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MINIMAL VECTORS IN ARBITRARY BANACH SPACES 3 0 0 VLADIMIR G. TROITSKY 2 n a Abstract. Weextendthemethodofminimalvectorstoarbitrary J Banach spaces. It is proved, by a variant of the method, that 3 certain quasinilpotent operators on arbitrary Banach spaces have 2 hyperinvariant subspaces. ] A F The method of minimal vectors was introduced by Ansari and Enflo . h in [AE98] in order to prove the existence of invariant subspaces for t a certain classes of operators on a Hilbert space. Pearcy used it in [P] to m [ prove a version of Lomonosov’s theorem. Androulakis in [A] adapted 1 the technique to super-reflexive Banach spaces. In [CPS] the method v 9 was independently generalized to reflexive Banach spaces. There has 6 been a hope that this technique could provide a positive solution to the 2 1 invariant subspace problem for these spaces. In this note we present a 0 3 version of the method of minimal vectors (based on [A]) that works for 0 / arbitrary Banach spaces. In particular, it applies in the spaces where h t there are known examples of operators without invariant subspaces, a m e.g., [E76, E87, R84, R85]. This shows that the method of minimal : v vectors alone cannot solve the invariant subspace problem for “good” i X spaces. r a Suppose that X is a Banach space. For simplicity, we assume that X is a real Banach space, though the results can be adapted to the complex case in the straightforward manner. In the following, B(x ,ε) 0 stands for the closed ball of radius ε centered at x while B◦(x ,ε) 0 0 stands for the open ball, and S(x ,ε) stands for the corresponding 0 sphere. 2000 Mathematics Subject Classification. Primary: 47A15. Key words and phrases. Invariant subspace, minimal vector. 1 2 V.G. TROITSKY Let Q be a bounded operator on X. Since we will be interested in the hyperinvariant subspaces of Q, we can assume without loss of generality that Q is one-to-one and has dense range, as otherwise kerQ or RangeQ would be hyperinvariant for Q. By {Q}′ we denote the commutant of Q. Fix a point x 6= 0 in X and a positive real ε < kx k. Let K = 0 0 Q−1B(x ,ε). Clearly, K is a convex closed set. Note that 0 ∈/ K and 0 K 6= ∅ because Q has dense range. Let d = inf kzk, then d > 0. It K is observed in [AE98, A] that if X is reflexive, then there exists z ∈ K with kzk = d, such a vector is called a minimal vector for x , ε 0 and Q. Even without reflexivity condition, however, one can always find y ∈ K with kyk 6 2d, such a y will be referred to as a 2-minimal vector for x , ε and Q. 0 The set K ∩B(0,d) is the set of all minimal vectors, in general this set may be empty. If z is a minimal vector, since z ∈ K = Q−1B(x ,ε) 0 then Qz ∈ B(x ,ε). As z is an element of minimal norm in K then, in 0 fact, Qz ∈ S(x ,ε). Since Q is one-to-one, we have 0 QB(0,d)∩B(x ,ε) = Q(cid:0)B(0,d)∩K) ⊆ S(x ,ε). 