On Burnside’s Theorem in Infinite-dimensional Spaces

V. Lomonosov

Abstract

The classical Burnside’s Theorem guarantees in a finite dimensional space the existence of invariant subspaces for a proper subalgebra of the matrix algebra. In this paper we give an extension of Burnside’s Theorem for a general Banach space, which also gives new results on invariant subspaces.

Introduction.

This preprint is an extension of the paper (10). New results are theorems 4 and 5.

The fundamental Burnside Theorem for finite-dimensional representations says that an algebra of linear operators on a finite-dimensional space F, without invariant subspace must be the algebra L(F) of all linear operators on the space. In other words, if an algebra R is strictly contained in L(F) then there exists x ∈ F and φ ∈ F^{∗} s.t. for every A ∈ R, (Ax, φ) = 0.

In this paper we will extend this result to Banach spaces in the following way. In Theorem 1 below, let B be a Banach space, L(B) the bounded linear operators on B and K(B) the compact operators on B. Let |||A||| denote the essential norm of A, i.e. the distance from A to the space of compact operators. A weakly closed algebra is an algebra closed in weak operator topology.

Theorem 1. Let R be a weakly closed subalgebra of L(B), R ≠ L(B). Then there exists x ∈ B^{∗∗} and y ∈ B^{∗}, x ≠ 0 and y ≠ 0, s.t. for every A ∈ R

|(x, A^{∗}y)| ≤ |||A|||.  (1)

If B is finite-dimensional then |||A||| = 0, so Theorem 1 gives Burnside’s theorem.

For a given subset S ⊂ L(B) and a vector x ∈ B put Sx = ⋃_{A∈S}Ax and S′ = ∪A^{∗}. We say that S is transitive if it doesn’t have a nontrivial invariant subspace and S is essentially transitive if the conclusion of Theorem 1 is false for S. We say that a subalgebra R ⊂ L(B) has the Pearcy-Salinas (P S) property if there exists net {A_{α}} ⊂ R and a nonzero operator A ∈ L(B) such that:

lim_{α}(x, A_{α}^{∗}y) = (x, A^{∗}y)  (2)

for every vector x ∈ B^{∗∗} and every functional y ∈ B^{∗} and

lim_{α} |||A_{α}||| = 0  (3)

Of course, every bounded operator is a weak limit of finite dimensional ones, so it is the assumption {A_{α}} ⊂ R that makes the condition nontrivial.

Corollary 1. Let R be a weakly closed proper subalgebra of L(B) with P S property. Then the algebra R′ is nontransitive.

Proof. Let x and y be as in Theorem 1. Then for every pair of operators T_{1}, T_{2} in R we have

|(x, T_{2}^{∗}A^{∗}T_{1}^{∗}y)| =

= lim_{α}|(x, T_{2}^{∗}A_{α}T_{1}^{∗}y)| ≤ \overline{lim_{α}}|||T_{2}A_{α}T_{1}||| ≤

≤ \overline{lim_{α}}||T_{2}∥ ∥T_{1}∥ |||A_{α}||| = 0

It is easy to see that one of the three subspaces ⋂_{T ∈R} ker(T^{∗}), \overline{R′y}, \overline{span_{T_{1},T_{2}∈R}(T_{2}^{∗}A^{∗}T_{1}^{∗}y)} is a nontrivial invariant subspace for the algebra R′.

If the algebra R contains a non-zero compact operator then the P S property is trivially true for R. So we get the results obtained by the author’s earlier techniques [8].

Corollary 2. If a weakly closed proper subalgebra R of L(B) contains a nonzero compact operator, then the algebra R′ is nontransitive.

Of course the existence of a hyperinvariant subspace in B^{∗} for operators that commute with a compact operator follows from Corollary 2. It should be mentioned that the following theorem by Pearcy and Salinas [6] is not a consequence of the author’s earlier techniques [see 7].

Theorem (Pearcy - Salinas): Let T be a bounded operator on Hilbert space. Assume that there is a sequence of rational functions (s_{n}) s.t. s_{n}(T) converges weakly to a non-zero operator A and s.t.

|||s_{n}(T)||| → 0 as n → ∞.

Then T has a non-trivial invariant subspace.

