Applications of fixed point theorems in the theory of invariant subspaces

Espínola and Lacruz Fixed Point Theory and Applications 2012, 2012:197 http://www.fixedpointtheoryandapplications.com/content/2012/1/197 R E S E A R C H Open Access Applications of fixed point theorems in the theory of invariant subspaces Rafa Espínola and Miguel Lacruz* *Correspondence: [email protected] Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Avenida Reina Mercedes, Seville, 41012, Spain Abstract We survey several applications of fixed point theorems in the theory of invariant subspaces. The general idea is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an operator on a Banach space. MSC: 47A15; 47H10 Keywords: invariant subspace; fixed point 1 Introduction One of the most recalcitrant unsolved problems in operator theory is the invariant subspace problem. The question has an easy formulation. Does every operator on an infinite dimensional, separable complex Hilbert space have a nontrivial invariant subspace? Despite the simplicity of its statement, this is a very difficult problem and it has generated a very large amount of literature. We refer the reader to the expository paper of Yadav [] foradetailedaccountofresultsrelatedtotheinvariantsubspaceproblem. In this survey we discuss some applications of fixed point theorems in the theory of invariant subspaces. The general idea is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an operator on a Banach space. In Section we consider the striking theorem of Lomonosov [] about the existence of invariant subspaces for algebras containing compact operators. The proof of this theorem is based on the Schauder fixed point theorem. In Section we present a recent result of Lomonosov, Radjavi, and Troitsky []aboutthe existence of invariant subspaces for localizing algebras. The proof of this result is based on the Ky Fan fixed point theorem for multivalued maps. The idea of using fixed point theorems for multivalued maps in the search for invariant subspaces was first introduced by Androulakis []. In Section we consider an extension of Burnside’s theorem to infinite dimensional Banach spaces. This result is originally due to Lomonosov []. We present a proof of it in a special case that was obtained independently by Scott Brown [] and that once again is based on the Schauder fixed point theorem. In Section we address the existence of invariant subspaces for operators on the Krein space of an indefinite product, and we present a result of Albeverio, Makarov, and Motovilov [] whose proof uses the Banach fixed point theorem. The rest of this section contains some notation, a precise statement of the invariant subspace problem, and a few historical remarks. ©2012 Espínola and Lacruz; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Espínola and Lacruz Fixed Point Theory and Applications 2012, 2012:197 Page 2 of 11 http://www.fixedpointtheoryandapplications.com/content/2012/1/197 Let Ebe an infinite dimensional, complex Banach space, and let B(E) denote the algebra of all bounded linear operators on E.AsubspaceofEis by definition a closed linear manifold in E. AsubspaceM⊆Eis said to be invariant under an operator T∈B(E) provided that TM ⊆M,andasubspaceM⊆Eis said to be invariant under a subalgebra R⊆B(H) provided that Mis invariant under every R∈R.AsubalgebraR⊆B(H)issaidtobe transitive provided that the only subspaces invariant under Rare the trivial ones, M={} and M=E. This is equivalent to saying that the subspace {Rx :R∈R}is dense in Efor each x∈E\{}. The commutant of a set of operators S⊆B(E)isthesubalgebraSof all operators R∈ B(E)suchthatSR =RS for all S∈S.AsubspaceM⊆Eis said to be hyperinvariant under an operator T∈B(E) provided that Mis invariant under {T}. The invariant subspace problem is the question of whether every operator in B(E)hasa nontrivial invariant subspace. This is one of the most important open problems in operator theory. The origin of this question goes back to , when von Neumann proved the unpublished result that any compact operator on a Hilbert space has a nontrivial invariant subspace. Aronszajn and Smith [] extended this result in  to general Banach spaces. Bernstein and Robinson [] used nonstandard analysis to prove in  that every polynomially compact operator on a Hilbert space has a nontrivial invariant subspace. Halmos [] obtained a proof of the same result using classical methods. Lomonosov [] proved in  that any nonscalar operator on a Banach space that commutes with a nonzero compact operator has a nontrivial hyperinvariant subspace. The result of Lomonosov came into the scene like a lightning bolt in a clear sky, generalizing all the previously known results and introducing the use of the Schauder fixed point theorem as a new technique to produce invariant subspaces. Enflo [] constructed in  the first example of an operator on a Banach space without nontrivial invariant subspaces. The example circulated in a preprint form and it did not appear published until , when it was recognized as correct work []. In the meantime, Beauzamy [] simplified the technique, and further examples were given by Read [, ]. Very recently, Argyros and Haydon [] constructed an example of an infinite dimensional, separable Banach space such that every continuous operator is the sum of a compact operator and a scalar operator, so that every operator on it has a nontrivial invariant subspace. However, after so many decades, the question about the existence of invariant subspaces for operators on Hilbert space is still an open problem. 2 Invariant subspaces for algebras containing compact operators We start with a fixed point theorem that is the key to the main result in this section. The use of this result is one of the main ideas in the technique of Lomonosov. We shall denote by conv(S)theclosedconvexhullofasubsetS⊆E. Proposition . [,Proposition] Let E be a Banach space,let C ⊆Ebeaclosedconvex set,and let :C→E be a continuous mapping such that (C)is a relatively compact subset of C.Then there is a point x∈Csuchthat(x)=x. Espínola and Lacruz Fixed Point Theory and Applications 2012, 2012:197 Page 3 of 11 http://www.fixedpointtheoryandapplications.com/content/2012/1/197 Proof Let Qdenote the closure of (C). It follows from a theorem of Mazur that conv(Q) is a compact, convex subset of E, and since Cis closed and convex, we have conv(Q)⊆C. Since (C)⊆Q,wehave(conv(Q)) ⊆Q⊆conv(Q), and now the result follows from the Schauder fixed point theorem.  Theorem . [,Theorem] Let R⊆B(E)be a transitive algebra and let K ∈B(E)be anonzerocompactoperator.Then there is an operator R ∈Rand there is a vector x∈E such that RKx=x. Proof We may assume, without loss of generality, that K=.Chooseanx∈Esuch that Kx>,sothatx> . Consider the closed ball B={x∈E:x–x≤}.Then, for each R∈R, consider the open set GR={y∈E:Ry –x<}.SinceRis a transitive algebra, we have  R∈R GR=E\{}. Since Kis a compact operator, KB is a compact subset of E, and since K=and Kx>,wehave /∈KB.Thus,thefamily{GR:R∈R}is an open cover of KB.Hence, there exist finitely many operators R,...,Rn∈Rsuch that KB ⊆ n  i= GRi. Next, for each y∈KB and i=,...,nwe define αi(y)=max{,  – Riy–x}.Then≤ αi(y)≤, and for each y∈KB,thereisani=,...,nsuch that y∈GRi,sothatαi(y)>. Thus, n i= αi(y)>foreachy∈KB, and we may define βi(y)= αi(y) n j= αj(y) for i=,...,nand y∈KB. Now, each βiis a continuous function from KB into R.Hence, we may define a continuous mapping :B→Eby the expression (x)= n  i= βi(Kx)RiKx. We claim that (B)⊆B. Indeed, for each x∈B,wehaven i= βi(Kx)=sothat  (x)–x =     n  i= βi(Kx)(RiKx –x)    ≤ n  i= βi(Kx)RiKx –x. If RiKx –x>,thenαi(Kx) =  and therefore βi(Kx)=.Hence,  (x)–x ≤ n  i= βi(Kx)=, Espínola and Lacruz Fixed Point Theory and Applications 2012, 2012:197 Page 4 of 11 http://www.fixedpointtheoryandapplications.com/content/2012/1/197 and this completes the proof of our claim. Finally, each operator RiKis compact so that each RiKB is relatively compact, and it follows from an earlier mentioned theorem of Mazur that Q=conv n i= RiKB is compact. Since (B)⊆Q,theset(B)isarelatively compact subset of B. Now, we apply Proposition . to find a vector x∈Bsuch that (x)=x.Since/∈B,wehavex= . Then we consider the operator defined by Rx = n  i= βi(Kx)Rix, and we conclude that R∈Rand RKx=x,aswewanted.  Corollary . [], [,Theorem]Every nonscalar operator that commutes with a nonzero compact operator has a nontrivial,hyperinvariant subspace. Proof Let T∈B(E) be a nonscalar operator and suppose that Tcommutes with a nonzero compact operator K. We must show that the commutant {T}is nontransitive. Suppose, on the contrary, that {T}is transitive. We can apply Theorem . to find an operator R∈{T}such that λ=  is an eigenvalue of the compact operator RK with an associated finite dimensional eigenspace F=ker(RK –I). Since Tcommutes with RK,weobserve that Tmaps Finto itself, and therefore, Tmust have an eigenvalue. Since Tis nonscalar, the corresponding eigenspace Mcannot be the whole E, and it is invariant under {T}.The contradiction has arrived.  3 Invariant subspaces for localizing algebras In this section we use the following fixed point theorem of Ky Fan []. Recall that if is a topological space and :→P()isapointtosetmapfromto the power set of , then is said to be upper semicontinuous if for every x∈and every open set U⊆ such that (x)⊆U, there is a neighborhood Vof xsuch that (x)⊆Ufor every x∈V. In terms of convergence of nets, this definition is equivalent to saying that for every x∈, for every net (xα)withxα→x,andforeveryyα∈(xα) such that the net (yα)converges to some y∈,wehavey∈(x). Theorem . (Ky Fan fixed point theorem []) Let C be a compact convex subset of a locally convex space,and let :C→P(C)be an upper semicontinuous mapping such that (x)is a nonempty,closed convex set for every x ∈C.Then there is an x∈Csuch that x∈(x). AsubalgebraR⊆B(H)issaidtobestrongly compact if its unit ball is precompact in the strong operator topology. An important example of a strongly compact algebra is the commutant of a compact operator with a dense range. We shall denote by ball(R)theunit ball of R. This notion was introduced by Lomonosov [] as a means to prove the existence of invariant subspaces for essentially normal operators on Hilbert spaces. Recall that an operator Ton a Hilbert space is said to be essentially normal if T∗T–TT∗is a compact operator. Lomonosov showed that if an essentially normal operator Thas the property that both its commutant {T}and the commutant of its adjoint {T∗}fail to be strongly compact, then Thas a nontrivial invariant subspace. Espínola and Lacruz Fixed Point Theory and Applications 2012, 2012:197 Page 5 of 11 http://www.fixedpointtheoryandapplications.com/content/2012/1/197 Thus, in order to solve the invariant subspace problem for essentially normal operators, it suffices to consider only operators with a strongly compact commutant. Lomonosov, Radjavi, and Troitsky [] obtained a result about the existence of invariant subspaces for an operator with a strongly compact commutant under the additional assumption that the commutant of the adjoint is a localizing algebra. AsubalgebraR⊆B(E)issaidtobelocalizing provided that there is a closed ball B⊆E such that  /∈Band such that for every