INVARIANT SUBSPACES OF POSITIVE QUASINILPOTENT OPERATORS ON ORDERED BANACH SPACES

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1 INVARIANT SUBSPACES OF POSITIVE QUASINILPOTENT OPERATORS ON ORDERED BANACH SPACES HAILEGEBRIEL E. GESSESSE AND VLADIMIR G. TROITSKY Abstract. In this aer we find invariant subsaces of certain ositive quasinilotent oerators on Krein saces and, more generally, on ordered Banach saces with closed generating cones. In the later case, we use the method of minimal vectors. We resent alications to Sobolev saces, saces of differentiable functions, and C*-algebras. 0. Introduction and notations Lomonosov roved in [Lom73] that if an oerator T on a Banach sace is not a multile of the identity and commutes with a non-zero comact oerator, then there is a closed (roer non-zero) subsace which is invariant under every oerator commuting with T. The following result shows that the situation is even better for ositive oerators on Banach lattices. Theorem 0.1 ([AAB94]). Let S and T be two ositive commuting oerators on a Banach lattice such that S is quasinilotent and dominates a non-zero ositive comact oerator. Then T has a closed invariant subsace. Moreover, this invariant subsace can be chosen to be a closed order ideal. Several variations of this result can be found in [AAB93, AAB94, AAB98, AA02, Drn01]. In articular, ST = T S may be relaced with ST T S. In [AE98], the method of minimal vectors was used to show that the resence of a comact oerator can be relaced with a weaker localization condition. In the resent aer we extend some of these results beyond Banach lattices. We show that many of them remain valid in ordered Banach saces with closed generating cones. We resent alications to saces of differentiable functions, C -algebras, and Sobolev saces. The aer is organized as follows. In Section 1, we consider invariant subsaces of oerators on Krein saces. In Sections 2, 3 and 4, we consider alications of the Date: August 20, Mathematics Subject Classification. Primary: 47A15; Secondary: 46B42, 47B65. Key words and hrases. Invariant subsace, minimal vector, ordered normed sace, ositive oerator, Sobolev sace. 1 Positivity, 12(2), 2008,

2 2 H. E. GESSESSE AND V. G. TROITSKY results of Section 1. In Section 5, we extend the method of minimal vectors to ordered Banach saces with generating cones, and in Section 6 we resent alications of this method. We now introduce some notations from the theory of ordered Banach saces; for further details we refer the reader to [AB85, AT07]. A vector sace X over R is an ordered vector sace if it is equied with an order relation such that a b imlies a + c b + c and λa λb for all a, b, c X and λ R +. The ositive cone of X is defined as X + = {x X : x 0}. Furthermore, X is said to be a vector lattice if the order is a lattice order, hence x y and x y exist for all x and y. In articular, x + = x 0, x = ( x) 0, and x = x ( x) are defined for every x X, and we have x = x + x and x = x + + x. We say that X is an ordered Banach sace if it is a Banach sace and an ordered vector sace such that X + is norm-closed. If, in addition, X = X + X +, then we say that X is a Banach sace with a generating closed cone. A lattice-ordered Banach sace is an ordered Banach sace with the order being a lattice order. A lattice-ordered Banach sace is a Banach lattice if x y imlies x y for all x, y X. A few examles can be mentioned. The classical saces L (µ), where µ is a measure and 1, as well as C 0 (Ω), where Ω is a locally comact Hausdorff sace, are Banach lattices. However, there are many imortant ordered Banach saces with generating cones which are not Banach lattices. One could mention, in articular, the saces C k [0, 1] of all k times continuously differentiable functions on [0, 1], as well as C*-algebras. Sobolev saces W k, (Ω) are ordered Banach saces. Moreover, we observe in Section 4 that for k = 1 they are lattice-ordered Banach saces. However, they fail to be Banach lattices as the norm is not monotone on X +. Throughout this aer, X will usually stand for an ordered vector or Banach sace. By a subsace of X we will always mean a linear subsace. By an oerator on X we will understand a linear oerator. If a < b in X, we write [a, b] = {x X : a x b}. Sets of this form are called order intervals. An oerator T on X is ositive if T x 0 whenever x 0. In this case we write T 0. We write T S if S T 0. If S and T are two oerators on X, we say that T is dominated by S if ±T x Sx whenever x 0. If X is a vector lattice, this is equivalent to T x S x for every x X because ±T x = ±(T x + T x ) Sx + + Sx = S x. A subsace E of X is said to be invariant under a collection S of oerators on X if {0} E X and T (E) E for every T S. For A, B X and u X we write u + B = {u + b : b B} and A + B = {a + b : a A, b B}. If A and B are two