However, from Corollary 1 we get the following strengthening of the Pearcy - Salinas Theorem as

Corollary 3. Let T be a bounded operator on Banach space. Assume that there is a nonzero operator A and net {A_{α}} of operators that commute with T s.t. (2) and (3) holds. Then T^{∗} has a nontrivial hyperinvariant subspace. The commutant of T is a proper subalgebra of L(B), so Corollary 3 is an immediate consequence of Corollary 1.

Corollary 4. Let S_{ϕ} ⊂ L(B) be defined by S_{ϕ} = {A ∈ R, |||A||| ≤ 1}. Then there exists a nonzero functional y ∈ B^{∗} s.t. the set S′_{ϕ}y is not dense in B^{∗}.

Proof. Let x and y be as in Theorem 1. Then sup_{|||A^{∗}|||<1}(x, A^{∗}y)| ≤ 1, so S′_{ϕ}y is not dense in B.

This corollary was pointed out to us by S.W. Brown, who has obtained it and similar results in the case of a commutative algebra.

In Hilbert space Theorem 1 has the following equivalent formulation.

Theorem 2. Let R be as in Theorem 1. Let B be Hilbert space, then there exist two bounded nets (x_{α}) in B and (y_{α}) in B s.t. x_{α} →^{w} x ≠ 0, y_{α} →^{w} y ≠ 0 and for every A ∈ R

< Ax_{α}, y_{α} >→ 0.

Proof of theorems 1 and 2. Let B be a Banach space, Q a compact Hausdorff space, R a subalgebra of L(B), W Q the Banach space of weak^{∗}-continuous functions Q → B^{∗} with sup-norm, SQ the subspace of W Q consisting of the strongly continuous functions, NR the algebra of norm-continuous functions Q → R′. Let C = C(Q) be the algebra of all complexvalued continuous functions on Q. Every function f ∈ C(Q) is a linear operator on the space W Q. We let f^{∗} denote its adjoint.

Definition. Let M be a subspace of W Q, invariant under the algebra C. We say that θ ∈ M^{∗} is a point functional if there exists a point q ∈ Q s.t.

f^{∗}θ = \overline{f(q)}θ  (4)

for every function f ∈ C.

If E is a subspace in a Banach space let U_{E} denote the unit ball of E and E^{⊥} denote the annihilator of E in the dual space. We say that a functional θ ∈ M^{∗} is a δ(q, x) functional or simply δ-functional if there exists a point q ∈ Q and an element x ∈ B^{∗∗} s.t. if h ∈ M then

θ(h) = (x, h(q))  (5)

Lemma 1. Let θ ∈ M^{∗} be a point functional and assume M ⊃ SQ. Then the restriction, χ = θ|_{SQ} is a δ-functional.

Proof. It is known [3] that the functional χ has the form

χ(h) = ∫_{Q}(dχ(q), h(q))  (6)

where dχ is a regular Borel measure on Q with values in B^{∗∗}. Let q ∈ Q be as in (4). For an open set V = Q/{q} and for a given ϵ > 0 there exists a closed set F ⊂ V s.t. ∥dχ|_{V} − dχ|_{F} ∥ < ϵ. Then there exist functions f ∈ C(Q) s.t. f(q) = 0, f|_{F} ≡ 1. If G is given Borel set and G ⊂ F, then dχ(G) = _{∫G}dχ = _{∫G}\overline{f}dχ = f^{∗}χ = \overline{f(q)}χ = 0, so dχ|_{F} = 0. This give for every ϵ > 0 ∥dχ|_{V} ∥ < ϵ, and so dχ|_{V} = 0. Q.E.D.

Let H be a Hilbert space with orthogonal basis {e_{n}}, LIM(α_{n}) a Banach limit in the space l^{∞}, Q compactification of integer numbers by {∞} .Then θ(h) = LIM(h(n), e_{n}) is an example of a point functional that is not δ-functional.

Let Q × U_{B} be the topological product of the compact Q and the ball U_{B} with norm topology. The functions s ∈ W Q and f ∈ C(Q) define complex valued functions L(s) and L_{1}(f) on Q × U_{B} for the formulas L(s)(q, x) = (x, s(q)), L_{1}(f)(q, x) = f(q).