sequence (xn)inBthere is a subsequence (xnj), and a sequence of operators (Rj)inRsuch that Rj≤and(Rjxnj)convergesinnormtosome nonzero vector. An important example of a localizing algebra is any algebra containing a nonzero compact operator. Proposition . [, Proof of Theorem .] Let R⊆B(E)be a transitive localizing algebra, let B ⊆Ebeaclosedballasabove,and let T ∈Rbe a nonzero operator.Then there exists an r >such that for every x ∈B we have r ball(R)(Tx)∩B=∅. Proof First, Tis one-to-one because Ris transitive and ker Tis invariant under R.If this is not so, then for every n≥, there is a vector xn∈Bsuch that R≥n,whenever R∈Rand RTxn∈B.SinceRis localizing, there is a subsequence (xnj) and a sequence (Rj)inRsuch that Rj≤and(Rjxnj) converges in norm to some nonzero vector x∈X. We have TRj=RjTfor all j≥, so that (RjTxnj)convergestoTx in norm. Now Tx = because Tis injective and x=.SinceRis transitive, there is an operator R∈Rsuch that RTx ∈int B. It follows that there is a j≥suchthatRRjTxnj∈int Bfor every j≥j.Since RRj∈R, the choice of the sequence (xn)impliesthatRRj≥njfor every j≥j,andthis is a contradiction because RRj≤Rfor every j≥.  If Eis a Banach space, then E∗denotes its dual space. If R⊆B(E)isasubalgebra,then R∗denotes the subalgebra of B(E∗) of the adjoints of the elements of R,thatis,R∗={R∗: R∈R}. Theorem . [,Theorem.]Let E be a complex Banach space,let R⊆B(E)be a strongly compact subalgebra such that R∗is a transitive localizing algebra and it is closed in the weak-∗operator topology.If T ∈Ris a nonzero operator,then there is an operator R∈Rand there is a nonzero vector x∗∈E∗such that R∗T∗x∗=x∗.Moreover,the operator T∗has a nontrivial invariant subspace. Proof We shall apply Proposition . to the algebra R∗.LetB∗⊆E∗be a closed ball as in the definition of a localizing algebra, let r>  be a positive number as in Proposition ., and define a multivalued map :B∗→P(B∗)bytheexpression x∗=rballR∗T∗x∗∩B∗. Then, (x∗) is a nonempty, convex subset of B∗.Also,(x∗)isweak-∗closed because ball(R∗)(T∗x∗)isweak-∗compact as the image of ball(R∗)underthemapR∗→R∗T∗x∗, which is continuous from B(E∗)withtheweak-∗operator topology into E∗with the weak-∗topology, and ball(R∗)iscompactintheweak-∗operator topology. We claim that is upper semicontinuous for the weak-∗topology. Indeed, let x∗,y∗∈B∗, and let (x∗ α)and(y∗ α)betwonetsinB∗with x∗ α→x∗,y∗ α→y∗in the weak-∗topology Espínola and Lacruz Fixed Point Theory and Applications 2012, 2012:197 Page 6 of 11 http://www.fixedpointtheoryandapplications.com/content/2012/1/197 and such that y∗ α∈(x∗ α). We must show that y∗∈(x∗). Since y∗ α∈(x∗ α), there is an R∗ α∈ball(R∗)suchthaty∗ α=rR∗ αT∗x∗ α.Sinceball(R) is precompact in the strong operator topology, there exists a subnet (Rαβ) that converges in the strong operator topology to some R∈B(E). Thus, R∗ αβ→R∗in the weak-∗operator topology. Notice that ball(R∗) is compact in this topology because ball(B(E∗)) is compact in this topology and R∗is closed in this topology. It follows that R∗∈ball(R∗). Let x∈Eand notice that TRαβx– TRx→. Then x,y∗ αβ=x,rR∗ αβT∗x∗ αβ =rTRαβx,x∗ αβ =rTRαβx–TRx,x∗ αβ+rTRx,x∗ αβ. We have TRαβx–TRx,x∗ αβ→andTRx,x∗ αβ→TRx,x∗=x,R∗T∗x∗,sothat x,y∗ αβ→x,rR∗T∗x∗. Since x∈Eis arbitrary, y∗ αβ→rR∗T∗x∗in the weak-∗topology, and it follows that y∗= rR∗T∗x∗. This shows that y∗∈(x∗), and the proof of our claim is complete. Since the map is upper semicontinuous and B∗is compact in the weak-∗topology, it follows from the Ky Fan fixed point theorem that there is a vector x∗∈B∗such that x∗∈(x∗); that is, there is an operator R∈ball(R)suchthatx∗=rR∗T∗x∗. Finally, consider the closed subspace defined as M=ker(R∗T∗–I). Notice that Mis invariant under T∗and M={}.IfT∗is not invertible then M=Eand we are done. If T∗is invertible, pick any λ∈σ(T∗)andputS=λ–T∗.ThenSis not invertible and the preceding argument applied to Sshows that Shas a nontrivial invariant subspace. It is clear that such subspace is also invariant under T∗. Corollary . [, Corollary .] Let T ∈B(E)be an operator such that {T}is a strongly compact algebra and {T∗}is a localizing algebra.Then T∗has a nontrivial invariant subspace. Proof If T∗has a hyperinvariant subspace then there is nothing to prove, and otherwise {T∗}is a transitive algebra so that Theorem . applies.  