Invariant subspaces and Deddens algebras

INVARIANT SUBSPACES AND DEDDENS ALGEBRAS MIGUEL LACRUZ Abstract. It is shown that if the Deddens algebra DTassociated with a quasinilpotent operator Ton a complex Banach space is closed and localizing then Thas a nontrivial closed hyperinvariant subspace. We shall represent by B(E) the algebra of all bounded linear operators on a complex Banach space E. Recall that the commutant of an operator T∈ B(E) is the subalgebra {T}0⊆ B(E) of all operators that commute with T. A subspace F⊆Eis said to be invariant under an operator T∈ B(E) provided that TF ⊆F. A subspace F⊆Eis said to be invariant under a subalgebra R ⊆ B(E) provided that Fis invariant under every R∈ R. A subspace F⊆Eis said to be hyperinvariant under an operator T∈ B(E) provided that Fis invariant under the commutant {T}0. A subalgebra R ⊆ B(E) is said to be transitive provided that the only closed subspaces invariant under Rare the trivial ones, that is, F={0}and F=E. As it turns out, this is equivalent to saying that for every x∈E\{0},the subspace {Rx:R∈ R} is dense in E. Recall that an operator T∈ B(E) is said to be quasinilpotent provided that σ(T) = {0}.According with the spectral radius formula, Tis quasinilpotent if and only if r(T) = lim n→∞ kTnk1/n = 0.(1) Let T∈ B(E) and consider the Deddens algebra DTassociated with T, that is, the family of those operators X∈ B(E) for which there is a constant M > 0 such that for every n∈Nand for every f∈E, kTnXfk ≤ MkTnfk.(2) When Tis invertible this is equivalent to saying that sup n∈N kTnXT −nk<∞.(3) It is easy to see that DTis indeed a unital subalgebra of B(E) with the nice property that {T}0⊆ DT. Also, DT=B(E) in case Tis an isometry. These algebras are named after Deddens because he first introduced them in the 1970s in the context of nest algebras [4]. The description of Deddens algebras associated with some special classes of operators has been recently obtained by Petrovic [14, 15]. Let T∈ B(E).A complex scalar λis said to be an extended eigenvalue for Tprovided that there exists a nonzero operator X∈ B(E) such that T X =λXT. Such an operator is called an extended eigenoperator for Tcorresponding to the extended eigenvalue λ. These notions became popular back in the 1970s when searching for invariant subspaces. Recently, the concepts of extended eigenvalue and extended eigenoperator have received a considerable amount of attention, both in the context of invariant subspaces [8, 9] and in the study of extended eigenvalues and extended eigenoperators for some special classes of operators [1, 2, 5, 12, 13, 14]. Date: March 21, 2014. 2010 Mathematics Subject Classification. Primary 47A15; Secondary 47L10 . Key words and phrases. Deddens algebra; Extended eigenvalue; Invariant subspace; Localizing algebra. This research was partially supported by Ministerio de Ministerio de Econom´ıa y Competitividad under grant MTM 2012-30748, and by Junta de Andaluc´ıa under grant FQM-3737. 1 arXiv:1403.5093v1 [math.FA] 20 Mar 2014 2 MIGUEL LACRUZ Let ET(λ) denote the set of extended eigenoperators of Tassociated with an extended eigenvalue λand let ETdenote the union of the sets ET(λ) when λruns through all the extended eigenvalues for Twith |λ| ≤ 1.It is easy to see that {T}0⊆ ET⊆ DT.Both inclusions may be proper; for instance, Petrovic [14] showed that if Wis an injective unilateral shift on a Hilbert space then boths inclusions {W}0⊂ EWand EW⊂ DWare proper. A subspace X ⊆ B(E) is said to be localizing provided that there is a closed