3 INVARIANT SUBSPACES OF POSITIVE OPERATORS 3 subsets of X, we say that A minorizes B if for each b in B there exists a A such that a b. Suose now that X is an ordered Banach sace. It follows from [a, b] = (a + X + ) (b X + ) that order intervals are closed sets. We will write L(X) for the sace of all bounded oerators on X. Note that L(X) is an ordered Banach sace, so that if T S in L(X), we write [T, S] = { R L(X) : T R S } (in articular, all the oerators in [T, S] are bounded). The commutant of T L(X) is the set {T } = {S L(X) : T S = ST }. Following [AA02], we define the suer leftcommutant Q] and the suer right-commutant of [Q of Q as follows: Q] = {T 0 : T Q QT } and [Q = {T 0 : T Q QT }. We will make use of the following well-known theorem (see, e.g., [AAB92]). Theorem 0.2 (Lozanovski). Every ositive oerator between ordered Banach saces with generating closed cones is bounded. Recall that a subsace E of a vector lattice X is said to be an (order) ideal if a E and x a imly x E. If A X, then I(A) stands for the smallest ideal in X containing A. It is easy to see that I(A) = { n x X : x λ i a i, λ 1,..., λ n R and a 1,..., a n A }. i=1 We now extend these concets to ordered vector saces. A subsace E in an ordered vector sace is said to be an (order) ideal if (i) x E imlies that there exists a ositive a E such that x a, and (ii) ±x a E imlies x E. Note that in a vector lattice this definition agrees with the usual one. Furthermore, if X is an ordered vector sace and A is a convex subset of X, we will write I 0 (A) = { x X : ±x λa for some λ R + and some a A }. Note that I 0 (A) need not contain A. The following three lemmas are elementary, we leave the roofs to the reader. Lemma 0.3. Suose that X is an ordered vector sace. (i) If A is a convex subset of X then I 0 (A) is an ideal; I 0 (A) is contained in every ideal containing A. (ii) A subset E X is an ideal iff E = a E + I 0 (a).

4 4 H. E. GESSESSE AND V. G. TROITSKY (iii) Suose that E and F are two ideals in X such that I 0 (a) I 0 (b) is an ideal whenever 0 a E and 0 b F. Then E F is an ideal. Lemma 0.4. Suose that X is a vector lattice and A X such that either (i) A is a convex subset of X +, or (ii) A is the range of a ositive oerator on X. Then I 0 (A) = I(A). Lemma 0.5. Let X be an ordered vector sace with a generating cone and Q a ositive oerator on X. Then I 0 (Range Q) is an ideal invariant under Q]. Note that if X is a Banach lattice and E is an ideal in X then E is still an ideal. This remains true if X is a lattice-ordered Banach saces with continuous lattice oerations. Since various lattice oerations can be exressed via each other, the continuity of any of the mas x x +, x x, x x, x x y, and x x y imlies the continuity of the other mas and joint continuity of x y and x y. Proosition 0.6. Suose that X is a lattice-ordered Banach sace with continuous lattice oerations and E is an ideal in X. Then E is again an ideal. Proof. Suose that x E. Take a sequence (x n ) in E such that x n x. Then E x n x, hence x E. Suose now that 0 y x E. Again, take a sequence (x n ) in E such that x n x. Then for every n we have E x n y x y = y, hence y E. 1. Positive quasinilotent oerators on Krein saces Suose that X is an ordered Banach sace (in articular, X + is closed). Recall that u X + is said to be a (strong) order unit if for every x X there exists λ > 0 such that x λu or, equivalently, if I 0 (u) = X. We make use of the following standard lemma. Lemma 1.1 (c.f. [AAB92, Lemma 3.2]). Suose that X is an ordered Banach sace and u X +. Then the following statements are equivalent: (i) u is an order unit; (ii) u Int X + ; (iii) λb 0 [ u, u] for some λ R +, where B 0 is the unit ball of X.

5 INVARIANT SUBSPACES OF POSITIVE OPERATORS 5 Proof. (iii) (i) is trivial. To show (ii) (iii), suose that u Int X +. Then u+λb 0 X + for some λ R +. It follows that for every x X with x λ we have u ± x 0, so that u x u, hence λb 0 [ u, u]. To show (i) (ii), suose that u is an order unit. Then X = n=1 [ nu, nu]. By the Baire Category Theorem, [ u, u] has non-emty interior, so that B(a, λ) [ u, u] for some a X and λ R +. Now suose that x B(u, λ), then a ± (x u) B(a, λ) [ u, u]. Now u a+(x u) imlies x+a 0, while a (x u) u imlies x a 0. Adding these two inequalities together we get x 0, so that B(u, λ) X +. An ordered Banach sace with an order unit is said to be a Krein sace. Clearly, the ositive cone in a Krein sace is generating. We would like to mention the following result, which is a corollary of Krein s theorem [KR48], see also [AAB92, SW99, OT05]. Theorem 1.2. If T is a ositive oerator on a Krein sace then there is a closed subsace invariant under {T }. Suose that X is an ordered Banach sace. Recall that w X + is said to be quasi-interior if I 0 (w) is dense in X. Clearly, every order unit is quasi-interior. Lemma 1.3. If X is a Krein sace then every quasi-interior oint is an order unit. Proof. Let w X + be quasi-interior. By Lemma 1.1 there exists u X + such that B 0 [ u, u]. It follows that 2u + B 0 [u, 3u]. Since I 0 (w) is dense, it meets 2u + B 0. It follows that there exists x B 0 such that u 2u + x λw for some λ R +. Therefore, w 1 u, hence I λ 0(w) I 0 (u) = X. Theorem 1.4. Suose that Q is a ositive quasinilotent oerator on a Krein sace X. Then Q] has a non-dense invariant ideal. Proof. Since X is a Krein sace, there is an order unit u in X +. Let w = Qu. We claim that w is not quasi-interior. Indeed, otherwise it would be a unit by Lemma 1.3, so that u λw for some λ R +. It follows that Qu 1 λ u, so that Qn u 1 λ n u, hence λ n Q n u is contained in u + X +. Since λq is quasinilotent, we have λ n Q n u 0. But u + X + is closed, hence 0 u + X +, a contradiction. Let x X, then ±x λu for some λ R +, so that ±Qx λw. It follows that Range Q I 0 (w), so that I 0 (Range Q) = I 0 (w), hence I 0 (Range Q) is not dense. Now aly Lemma 0.5.