Lemma 2 L(s), L_{1}(f) are continuous functions on

Q × U_{B}

Proof. Let q_{0} ∈ Q and x_{0} ∈ U_{B}. Let V_{g_{0}}, V_{x_{0}} be open subsets defined by

\begin{matrix}V_{q_{0}} = \{q ∈ Q : |(x_{o}, s(q) − s(q_{o}))| < \frac{ϵ}{2}\} \\ V_{x_{0}} = \{x ∈ U_{B} : ∥x − x_{o}∥ < \frac{ϵ}{2|s|}\} \end{matrix}

then for all (q, x) ∈ V_{x_{0}} × V_{q_{0}} we have

\begin{matrix}|(x, s(q)) − (x_{o}, s(q_{o}))| ≤ |((x − x_{o}, s(q))|+ \\ +|(x_{o}, s(q) − s(q_{o}))| ≤ \frac{ϵ}{2|s|}|s| + \frac{ϵ}{2} = ϵ, \end{matrix}

so L(s) is continuous. The continuity of L_{1}(f) is obvious. U_{B} is a completely regular space, so there exists the Cheh compactification F of the space Q × U_{B}. The functions L_{1}(f) and L(s) in lemma 2 have continuous extensions to all of F. Thus L(M), L(N), L_{1}(C(Q)) are subspaces of C(F), L(M) ⊃ L(N), and L(M) and L(N) are invariant for L_{1}(C(Q)). Moreover the operators L : M → C(F) and L_{1} : C(Q) → C(F) are isometries.

Lemma 3. Let φ ∈ L(M)^{∗}, f ∈ C(Q) and

(L_{1}f)^{∗}φ = γφ

then f^{∗}(L^{∗}φ) = γ(L^{∗}φ)

Proof. For a given function h ∈ W Q we have (h, f^{∗}(L^{∗}φ)) = (fh, L^{∗}φ) = (L(fh), φ) = (L_{1}(f)L(h), φ) = (L(h), L_{1}(f)^{∗}φ) = (L(h), γφ) = (h, γL^{∗}φ). Q.E.D.

Lemma 4. Let M, N be subspaces of W Q, invariant for C(Q). Let M ⊃ N and let θ ∈ M^{∗} be an extreme point of U_{N^{⊥}}. Then θ is a point-functional.

The proof of this lemma is similar to De Branges famous proof of Stone-Weierstrass Theorem [2]

Proof. Let θ̂ be a functional in L(M)^{∗} s.t. L^{∗}(θ̂) = θ. Then, obviously 1 = |θ̂| = |θ|. By Hahn-Banach and Riesz Theorems there exists a measure dμ ∈ C(F)^{∗} which is an extension of θ̂ to space C(F). Let f be a function in C(Q), s.t.

0 ≤ f ≤ 1  (7)

and f̂ = L_{1}(f), dμ_{1} = f̂dμ, dμ_{2} = (1−f̂)dμ, m_{1} = ∥dμ_{1}∥, m_{2} = ∥dμ_{2}∥. Then {dμ_{1}, dμ_{2}, dμ} ⊂ L(N)^{⊥} and m_{1} +m_{2} = ∫_{F} f̂|dμ|+ ∫_{F} (1−f̂)|dμ| = ∫_{F} |dμ| = 1. If θ_{1} = dμ_{1}|_{L(M)}, θ_{2} = dμ_{2}|_{L(M)}, then |θ_{1}| + |θ_{2}| ≥ |θ_{1} + θ_{2}| = 1 and |θ_{1}| + |θ_{2}| ≤ m_{1} + m_{2} = 1, so we have |θ_{2}| = m_{2}, |θ_{1}| = m_{1}. If m_{1} = 0, then (L_{1}f)^{∗}θ̂ = 0, so by lemma 3 f^{∗}θ = 0. If m_{2} = 0 then (1 − f)^{∗}θ = 0. If m_{1} ≠ 0 and m_{2} ≠ 0 then we have

θ̂ = m_{1} \frac{θ_{1}}{m_{1}} + m_{2} \frac{θ_{2}}{m_{2}} , ∥\frac{θ_{1}}{m_{1}} ∥ = 1, ∥\frac{θ_{2}}{m_{2}} ∥ = 1

The functional θ̂ is an extreme point in the ball U_{L(N)^{⊥}}, so we get \frac{θ_{1}}{m_{1}} = θ̂ or by lemma 3 \frac{f^{∗}θ}{m_{1}} = θ. So for any function f with property (7) the functional θ is an eigenvector and the functions of this type obviously generate C(Q). The corresponding eigenvalue is a multiplicative functional on C(Q), so Gelfand’s theorem now gives the lemma. Q.E.D.