Notice that the assumptions of Corollary . are met whenever Tis a compact operator with a dense range. 4 An infinite dimensional version of Burnside’s theorem Burnside’s classical theorem is the assertion that for a finite dimensional linear space F, the only transitive subalgebra of B(F)isthewholealgebraB(F). Lomonosov []obtaineda generalization of Burnside’s theorem to infinite dimensional Banach spaces. Scott Brown [] proved the same result independently for the special case of a Hilbert space and a commutative algebra. Lindström and Schlüchtermann [] provided a relatively short proof of the Lomonosov result in full generality. In this section we present a proof of the Scott Brown result that is based on the Schauder fixed point theorem. Espínola and Lacruz Fixed Point Theory and Applications 2012, 2012:197 Page 7 of 11 http://www.fixedpointtheoryandapplications.com/content/2012/1/197 Let Hbe a complex, infinite dimensional, and separable Hilbert space. Let T∈B(H), and let Tedenote the essential norm of T,thatis,thedistancefromTto the space of compact operators. Theorem . [, Theorem .] Let Rbe a commutative subalgebra of B(H). Then there exist nonzero vectors x,y∈HsuchthatforanyR∈Rwe have |Rx,y|≤Re. Proof Consider the set E={R∈R:Re≤/}. We claim that there is some x∈H\{} such that the set Exis not dense in H. The result then follows easily because in that case there is some y∈H\{}such that |Rx,y|≤ for all R∈E. Now, for the proof of our claim, we proceed by contradiction. Suppose that the set Exis dense in Hfor every x∈H\{}. Choose x∈Hwith x=  and consider the closed ball B={x∈H:x–x≤}.Then, for every vector x∈B, there is an operator Rx∈Esuch that Rxx–x< /. Next, there is a bounded operator Txand a compact operator Kxsuch that Rx=Tx+Kxand Tx≤ /. Since Kxis a compact operator, it is weak-to-norm continuous on bounded sets so that there exists an open neighborhood of xin the weak topology, say Vx⊆H,suchthat Kxy–Kxx< / for all y∈Vx∩B. Then consider the set Ux=Vx∩Band notice that Ux is an open neighborhood of xin the weak topology relative to B.Moreover,fory∈Uxwe have Rxy–Rxx≤Txy–Txx+Kxy–Kxx<· + = , and therefore Rxy–x<.Hence,RxUx⊆B.SinceBis compact in the weak topology, there exist finitely many vectors x,...xn∈Bsuch that B⊆ n  j= Uxj. Choose some weakly continuous functions f,...,fnon Bsuch that supp(fj)⊆Uj,≤ fj(x)≤, and n  j= fj(x) =  for all x∈B. Define a weakly continuous mapping :B→Bby the expression (x)= n  j= fj(x)Rxjxfor all x∈B, and apply the Schauder fixed point theorem to find a vector y∈Bsuch that (y)=y. Finally, consider the operator R∈Rdefined by the expression R= n  j= fj(y)Rxj. Hence, Ry=y.NoticethatR=Ibecause Re≤/. Then, the eigenspace M={x∈H: Rx =x}is a closed nontrivial invariant subspace for the algebra R.Thus,anyvectorx∈M has the property that the set Exis not dense in H. The contradiction has arrived.  