Radjavi, Heydar

English

Radjavi Heydar

University of Szeged

Acta Sci. Math., 47 (1984), 113—115 On minimal invariant manifolds and density of operator algebras HEYDAR RADJAVI Let X be a complex Banach space and let 91 be a subalgebra of 38 (X), the algebra of all bounded linear operators on X. When is 9f strongly dense in á?(3E), i.e., dense in the topology of pointwise convergence? This question can sometimes be answered by examining various lattices associated with 91. The first result of this sort was obtain-ed by R ICKART and Y O O D [2 , p. 6 2 ] , a consequence of which is: if the only linear manifolds (i.e., not necessarily closed subspaces) of X invariant under all members of 91 are {0} and X, then 9Í is strongly dense in 33 {X). If we denote by i f (91) the lattice of all linear manifolds invariant under 91, the hypothesis in this assertion amounts to saying that if (91) is trivial, that is, the only nonzero element of JS?(9I) is X. We shall prove the following result. Theorem 1. Let 91 be an algebra of bounded linear operators ón the Banach space X. If the nonzero elements of JSf (9t) have a dense intersection, then 91 is strongly dense in SS(X). . Note that the hypothesis of the theorem implies that i f (91) has no closed mem-bers other than {0} and X, i.e., 91 is topológically transitive. It is not known whether topological transitivity for 91 is sufficient for strong density if X is a reflexive Banach space. (This is the Transitive Algebra Problem; see [3]). Many examples of algebras satisfying the hypothesis of Theorem 1 exist. Here is a simple, nontrivial example. Fix an orthonormal basis j for a separable, infinite-dimensional Hilbert space Let 91 be the set of all those operators on § whose matrices relative to are "column-finite", i.e., each of their columns have finitely hiány nonzero entries. It is easy to see that 91 is in fact an algebra, that every operator in 91 leaves the linear span "V of i invariant, and that, furthermore, "f is con-tained in every invariant linear manifold of 91. Another example is the subalgebra 9l„ of the above 91 consisting of finite-rank operators; is still the intersection of all nonzero members of i f (9I0). Received July 19, 1982. 8 114 H. Radjavi To prove Theorem 1, we shall need a lemma, which seems to be of independent interest, and which is itself an extension of the Rickart—Yood result referred to above. We first recall some definitions: An algebra 91 of linear operators on any vector space "V is called strictly transitive if for XT^O and y in "V there exists A in 91 with Ax=y. More generally, 91 is strictly n-fold transitive if for any independent vectors in -Vand arbitrary yt, ..., y„ in V there exists A in 91 with Axt=y( for / = 1, ..., n. If 91 is strictly «-fold transitive for every «, then it is called strictly dense. The well-known theorem of JACOBSON [1] (see [2, p. 50]) states that 2-fold transitivity implies strict density. In general, 1-fold transitivity does not imply strict density. Lemma 1. Let 91 be an algebra of bounded linear operators on the (not neces-sarily complete) complex normed linear space X. If 91 is strictly transitive, then it is strictly dense. (Note that 91 is not assumed to be closed in any topology.) Proof . Suppose 91 is strictly transitive but not strictly dense. Then by [1], 91 is not 2-fold transitive. It follows (as in [2, p. 62]), that there exists a (not necessarily bounded) non scalar linear transformation T of X onto X that commutes with every A in 91. Now (T-olI)A = A(T-aI) for every scalar a, and thus the nullspace and range of T—al are invariant linear manifolds for 91. It follows from the transitivity hypothesis that T—al is bijective for every a. Thus r(T) is a bijective linear transformation for every rational function r, and r(T)A—Ar(T) for all A in 91. Fix a nonzero x„ in X and let X0 be the linear manifold (r(T)x0: r a rational function}. Let 9i0={>4i9l: Ax0£X0}. Observe that 3c0 is invariant under 9l0. For each A in 9I0 there is a rational function rA such that Ax0=rA(t)x„ {rA is unique because of the bijectivity of r(T) for nonzero r); thus it follows from Ar(T)x0 = r(T)Ax0 = r(T)rA(T)x0 = rA(T)r(T)x0 that the restriction of A to X0 is just that of rA(T). Conversely, by the transitivity hypothesis, every r(T) coincides with rA(T) for some A in 9I0. Hence the restriction of 9I0 to X0 is a field. Since this restriction consists of bounded operators on X0, the Gelfand—Mazur theorem implies that T is a scalar on X0: T\X0=ctl, which contradicts the bijectivity of T—al on X. This proves that 21 is strictly dense. P roof of Theorem 1. Let •f be the intersection of all nonzero invariant linear manifolds of 91. Then the restriction of 91 to "f is clearly strictly transitive and thus strictly dense. Also, 91 has no closed invariant subspaces, because "V is dense in Minimal invariant manifolds and density of operator algebras 115 X. We use the notation and techniques described in [3, Chapter 8]. As in the proofs of Arveson's Lemma (Lemma 8.8) and Lemma 8.11 of [3], it suffices to show that each graph transformation for 91 has an eigenvalue; this will imply the strong density of 91 in 3t (£). Let {xffi T^x©... © Tnx\ be an invariant graph subspace for 9I(n+1) for some positive integer n. We must show that each linear transformation Ti has an eigenvalue. If Tt has nonzero null space, we are done. Otherwise, observe that the 9l-invari-ant linear manifolds X) and TtT> both contain % by hypothesis. Hence the identities ATi = TtA on D and ATf^ = Tc x A on 7J35 (for all A in 91) hold on "K This implies that T^and Tr^y are also invariant under 91, and thus contain Y. This yields T{V=~f. Now 91 is strictly dense on and com-mutes with the linear transformation Tt on We conclude that Tt is a scalar on "V and complete the proof. We conclude with a question an affirmative answer to which would be a generali-zation of Theorem 1. Ques t ion . If 91 is a topologically transitive algebra of bounded linear operators on X and if JS?(9I) has minimal nonzero elements, is 91 necessarily strongly closed in ^(X)? References [1] N. JACOBSON, Structure theory of simple rings without finiteness assumptions, Trans. Amer, Math. Soc., 5 7 (1944) , 2 2 8 — 2 4 5 . [2] C. E. RICKART, General Theory of Banach Algebras, D . Van Nostrand (Princeton, 1960) . [3] H. RADJAVI and P. ROSENTHAL, Invariant Subspaces, Springer-Verlag (Berlin—Heidelberg—New York, 1973). DALHOUSIE UNIVERSITY HALIFAX. N.S., CANADA 8* 

On minimal invariant manifolds and density of operator algebras

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On minimal invariant manifolds and density of operator algebras

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