Lemma 5. Let T ∈ K(B) and h ∈ W Q. Then T^{∗}h ∈ SQ.

We omit the simple proof.

Let R be an algebra in L(B) and let h ∈ W Q, ∥h∥ = 1. By applying the algebra NR to h we get subspaces

N = NR(h), M = span (N, SQ).

then M ⊃ N and M and N are invariant for the algebra C(Q). Let φ be a point functional, φ ∈ M^{∗}. Then, by Lemma 1 , φ|_{SQ} is a δ(q, x) functional

Lemma 6. If functional φ ∈ N^{⊥} then for every operator A ∈ R

|(x, A^{∗}h(q))| ≤ 2|φ| |||A||| |h|  (8)

Proof. Let T be a compact operator s.t. ∥A − T ∥ ≤ |||A||| + ϵ. Then we have

|(x, A^{∗}h(q))| = |δA^{∗}h| =
= |δ(A^{∗}h) − φ(A^{∗}h)| = |(δ − φ)A^{∗}h| =
≤ |(δ − φ)(A^{∗} − T^{∗})h| + |(δ − φ)T^{∗}h|

By Lemma 5 T^{∗}h ∈ SQ, so by Lemma 1 δ(T^{∗}h) − φ(T^{∗}h) = 0. It is clear that |δ| ≤ |φ|, so we get

|(x, A^{∗}h(q))| ≤ 2|φ| ∥A − T ∥ |h| < 2|φ|(|||A||| + ϵ)|h|.

Since ϵ is arbitrary,

|(x, A^{∗}h(q))| ≤ 2|φ| |||A||| |h| Q.E.D.

Let σ_{F} (T) be the subset of the spectrum of an operator T ∈ L(B) consisting of the isolated points of finite multiplicity.

Lemma 7. Let the subspace L ⊂ B have finite codimension. Assume α > 0 and

sup_{x∈L} \frac{∥T x∥}{∥x∥} < α  (8)

Assume γ ∈ σ(T) and |γ| > α. Then γ ∈ σ_{F} (T)

Proof. It is sufficient to prove this lemma in the case that γ is a boundary point of σ(T). Let P be a bounded projection on L. Then the operator T_{1} = (T − γI)P is semi-Fredholm.

It is clear that kerT_{1} = kerP and if x ∈ L, then |T_{1}x| = |(T − γI)P x| = |(T − γI)x| ≥ (|γ|−α)|x|. So T_{1} has finite dimensional kernel and closed range. T −γI is a finite dimensional perturbation of T_{1}, so by [1, Corollary 1.3.7] T −γI is also semi-Fredholm. Moreover, T −γI is a limit of invertible operators so it is a Fredholm operator with index zero by [1, Theorem 4.2.1., Corollary 3.2.10]. Now by [9, Corollary 3.11] T − γI is a compact perturbation of an invertible operator, so by Weil’s theorem γ is isolated and has finite multiplicity Q.E.D.

Lemma 8. If R is an essentially transitive subalgebra, then R contains an operator T with the following properties: 1 ∈ σ_{F} (T) and if γ ∈ σ(T)\σ_{F} (T) then |γ| < \frac{1}{2}.

