(0)=0 andIj=[Gj], the invari- ant subspace generated by Gj. Let I=I1∨...∨In, then I is an invariant subspace with the property dimIzI=n[7, Theorem 6.4].
LetN=zI∨CG2∨...∨CGn. SinceI=zI∨CG1∨CG2∨...∨CGn and G1, ..., Gn
are linearly independent [7, Theorem 6.4], N is of codimension 1 in I and thus maximal inI. Note that the functions inI have no common zeros, son0=0, where n0is as defined in (3.1). We claim thatN is a maximal invariant subspace ofIthat cannot be written as{f∈I:f(λ)=0}for anyλ∈D. In fact, sinceG2(0)=0,N cannot be written as{f∈I:f(0)=0}. Moreover, N cannot be written as {f∈I:f(λ0)=0} for any 0=λ0∈D either, because the functions inzI have no other common zeros except 0.
Now we turn to another operator whose invariant subspaces are also closely related to the invariant subspace problem, and see what the main theorem tells us about this operator.
LetF1 denote the Fock type space F1=
h∈Hol(C) :h2= 1 2π
C|h(z)|2e−2|z|dA(z)<∞
,
wheredA is the area measure over the complex planeC. The translation operator Tb onF1 is defined by
Tbf(z) =f(z+b) forf∈ F1.
LetD=d/dzbe the differential operator. By [4], bothTb andDare bounded onF1. A closed subspace M of F1 is called translation invariant if it is invariant for all the translation operators. SinceTb=ebD, one can verify that a closed subspace M is translation invariant if and only if it is invariant forD [4].
It is shown in [4] that the invariant subspace problem is equivalent to the following question on translation invariant subspaces ofF1.
Question 3. Given two translation invariant subspaces M and N of F1 with N⊆M and dimMN≥2, is there another translation invariant subspace L such thatNLM?
Intuitively, Question3is similar to Question2and we may get similar conclu- sions as in Example3.1. But the following theorem contradicts the intuition.
Theorem 3.4. The space F1 has no maximal translation invariant subspace.
Proof. Suppose thatN is a maximal translation invariant subspace ofF1. This turns out to be equivalent to the fact thatN is a maximal invariant subspace for the differential operator D. Obviously, the operator D is Fredholm. Leten=zn/zn. Then{en}∞n=0 is an orthonormal basis ofF1. A direct computation shows that
1−DD∗=
∞
n=0
1
2n+3en⊗en.
So 1−DD∗∈Spfor anyp>1. Note that dimF1DF1=0<∞, and hence if there ex- ists a maximal invariant subspace, sayN, then by Theorem1.1,N is of codimension 1 inF1, that is, dimN⊥=1.
To reach a contradiction, let F2(D) =
f=
∞
n=0
anzn∈Hol(D) :f2F2(D)=|a0|2+
∞
n=1
|a√n|2 n <∞
.
ThenMz is a bounded multiplication operator defined by the coordinate function.
By [4], there is an invertible operatorS:F1→F2(D) such that SD∗S−1=Mz.
Since N is invariant for D and N=F1, SN⊥ is a nontrivial invariant subspace for Mz, and thus dimSN⊥=∞. This implies that dimN⊥=∞, a contradiction.
Therefore, the space F1 has no maximal translation invariant subspace, which is the desired conclusion.
In view of the above theorem, it is not feasible to study maximal translation invariant subspaces ofF1. However, the following definition is natural in this setting.
Let T∈B(H) and let N be an invariant subspace for T. A T-invariant subspace M is calledminimal overN ifNM and there is noT-invariant subspace Lsuch that NLM. We have the following proposition which is a direct consequence of Theorem1.1.
Proposition 3.5. SupposeT is a Fredholm operator and1−T T∗∈Spfor some p≥1. Let M and N be two T-invariant subspaces. If M is minimal over N and dimN⊥T∗N⊥<∞,then dimMN=1.
Applying the above proposition to the differential operator D on F1, we get that if M and N are two translation invariant subspaces of F1, M is minimal over N and dimN⊥D∗N⊥<∞, then dimMN=1. This answers Question 3 affirmatively in the case dimN⊥D∗N⊥<∞.
Given a translation invariant subspaceN ofF1 with the property that dimN⊥D∗N⊥<∞,
what do the minimal translation invariant subspaces over N look like? Here are two examples.
Example 3.6. The minimal translation invariant subspaces of F1 are exactly those minimal translation invariant subspaces over {0}. By Proposition 3.5, they are all one-dimensional. An easy computation shows that they are of the formCeλz forλ∈D.
Example 3.7. By [4], the space F1 possesses a class of translation invariant subspaces of the form
N= span
∞
k=1
eλkzPnk
, (3.2)
where Λ={λk}∞k=1 is a sequence of distinct points in the unit diskD,{nk}∞k=1 is a sequence of nonnegative integers and Pnk is the set of polynomials of degree less than or equal tonk. SupposeN=F1. We will describe all the minimal translation invariant subspaces overN.
To this end, recall that in the proof of Theorem 3.4, there is an invertible operatorS:F1→F2(D) such that
SD∗S−1=Mz. One can also verify that
f, zleλz=dl(Sf)
dzl (¯λ) forf∈ F1. (3.3)
By the above equality, we can deduce that ifN is a translation invariant subspace of the form (3.2), then
SN⊥={g∈F2(D) :g(¯λk) =...=g(nk)(¯λk) = 0, k= 1,2, ...},
andSN⊥is invariant forMz. Noting thatN=F1, we obtain thatSN⊥is nontrivial.
SinceSN⊥is zero based, we have dimSN⊥MzSN⊥=1. Similar to the proof of Theorem3.2, every maximal invariant subspaceLofSN⊥is of the formϕλSN⊥ for someλ∈D. More precisely, ifλ /∈Λ, then
L={g∈F2(D) :g(¯λ) =g(¯λk) =...=g(nk)(¯λk) = 0, k= 1,2, ...}. Ifλ∈Λ, say λ=λk0 for somek0, then
L={g∈F2(D) :g(nk0+1)(¯λk0) =g(¯λk) =...=g(nk)(¯λk) = 0, k= 1,2, ...}. Applying equality (3.3) again, we get that ifλ /∈Λ, then the minimal translation invariant subspaceM overN is
M= span
∞
k=1
eλkzPnk, eλz
.
Ifλ=λk0 for somek0, the minimal translation invariant subspaceM overN is M= span
∞
k=1
eλkzPnk, eλk0zPnk0+1
.
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Kunyu Guo
School of Mathematical Sciences Fudan University
Shanghai P.R. China
kyguo@fudan.edu.cn Wei He
Department of Mathematics Southeast University Nanjing, Jiangsu P.R. China
051018010@fudan.edu.cn
Shengzhao Hou
Department of Mathematics Suzhou University
Suzhou, Jiangsu P.R. China shou@suda.edu.cn
Received September 23, 2008 published online October 1, 2009