0 0 It follows that QB(0,d) and B◦(x ,ε) are two disjoint convex sets. 0 Since one of them has non-empty interior, they can be separated by a continuous linear functional (see, e.g., [AB99, Theorem 5.5]). That is, there exists a functional f with kfk = 1 and a positive real c such that f|QB(0,d) 6 c and f|B◦(x0,ε) > c. By continuity, f|B(x0,ε) > c. We say that f is a minimal functional for x , ε, and Q. 0 We claim that f(x ) > ε. Indeed, for every x with kxk 6 1 we 0 have x − εx ∈ B(x ,ε). It follows that f(x − εx) > c, so that 0 0 0 f(x ) > c+εf(x). Taking sup over all x with kxk 6 1 we get f(x ) > 0 0 c+εkfk > ε. Observe that the hyperplane Q∗f = c separates K and B(0,d). In- deed, if z ∈ B(0,d), then (Q∗f)(z) = f(Qz) 6 c, and if z ∈ K then Qz ∈ B(x ,ε) so that (Q∗f)(z) = f(Qz) > c. For every z with kzk 6 1 0 wehavedz ∈ B(0,d),sothat(Q∗f)(dz) 6 c,itfollowsthat(cid:13)Q∗f(cid:13) 6 c. (cid:13) (cid:13) d MINIMAL VECTORS IN BANACH SPACES 3 On the other hand, for every δ > 0 there exists z ∈ K with kzk 6 d+δ, then (Q∗f)(z) > c > c kzk, whence (cid:13)Q∗f(cid:13) > c . It follows that d+δ (cid:13) (cid:13) d+δ (cid:13)Q∗f(cid:13) = c. For every z ∈ K we have (Q∗f)(z) > c = d(cid:13)Q∗f(cid:13). In (cid:13) (cid:13) d (cid:13) (cid:13) particular, if y is a 2-minimal vector then (1) (Q∗f)(y) > 1(cid:13)Q∗f(cid:13)kyk. 2(cid:13) (cid:13) We proceed to the main theorem Theorem. Let Q be a quasinilpotent operator on a Banach space X, and suppose that there exists a closed ball B such that 0 ∈/ B and for every sequence (x ) in B there is a subsequence (x ) and a sequence n ni (K ) in {Q}′ such that kK k 6 1 and (K x ) converges in norm to a i i i ni non-zero vector. Then Q has a hyperinvariant subspace. Remark. The hypothesis of the theorem is slightly weaker than the condition (∗) in [A], where it is required that for every ε ∈ (0,1) there exists x of norm one such that the ball B(x ,ε) satisfies the rest of 0 0 the condition. Proof. Without loss of generality Q is one-to-one and has dense range. Let x 6= 0 and ε ∈ (cid:0)0,kx k(cid:1) be such that B = B(x ,ε). For every 0 0 0 n > 1 choose a 2-minimal vector y and a minimal functional f for n n x , ε, and Qn. 0 Since Q is quasinilpotent, there is a subsequence (y ) such that ni kyni−1k → 0. Indeed, otherwise there would exist δ > 0 such that kynik kyn−1k > δ for all n, so that ky k > δky k > ... > δnky k. Since kynk 1 2 n+1 Qny ∈ Q−1B then n+1 ky k δn (cid:13)Qny (cid:13) > d > 1 > ky k. n+1 n+1 (cid:13) (cid:13) 2 2 It follows that kQnk > δn/2, which contradicts the quasinilpotence of Q. Since kf k = 1 for all i, we can assume (by passing to a further ni subsequence), that (f ) weak*-converges to some g ∈ X∗. Since ni f (x ) > ε for all n, it follows that g(x ) > ε. In particular, g 6= 0. n 0 0 4 V.G. TROITSKY Consider the sequence (Qni−1y )∞ . It is contained in B, so that ni−1 i=1 bypassing toyet a further subsequence, ifnecessary, we finda sequence (K ) in {Q}′ such that kK k 6 1 and K Qni−1y converges in norm i i i ni−1 to some w 6= 0. Put Y = {Q}′Qw = (cid:8)TQw | T ∈ {Q}′(cid:9). One can easily verify that Y is a linear subspace of X invariant under {Q}′. Notice that Y is non-trivial because Q is one-to-one and 0 6= Qw ∈ Y. We will show that Y ⊆ kerg, so that Y is a proper Q-hyperinvariant subspace. Take T ∈ {Q}′; we will show that g(TQw) = 0. It follows from (1) that (cid:0)Q∗nif (cid:1)(y ) 6= 0 for every i, so that X = span{y } ⊕ ni ni ni ker(cid:0)Q∗nif (cid:1). Then one can write TK y = α y +r , where α is a ni i ni−1 i ni i i scalar and r ∈ ker(cid:0)Q∗nif (cid:1). We claim that α → 0. Indeed, i ni i (2) (cid:0)Q∗nif (cid:1)(cid:0)TK