Proof. Let y be a functional in B^{∗} |y| = 3, and let Q be a ball of a radius 2 with center in y. Then Q is weak^{∗} compact and {0} ∉ Q. The function h(q) ≡ q if obviously weak^{∗} continuous, so h(q) ∈ W Q and |h(q)| = 5. Let M and N are subspaces of W Q as in Lemma 6. If N is not dense in M, then there exists a functional θ ∈ M^{∗} which is an extreme point of U_{N^{⊥}}. Then by Lemma 6 there exists a point q_{0} ∈ Q and a nonzero element x ∈ B^{∗∗} s.t. the inequality

|(x, A^{∗}h(q_{0}))| ≤ 2|||A||| |h|

holds for every operator A ∈ R. For y = \frac{h(q_{0})}{2|h|} we get inequality (1). This contradiction gives that N is dense in M. Let g be a function in SQ defined by g(q) ≡ y. Then there exists an operator-valued function A(q) ∈ NR s.t. |A(q)h(q) − g(q)| < \frac{1}{8}. Thus the function A(q) generates a continuous map Ψ : Q → Q by the formula Ψ(q) = A(q)q. By Tychonoff’s fixed point theorem there exists a point q_{0} ∈ Q s.t. A(q_{0})q_{0} = q_{0}. We let T denote A(q_{0}). Let V be open set in Q defined by

V = \{q ∈ Q : ∥A(q) − T ∥ < \frac{1}{48}\}.

Then

sup_{q∈V}|T q − y| ≤ sup_{q∈V}|(T − A(q))q|+

+sup_{q∈V}|(A(q)q) − y| ≤ \frac{5}{48} + \frac{1}{8} < \frac{1}{4}

If V_{1} = {z : |z| ≤ 1, z + q_{0} ∈ V } then V_{1} is a weak^{∗} open subset of the unit ball U_{B^{∗}} that contains {0} and

sup_{z∈V_{1}}∥T z∥ = sup_{z∈V_{1}}∥T (z + q_{0}) − T q_{0}∥ ≤

≤ sup_{q∈V}|T q − T q_{0}| ≤ sup_{q∈V}|T q − y| + |T q_{0} − y|

< \frac{1}{4} + \frac{1}{4} = \frac{1}{2}

By definition of weak^{∗} topology there exists a subspace L ⊂ B^{∗} of finite codimension s.t.

L∩U_{B^{∗}} ⊂ V_{1}. We have sup_{z∈U_{L}}|T z| ≤ sup_{z∈V_{1}}|T z| < \frac{1}{2}, so by Lemma 7 every point γ ∈ σ(T), |γ| ≥

\frac{1}{2} is an isolated point in σ(T) of finite multiplicity. Q.E.D.

Lemma 9. Let R be a uniformly closed essentially transitive subalgebra L(B), then R contains a nonzero finite dimensional projection.

Proof. Let T be an operator as in Lemma 8. Since {γ : (T − γI)^{−1} ∈ R} is a component of the resolvent set of T it follows that if (T − γI)^{−1} exists and |γ| ≥ \frac{1}{2} then (T − γI)^{−1} ∈ R.

By Lemma 8 there exists circle α ⊂ C s.t. 1 is the only point of σ(T) inside α and for every point γ ∈ α (T − γI)^{−1} ∈ R. By Riesz theorem P = \frac{1}{−2πi}_{∫α}(T − γI)^{−1}dγ is a non-zero finite dimensional projection and P ∈ R′. If P = P_{1}^{∗} then P_{1} is finite dimensional projection in R. Q.E.D.

Now to finish the proof of Theorem 1 we need the folloving well-known fact [8].

Lemma 10. If a transitive algebra R contains a non-zero finite dimensional operator, then R is weakly dense in L(B).

Essential transitivity implies transitivity, so by combining Lemma 9 and Lemma 10 we finish the proof of Theorem 1.

Of course Lemma 9 gives the possibility to obtain different results on density. For example one of them is

Theorem 3. If H is a Hilbert space and R is uniformly closed essentially transitive subalgebra of L(H) then R contains all compact operators in L(H).

Now we’ll prove the equivalence of Theorem 1 and Theorem 2. in Hilbert space. We first prove that Theorem 2 implies Theorem 1 if B is Hilbert space.

Assume that Theorem 2 holds. Let C = \overline{lim_{α}}|x_{α}| · |y_{α}|. If A ∈ R then

0 = lim_{α}(Ax_{α}, y_{α}) =

lim_{α}(A(x_{α} − x), y_{α} − y)) − (Ax, y)+

+lim_{α}(Ax, y_{α}) + lim_{α}(Ax_{α}, y)

where

lim_{α}(Ax, y_{α}) = lim_{α}(Ax_{α}, y) = (Ax, y).