Espínola and Lacruz Fixed Point Theory and Applications 2012, 2012:197 Page 8 of 11 http://www.fixedpointtheoryandapplications.com/content/2012/1/197 5 Invariant subspaces for operators on the Krein space Let H,Hbe two Hilbert spaces and consider the orthogonal direct sum H=H⊕H.Let P,Pdenote the orthogonal projections from Honto H,H, respectively. Consider the operator J:= P–P.TheKrein space is the space Hprovided with the indefinite product [x,y]:=Jx,y,x,y∈H. Notice that Jis a selfadjoint involution, that is, J∗=Jand J=I. The operator Jis sometimes called the fundamental symmetry of the Krein space. Avectorx∈His said to be nonnegative provided that [x,x]≥, and a subspace M⊆H is said to be nonnegative provided that [x,x]≥ for all x∈M. Every operator T∈B(H)hasamatrixrepresentation T=T T T T with respect to the decomposition H=H⊕H. There is a natural, one-to-one and onto correspondence between the maximal nonnegative invariant subspaces Mof an operator T∈B(H) and the contractive solutions X∈B(H,H) of the so-called operator Riccati equation XTX+XT –TX–T =. The correspondence X↔Mis given by M={x⊕Xx:x∈H},whereX≤. The operator Tis usually called the Hamiltonian operator of the operator Ricatti equation. An operator T∈B(H)issaidtobeJ-selfadjoint provided that [Tx,y]=[x,Ty]forevery x,y∈H.ThisisequivalenttosayingthatJT =T∗J,orinotherwords,T∗  =T,T∗  =T, and T∗  =–T. A classical theorem of Krein is the assertion that if the Hamiltonian operator Tis J-selfadjoint and the corner operator T is compact, then there exists a maximal nonnegative invariant subspace for T. Albeverio, Makarov, and Motovilov [] addressed the question of the existence and uniqueness of contractive solutions to the operator Riccati equation under the condition that the diagonal entries in the Hamiltonian operator have disjoint spectra, that is, σ(T)∩σ(T)=∅. They proved the following Theorem . [, Theorem . and Lemma .] There is some universal constant c > such that whenever the corner operator T satisfies the condition T<c·distσ(T),σ(T), there is a unique solution X to the operator Riccati equation with X≤. An earlier result in this direction was given by Motovilov [, Corollary ] with the stronger assumption that the corner operator T is Hilbert-Schmidt. Adamjan, Langer, and Tretter [] extended the technique to the case that the Hamiltonian operator is not Espínola and Lacruz Fixed Point Theory and Applications 2012, 2012:197 Page 9 of 11 http://www.fixedpointtheoryandapplications.com/content/2012/1/197 J-selfadjoint. Kostrykin, Makarov, and Motovilov [] adopted the assumption that σ(T) lies in a gap of σ(T) and they showed that the best constant, in that context, is c=√. We present a proof of Theorem . that is based on the Banach fixed point theorem. This method can be found in the paper of Albeverio, Motovilov, and Shkalikov [,Theorem .]. A basic tool is the bounded linear operator Rdefined for X∈B(H,H)bythe expression R(X):=TX–XT. It follows from the Rosenblum theorem that the map Ris invertible. The main result is the following Theorem . [, Theorem .] If the operators T,T have disjoint spectra and the corner operator T satisfies the estimate T< R–, then there is a unique solution X to the operator Riccati equation with X≤. The following upper bound on the norm of the inverse R– canbefoundinthework of Albeverio, Makarov, and Motovilov [, Theorem .]. See also the paper by Bhatia and Rosenthal [, p.] for this interesting result and other related issues. Theorem . [,Theorem.] If the operators T,T have disjoint spectra,then  R– ≤π · dist[σ(T), σ(T)]. Notice that Theorem . becomes a corollary of Theorem . and Theorem . with the constant c=/π. Proof of Theorem . Consider the quadratic map Qdefined for X∈B(H,H)bytheexpression Q(X):=XTX–T. It is clear that the operator Riccati equation can be expressed as Q(X)–R(X)=, or equivalently, X=R–(Q(X)). Thus, the solutions of the operator Riccati equation are the fixed points of the map S:= R– ◦Q. Now, let us check that the map Stakes the unit ball of B(H,H) into itself. Indeed, if X≤, then  S(X) = R–Q(X) ≤ R– · Q(X)  ≤ R– ·T·X+T ≤ R– ·T+T=  R– ·T<.