y (cid:1) = α (cid:0)Q∗nif (cid:1)(y ), ni i ni−1 i ni ni and, combining this with (1) we get |α | (3) (cid:12)(cid:0)Q∗nif (cid:1)(cid:0)TK y (cid:1)(cid:12) > i (cid:13)Q∗nif (cid:13)ky k. (cid:12) ni i ni−1 (cid:12) 2 (cid:13) ni(cid:13) ni On the other hand, (4) (cid:12)(cid:0)Q∗nif (cid:1)(cid:0)TK y (cid:1)(cid:12) 6 (cid:13)Q∗nif (cid:13)·kTk·ky k. (cid:12) ni i ni−1 (cid:12) (cid:13) ni(cid:13) ni−1 It follows from (3) and (4) that ky k |α | 6 2kTk ni−1 → 0. i ky k ni Then (2) yields that (cid:12)f (cid:0)QniTK y (cid:1)(cid:12) = (cid:12)α f (cid:0)Qniy (cid:1)(cid:12) (cid:12) ni i ni−1 (cid:12) (cid:12) i ni ni (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 6 |α |·kf k·kQniy k 6 |α |·1·(cid:0)kx k+ε(cid:1) → 0, i ni ni i 0 so that f (cid:0)QniTK y (cid:1) → 0. On the other hand, since T,K ∈ {Q}′ ni i ni−1 i we have QniTK y = TQK Qni−1y → TQw i ni−1 i ni−1 in norm, while f −w→∗ g, so that f (cid:0)QniTK y (cid:1) → g(TQw). Hence, ni ni i ni−1 g(TQw) = 0. (cid:3) MINIMAL VECTORS IN BANACH SPACES 5 Clearly, the argument will work as well for λ-minimal vectors for any λ > 1. Suppose that Q is a quasinilpotent operator commuting with a compact operator K. Then Q satisfies the hypothesis of Theorem . In- deed, without loss of generality kKk = 1. Fix ε = 1, there exists x 3 0 with kx k = 1 such that kKx k > 2, then 0 ∈/ KB(x ,ε). For every 0 0 3 0 sequence (x ) in B(x ,ε), the sequence (Kx ) has a convergent sub- n 0 n sequence (Kx ). Take K = K for all i; since 0 ∈/ KB(x ,ε) then ni i 0 lim K x 6= 0. It follows from Theorem that if Q is a quasinilpotent i i ni operator on a real or complex Banach space commuting with a non-zero compact operator, then Q has a hyperinvariant subspace. This fact is not new though: for complex Banach spaces it is a special case of the celebrated Lomonosov’s theorem [L73], and for real Banach spaces it follows from Theorem 2 of [H81]. Acknowledgements. Thanks are due to A. Litvak, N. Tomczak- Jaegermann, andR.Vershynin fortheirinterest intheworkandhelpful suggestions. The author would also like to thank the Department of Mathematics of the University of Alberta for its support and hospital- ity. References [AB99] C. D. Aliprantis and K. C. Border, Infinite-dimensional analysis, a hitch- hiker’s guide, second ed., Springer-Verlag,Berlin, 1999. MR 2000k:46001 [AE98] S. Ansari and P. Enflo, Extremal vectors and invariant subspaces, Trans. Amer. Math. Soc. 350 (1998), no. 2, 539–558. MR 98d:47019 [A] G.Androulakis,Anoteonthemethodofminimalvectors,TrendsinBanach spacesandoperatortheory,ContemporaryMathematics(EditorA.Kamin- ska), to appear. [CPS] I. Chalendar, J. R. Partington, and M. Smith, Approximation in reflexive banach spaces and applications to the invariant subspace problem, Proc. Amer. Math. Soc., to appear. [E76] P. 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MR 86f:47005 [R85] , A solution to the invariant subspace problem on the space ℓ1,Bull. London Math. Soc. 17 (1985), no. 4, 305–317. MR 87e:47013 [RR73] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, New York, 1973, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77. MR 51:3924 Department ofMathematics, University ofAlberta, Edmonton,AB, T6G2G1. Canada. E-mail address: [email protected]