ball B⊆Esuch that 0/∈Band such that for every sequence of vectors (fn) in Bthere is a subsequence (fnj) and a sequence of operators (Xj) in Xsuch that kXjk ≤ 1 and such that the sequence (Xjfnj) converges in norm to some nonzero vector. This notion was introduced by Lomonosov, Radjavi, and Troitsky [11] as a side condition to build invariant subspaces. A typical example of a localizing algebra is a subalgebra R ⊆ B(E) such that the closure in the weak operator topology of the unit ball of Rcontains a nonzero compact operator. Rodr´ıguez-Piazza and the author studied some properties of localizing algebras in a recent paper [8]. They also obtained a result on the existence of invariant subspaces that extends and unifies previous results of Scott Brown [3] and Kim, Moore and Pearcy [6], on the one hand, and Lomonosov, Radjavi and Troitsky [11], on the other hand. The result goes as follows. Theorem 1. Let T∈ B(E)be a nonzero operator, let λ∈Cbe an extended eigenvalue of Tsuch that the subspace ET(λ)of all associated extended eigenoperators is localizing and suppose that either (1) |λ| 6= 1,or (2) |λ|= 1 and Tis quasinilpotent. Then Thas a nontrivial closed hyperinvariant subspace. The aim of this note is to provide an extension of part (2) in Theorem 1 by replacing the assumption that the subspace ET(λ) be localizing with the assumption that the Deddens algebra DTbe closed and localizing. Our main result can be stated as follows. Theorem 2. Let T∈ B(E)be a nonzero quasinilpotent operator. If the Deddens algebra DTis closed and localizing then Thas a nontrivial closed hyperinvariant subspace. Notice that, under the assumption that DTbe closed, part (2) of Theorem 1 is a consequence of Theorem 2 since ET(λ)⊆ DT,and that Theorem 2 is strictly more general than part (2) of Theorem 1, because the inclusion ET⊆ DTis proper in general. Let us start with a general result about Deddens algebras before we proceed with the proof of Theorem 2. This result characterizes when the Deddens algebra DTis closed in the operator norm. The corresponding result for spectral radius algebras was obtained by Lambert and Petrovic [7]. Lemma 3. Let T∈ B(E).The following conditions are equivalent: (1) The Deddens algebra DTis closed in the operator norm topology. (2) There is a constant M > 0such that for every X∈ DT,for all n∈Nand for all f∈Ewe have kTnXfk ≤ MkXk·kTnfk(4) Proof of Lemma 3. Suppose DTis closed and consider for every k∈Nthe closed set Fkof those operators X∈ DTthat satisfy the inequality kTnXfk ≤ kkTnfkfor all n∈Nand for all f∈E. Then we have DT=[ k∈N Fk. It follows from Baire’s theorem that there is some k0∈Nsuch that Fk0has nonempty interior, that is, there is some X0∈ Fk0and there is some ε > 0 such that {X∈ DT:kX−X0k ≤ ε}⊆Fk0.Let Y∈ DT INVARIANT SUBSPACES AND DEDDENS ALGEBRAS 3 such that kYk ≤ 1 and let X=X0+εY. Then we have εkTnY fk ≤ kTnX0fk+kTnXfk ≤ 2k0kTnfk. Finally, we conclude that for every X∈ DT,for all n∈Nand for all f∈ DTwe have kTnXfk ≤ 2k0 εkXk·kTnfk. The converse is trivial because if such a constant M > 0 exists then DT=\ n∈N \ f∈E {X∈ B(E): kTnXfk ≤ MkXk·kTnfk}. so that DTis closed since it is the intersection of a family of closed sets.  