Applications of fixed point theorems in the theory of invariant subspaces

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.


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 non trivial 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 [1] for a detailed account of results related to the invariant subspace problem.
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 2 we consider the striking theorem of Lomonosov [2] 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 3 we present a recent result of Lomonosov, Radjavi, and Troitsky [3] about the 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 [4].
In Section 4 we consider an extension of Burnside's theorem to infinite dimensional Banach spaces. This result is originally due to Lomonosov [5]. We present a proof of it in a special case that was obtained independently by Scott Brown [6] and that once again is based on the Schauder fixed point theorem.
In Section 5 we address the existence of invariant subspaces for operators on Krein space of indefinite product, and we present a result of Albeverio, Makarov, and Motovilov [7] 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.
Let E be an infinite dimensional, complex Banach space and let B(E) denote the algebra of all bounded linear operators on E. A subspace of E is by definition a closed linear manifold in E.
A subspace M ⊆ E is said to be invariant under an operator T ∈ B(E) provided that T M ⊆ M, and a subspace M ⊆ E is said to be invariant under a subalgebra R ⊆ B(H) provided that M is invariant under every R ∈ R. A subalgebra R ⊆ B(H) is said to be transitive provided that the only subspaces invariant under R are the trivial ones, M = {0} and M = E. This is equivalent to say that the subspace {Rx : R ∈ R} is dense in E for each x ∈ E\{0}.
The commutant of a set of operators S ⊆ B(E) is the subalgebra S ′ of all operators R ∈ B(E) such The invariant subspace problem is the question of whether every operator in B(E) has a non trivial invariant subspace. This is one of the most important open problems in operator theory.
The origin of this question goes back to 1935, when von Neumann proved the unpublished result that any compact operator on a Hilbert space has a non trivial invariant subspace. Aronszajn and Smith [8] extended this result in 1954 to general Banach spaces. Bernstein and Robinson [9] used non standard analysis to prove in 1966 that every polynomially compact operator on a Hilbert space has a non trivial invariant subspace.
Halmos [10] obtained a proof of the same result using classical methods.
Lomonosov [2] proved in 1973 that any non scalar operator on a Banach space that commutes with a non zero compact operator has a non trivial 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 [11] constructed in 1976 the first example of an operator on a Banach space without non trivial invariant subspaces. The example circulated in preprint form and it did not appear published until 1987, when it was recognized as correct work [12]. In the meantime, Beauzamy [13] simplified the technique, and further examples were given by Read [14,15].
Very recently, Argyros and Haydon [16] 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 non trivial 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.

Invariant subspaces for algebras containing compact operators
We start with a fixed point theorem that is the key for 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) the closed convex hull of a subset S ⊆ E. Proposition 2.1. [17, Proposition 1] Let E be a Banach space, let C ⊆ E be a closed convex set, and let Φ : C → E be a continuous mapping such that Φ(C) is a relatively compact subset of C. Then there is a Proof. Let Q denote the closure of Φ(C). It follows from a theorem of Mazur that conv(Q) is a compact, Since K is a compact operator, KB is a compact subset of E, and since K = 1 and Kx 0 > 1, we have 0 / ∈ KB. Thus, the family {G R : R ∈ R} is an open cover of KB. Hence, there exist finitely many operators Next, for each y ∈ KB and i = 1, . . . , n we define α i (y) > 0 for each y ∈ KB, and we may define for i = 1, . . . , n and y ∈ KB. Now, each β i is a continuous function from KB into R. Hence, we may define a continuous mapping Φ : B → E by the expression and this completes the proof of our claim. Finally, each operator R i K is compact, so that each R i KB is relatively compact, and it follows from an earlier mentioned theorem of Mazur that Q = conv is a relatively compact subset of B. Now we apply Proposition 2.1 to find a vector x 1 ∈ B such that Φ(x 1 ) = x 1 . Since 0 / ∈ B, we have x 1 = 0. Then we consider the operator defined by and we conclude that R ∈ R and RKx 1 = x 1 , as we wanted.