Thus

(Ax, y) = lim_{α}(A(x − x_{α}), y_{α} − y)).

If K is a compact operator, then

|(Ax, y)| ≤ \overline{lim_{α}}|((A − K)(x − x_{α}), (y_{α} − y))|

+\overline{lim_{α}}|(K(x − x_{α}), (y_{α} − y))| ≤

≤ ∥A − K∥4C.

Now let z = \frac{y}{4C}. Then

|(Ax, z)| ≤ inf_{K}∥A − K∥ = |||A|||

and so Theorem 1 follows from Theorem 2.

To prove that Theorem 1 implies Theorem 2 we use the following result (by Glimm). [5] B is assumed to be Hilbert space

Lemma 11 . Let Ψ ∈ L(B)^{∗} and Ψ ∈ K(B)^{⊥}. Then there is a pair of bounded nets (x_{α}) and (y_{α}) s.t.

1. w − lim_{α}x_{α} = 0, w − lim_{α}y_{α} = 0

2. For every A ∈ L(B), Ψ(A) = lim_{α}(Ax_{α}, y_{α}).

Now let Ψ_{1}(A) = (Ax, y) where x, y are as in Theorem 1, A ∈ R. Theorem 1 gives that Ψ_{1} can be extended to span (R, K(B)) by putting Ψ_{1}(T) = 0 for T ∈ K(B). Now let Ψ be the Hahn-Banach extension of the Ψ_{1} to all of L(B). We now use the Lemma 1 and we get Theorem 2 with x_{α} = x + x_{α} and y_{α} = y − y_{α}.

Theorem 4. Let H be a Hilbert space and let R be a norm closed proper subalgebra of L(H). Then there exists a nonzero functional φ on L(H) which annihilates R and there exist two bounded nets (x_{α}) in H and (y_{α}) in H s.t. for every operator A ∈ L(H)

φ(A) = lim_{α} < Ax_{α}, y_{α} > .  (9)

Proof. If R is essentially transitive then by Theorem 3 it contains all compact operators. Let φ be any nonzero functional on L(H) which annihilates R. By Glimm Theorem it has the representation (9). If R is not essentially transitive then the conclusion of Theorem 1 is true for R and the representation (9) follows from the Theorem 2.

It is natural to ask if the previous theorem is true for an arbitrary proper subalgebra of L(H). The answer is negative, because of the following statement.

Theorem 5. There exists a norm dense proper subalgebra of L(H). Proof. Let {e_{i}} be an orthonormal basis in H and let M be the set of finite linear combinations of vectors {e_{i}}. Consider the set R of all bounded linear operators on H which leave the set M invariant. R is obviously a subalgebra of L(H). We’ll show that R is norm dense in L(H). Let T be a bounded linear operator on H and let ϵ be an arbitrary positive number. For each integer i there exists a vector x_{i} ∈ M such that ||x_{i} − T e_{i}|| ≤ ϵ 2^{−i}. Let Q be a linear operator on M defined by Qe_{i} = T e_{i} −x_{i}. Then ||Qe_{i}|| ≤ ϵ 2^{−i}, so ∑ ||Qe_{i}|| < ∞. For this reason the operator Q has an extension \overline{Q} which is a bounded linear operator on H and ||\overline{Q}|| ≤ ϵ ∑ 2^{−i} = ϵ. Put P = T − \overline{Q}. Then ||T − P || ≤ ϵ and obviously M is invariant under T.

Finally we will mention that in the case of a nonreflexive Banach space Theorem 1 gives invariant subspace corollaries only in the dual space B^{∗}.

Since Enflo in 1976 [4] showed that there are counter examples to the invariant subspace problem in general Banach spaces, this may be a sign that the following result can be true: If A is a bounded linear operator in a Banach space then A^{∗} has a non-trivial invariant subspace.

The known counterexamples do not contradict this conjecture. By corollary 1, this would be true if the following is true: If A is a bounded linear operator in a Banach space then there exists an algebra with PS property which contains the operator A.

Acknowledgments. I am very gratefull to Max-Planck-Institut für Mathematik for hospitality and very stumulating working atmosphere.

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Address of the author: Dept. of Math. Kent State University Kent OH 44242