An easy proof of the nontrivial part of Lemma 3 can be obtained from the uniform boundedness principle in the special case that Tis an invertible operator. The proof goes as follows. Proof of Lemma 3 when Tis invertible. Consider the operator Φ: DT→ DTdefined by the expression Φ(X) = TXT −1for all X∈ DT.Notice that Φn(X) = TnXT−nfor all n∈N,so that supnkΦn(X)k<∞. Now it follows from the uniform boundedness principle that supnkΦnk<∞.This means that there is a constant M > 0 such that kTnXT−nk ≤ MkXkfor all n∈Nand for all X∈ DT.Therefore kTnXT −ngk ≤ MkXk·kgkfor all g∈E, and taking g=Tnfwe get kTnXfk ≤ MkXk·kTnfk. The key for the proof of Theorem 2 is a lemma that we have extracted from the proof of Theorem 2.3 in the paper of Lomonosov, Radjavi and Troitsky [11]. This lemma can be stated as follows. Lemma 4. Let T∈ B(E)be a nonzero operator such that {T}0is a transitive algebra, let R⊆B(E)be a localizing algebra such that {T}0⊆ R,and let B⊆Ebe a closed ball that makes Ra localizing algebra. There is a constant c > 0such that for every f∈Bthere is an X∈ R such that TXf ∈Band kXk ≤ c. Proof of Lemma 4. Assume the commutant {T}0is a transitive algebra. Since the closed subspace ker T is invariant under {T}0and since T6= 0,we must have ker T={0},so that Tis injective. We ought to show that there is some constant c > 0 such that for every f∈Bthere is an X∈ R such that kXk ≤ c and TXf ∈B. We proceed by contradiction. Otherwise, for every n∈Nthere is an fn∈Bsuch that the condition X∈ R and TXfn∈Bimplies kXk> n. Since Ris localizing, there is a subsequence (fnj) and there is a sequence (Xj) in Rwith kXjk ≤ 1,and such that (Xjfnj) converges in norm to some nonzero vector f∈E. Therefore, (TXjfnj) converges in norm to Tf. Since Tis injective, Tf 6= 0,and since {T}0is transitive, there is an R∈ {T}0such that RTf ∈int B. Hence, there is some j0≥1 such that RTXjfnj∈Bfor all j≥j0.Since RT =T R, we have T RXjfnj∈Bfor all j≥j0.Since RXj∈ R, the choice of the sequence (fn) implies kRXjk> njfor all j≥j0.Finally, this leads to a contradiction, because kRXjk≤kRkfor all j≥1.This completes the proof of Lemma 4.  The technique for the proof of Theorem 2 is an iterative procedure that is reminiscent of an argument at the end of the proof in Hilden’s simplification for the striking theorem of Lomonosov [10] that a nonzero compact operator on a complex Banach space has a nontrivial hyperinvariant subspace. We recommend the book of Rudin [16] for an exposition of this argument. Proof of Theorem 2. Start with any vector f0∈Band use Lemma 4 to choose an operator X1∈ DTsuch that kX1k ≤ cand such that TX1f0∈B. Now use again Lemma 4 to choose another operator X2∈ DT such that kX2k ≤ cand TX2TX1f0∈B. Continue this ping pong game to obtain a sequence of vectors fn∈Band a sequence of operators Xn∈ DTsuch that kXnk ≤ cand such that fn=T Xn· · · T X1f0. Now apply Lemma 3 to find a constant M > 0 such that kTnXfk ≤ MkXk · kTnfkfor every X∈ DT, for all n∈Nand for all f∈H. Notice that kf1k=kTX1f0k ≤ cMkTf0k.Also, notice that kf2k=kTX2TX1f0k ≤ cMkT2X1f0k ≤ (cM)2kT2f0k, 4 MIGUEL LACRUZ and in general kfnk ≤ (cM)nkTnf0k.Let d= min{kxk:x∈B}.It is plain that d > 0 because 0 /∈B. Then, for all n∈Nwe have 0 < d ≤ kfnk ≤ (cM)nkTnf0k,and this gives information on the spectral radius of T, namely, r(T) = lim n→∞ kTnk1/n ≥1 cM >0. We arrived at a contradiction because Tis quasinilpotent. This completes the proof of Theorem 2.  References [1] A. Biswas, A. Lambert, and S. Petrovic, Extended eigenvalues and the Volterra operator, Glasg. Math. J. 44 (2002), 521–534, MR 1956558. [2] A. Biswas and S. Petrovic, On extended eigenvalues of operators, Integr. Equ. Oper. 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Lacruz, Facultad de Matem´ aticas, Universidad de Sevilla, Avenida Reina Mercedes, 41012 Seville (Spain) E-mail address:[email protected]