Invariant subspaces for localizing algebras
In this section we use the following fixed point theorem of Ky Fan [18]. Recall that if Ω is a topological space and Φ : Ω → P(Ω) is a point to set map from Ω to the power set of Ω, then Φ is said to be upper In terms of convergence of nets, this definition is equivalent to say that for every x ∈ Ω, for every net (x α ) with x α → x, and for every y α ∈ Φ(x α ) such that the net (y α ) converges to some y ∈ Ω, we have y ∈ Φ(x).
Theorem 3.1 (Ky Fan fixed point theorem [18]). Let C be a compact convex subset of a locally convex space and let Φ : C → P(C) be an upper semicontinuous mapping such that Φ(x) is a non empty, closed convex set for every x ∈ C. Then there is an x 0 ∈ C such that x 0 ∈ Φ(x 0 ).
A subalgebra R ⊆ B(H) is said to be strongly 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 dense range. We shall denote by ball(R) the unit ball of R. Lomonosov, Radjavi and Troitsky [3] 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.
A subalgebra R ⊆ B(E) is said to be localizing provided that there is a closed ball B ⊆ E such that 0 / ∈ B and such that for every sequence (x n ) in B there is a subsequence (x nj ) and a sequence of operators (R j ) in R such that R j ≤ 1 and (R j x nj ) converges in norm to some non zero vector. An important example of a localizing algebra is any algebra containing a non zero compact operator. Proof. First, T is one to one, because R is transitive and ker T is invariant under R. If this is not so, then for every n ≥ 1 there is a vector x n ∈ B such that R ≥ n, whenever R ∈ R and RT x n ∈ B. Since R is localizing, there is a subsequence (x nj ) and a sequence (R j ) in R such that R j ≤ 1 and (R j x nj ) converges in norm to some nonzero vector x ∈ X. We have T R j = R j T for all j ≥ 1, so that (R j T x nj ) converges to T x in norm. Now T x = 0 because T is injective and x = 0. Since R is transitive, there is an operator R ∈ R such that RT x ∈ int B. It follows that there is a j 0 ≥ 1 such that RR j T x nj ∈ int B for every j ≥ j 0 .
Since RR j ∈ R, the choice of the sequence (x n ) implies that RR j ≥ n j for every j ≥ j 0 , and this is a contradiction because RR j ≤ R for every j ≥ 1.
If E is a Banach space then E * denotes its dual space. If R ⊆ B(E) is a subalgebra, then R * denotes the subalgebra of B(E * ) of the adjoints of the elements of R, that is R * = {R * : R ∈ R}.

Theorem 3.3. [3, Theorem 2.3] 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 non zero operator then there is an operator R ∈ R and there is a non zero vector x * ∈ E * such that R * T * x * = x * . Moreover, the operator T * has a non trivial invariant subspace.
Proof. We shall apply Proposition 3. Then, Φ(x * ) is a non empty, convex subset of B * . Also, Φ(x * ) is weak- * closed because ball(R * )(T * x * ) is weak- * compact as the image of ball(R * ) under the map R * → R * T * x * , which is continuous from B(E * ) with the weak- * operator topology into E * with the weak- * topology, and ball(R * ) is compact in the weak- * operator topology.
We claim that Φ is upper semicontinuous for the weak- * topology. Indeed, let x * , y * ∈ B * , and let (x * α ) and (y * α ) be two nets in B * with x * α → x * , y * α → y * in the weak- * topology, and such that y * α ∈ Φ(x * α ). We must show that y * ∈ Φ(x * ). Since y * α ∈ Φ(x * α ), there is an R * α ∈ ball(R * ) such that y * α = rR * α T * x * α . Since ball(R) is precompact in the strong operator topology, there exists a subnet (R α β ) that converges in the strong operator topology to some R ∈ B(E). Thus, R * α β → 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 ∈ E and notice that T R α β x − T Rx → 0. Then We have T R α β x − T Rx, x * α β → 0 and T Rx, x * α β → T Rx, x * = x, R * T * x * , so that x, y * α β → x, rR * T * x * .

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 ) is the whole algebra B(F ). Lomonosov [5] obtained a generalization of Burnside's theorem to infinite dimensional Banach spaces. Scott Brown [6] proved the same result independently for the special case of a Hilbert space and a commutative algebra. Lindström and Schlüchtermann [20] 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 Schauder fixed point theorem.
Let H be a complex, infinite dimensional, separable Hilbert space. Let T ∈ B(H) and let T e denote the essential norm of T, that is, the distance from T to the space of compact operators. Proof. Consider the set E = {R ∈ R : R e ≤ 1/16}. We claim that there is some x ∈ H\{0} such that the set Ex is not dense in H. The result then follows easily because in that case there is some y ∈ H\{0} such that | Rx, y | ≤ 1 for all R ∈ E. Now, for the proof of our claim, we proceed by contradiction. Suppose that the set Ex is dense in H for every x ∈ H\{0}. Choose x 0 ∈ H with x 0 = 2 and consider the closed ball B = {x ∈ H : x − x 0 ≤ 1}. Then, for every vector x ∈ B there is an operator R x ∈ E such that R x x− x 0 < 1/2. Next, there is a bounded operator T x and a compact operator K x such that R x = T x + K x and T x ≤ 1/8. Since K x is a compact operator, it is weak to norm continuous on bounded sets, so that there exists an open neighborhood of x in the weak topology, say V x ⊆ H, such that K x y − K x x < 1/4 for all y ∈ V x ∩ B. Then consider the set U x = V x ∩ B and notice that U x is an open neighborhood of x in the weak topology relative to B. Moreover, for y ∈ U x we have Notice that J is a selfadjoint involution, that is, J * = J and J 2 = I. The operator J is sometimes called the fundamental symmetry of the Krein space.
A vector x ∈ H is said to be non negative provided that [x, x] ≥ 0, and a subspace M ⊆ H is said to be non negative provided that [x, x] ≥ 0 for all x ∈ M.
Every operator T ∈ B(H) has a matrix representation There is a natural, one to one and onto correspondence between the maximal non negative invariant subspaces M of an operator T ∈ B(H) and the contractive solutions X ∈ B(H 1 , H 2 ) of the so called operator Riccati equation The correspondence X ↔ M is given by M = {x 1 ⊕ Xx 1 : The operator T is usually called the hamiltonian operator of the operator Ricatti equation.
An operator T ∈ B(H) is said to be J-selfadjoint provided that [T x, y] = [x, T y] for every x, y ∈ H. This is equivalent to say that JT = T * J, or in other words, T * 11 = T 11 , T * 22 = T 22 , and T * 12 = −T 21 . A classical theorem of Krein is the assertion that, if the hamiltonian operator T is J-selfadjoint and the corner operator T 12 is compact, then there exists a maximal non negative invariant subspace for T.
Albeverio, Makarov, and Motovilov [7] 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 11 ) ∩ σ(T 22 ) = ∅. They proved the following Theorem 5.1. [7, Theorem 3.6 and Lemma 3.11] There is some universal constant c > 0 such that whenever the corner operator T 12 satisfies the condition there is a unique solution X to the operator Riccati equation with X ≤ 1.
An earlier result in this direction was given by Motovilov [21,Corollary 1] with the stronger assumption that the corner operator T 12 is Hilbert-Schmidt. Adamjan, Langer, and Tretter [22] extended the technique to the case that the hamiltonian operator is not J-selfadjoint. Kostrykin, Makarov, and Motovilov [23] adopted the assumption that σ(T 11 ) lies in a gap of σ(T 22 ) and they showed that the best constant, in that context, We present a proof of Theorem 5. It follows from the Rosenblum theorem that the map R is invertible. The main result is the following Indeed, if X ≤ 1 then S(X) = R −1 (Q(X)) ≤ R −1 · Q(X) ≤ R −1 · ( T 12 · X 2 + T 21 ) ≤ R −1 · ( T 12 + T 21 ) = 2 R −1 · T 12 < 1.
Also, the map S is contractive, for if X , Y ≤ 1 then and from this inequality it follows that so that the map S satisfies a Lipschitz condition with a Lipschitz constant 2 R −1 · T 12 < 1. The result now follows at once as a consequence of the Banach fixed point theorem.