6 6 H. E. GESSESSE AND V. G. TROITSKY 2. Alications to C(K) and C k (Ω) saces and to C*-algebras In this section we aly the results of the receding sections to unital uniform algebras. We will see that these algebras are Krein saces, and aly Theorem 1.4 to find invariant closed subsaces for ositive quasinilotent oerators on them. Moreover, these subsaces can be chosen to be the closures of non-dense invariant order ideals. It should be ointed out that in this setting one has to distinguish between the algebraic ideals and the order ideals. The non-unital case will be considered in Section 6. Clearly, for every comact Hausdorff sace K, the sace C(K) is a Krein sace with 1 being an order unit. Hence, Theorem 1.4 yields the following. Corollary 2.1. If Q is a ositive quasinilotent oerator on C(K) where K is a comact Hausdorff sace, then Q] has a closed invariant ideal. Let Ω be an oen connected subset of R n. Recall that the sace C k (Ω) consists of all real-values functions f on Ω such that D α f is bounded and uniformly continuous on Ω whenever α k. With the norm defined by f = α k D αf su, the set C k (Ω) is an ordered Banach sace, see, e.g., [KJF77]. Clearly, 1 is an order unit in C k (Ω), hence C k (Ω) is a Krein sace. Therefore, by Theorem 1.4 we have the following. Corollary 2.2. If Q is a ositive quasinilotent oerator on C k (Ω) then Q] has an invariant non-dense order ideal. In articular, it has a closed invariant subsace. Before we roceed to C*-algebras, we would like to make a remark about alicability of our techniques to comlex Banach saces. In all our revious results we assumed X to be an ordered Banach sace over real scalars. By a comlex ordered Banach sace we understand the comlexification X c = X + ix of an ordered Banach sace X over R (see [AA02] on comlexifications of ordered Banach saces). An oerator T L(X c ) is said to be ositive if T (X + ) X +, where X + is the ositive cone of X. Suose that X + is generating in X. Then every ositive oerator T in L(X c ) is the comlexification of its restriction to X, that is, T = (T X ) c. It is easy to see that if a subsace V of X is invariant under S L(X), then V + iv is a subsace of X c invariant under S c. Hence, in order to rove that a ositive oerator T on X c has an invariant subsace, it suffices to find an invariant subsace for the restriction of T to X. Let A be a C*-algebra. Then A is a Banach sace over C. Observe that its selfadjoint art A sa is a real Banach sace. Recall that for x A sa we write x 0 when σ(x) [0, + ). With this order, the ositive cone A + is closed and generating

7 INVARIANT SUBSPACES OF POSITIVE OPERATORS 7 (generally, A sa is not a lattice). Furthermore, A can be viewed as the comlexification of A sa. Suose that T is a ositive oerator on A. By the receding aragrah, if its restriction to A sa has an invariant subsace in A sa, then the T has an invariant subsace in A. Hence, it suffices to look for invariant subsaces of ositive oerators on A sa. If, in addition, A is unital, then A sa is a Krein sace. Indeed, suose that x A sa with x 1. Then σ(±x) [ 1, 1] and the Sectral Maing Theorem yields σ(e ± x) [0, 2] R +, so that ±x e and, therefore, x [ e, e]. It follows from Lemma 1.1 that A sa is a Krein sace. Hence, Theorem 1.4 yields the following result. Theorem 2.3. If Q is a ositive quasinilotent oerator on a unital C*-algebra then Q] has an invariant non-dense order ideal. In articular, it has a closed invariant subsace. 3. Alications to Sobolev saces W k, (Ω) Throughout this section we assume that Ω is a bounded oen subset of R N, 1, and k N. The Sobolev sace W k, (Ω) is defined as the set of all those functions f L (Ω) for which the weak artial derivatives D α f exist and are in L (Ω) for each multi-index α with α k. The norm on W k, (Ω) is given by [ Dα f ] 1 L if <, f = α k Dα f L if =, where L α k is the norm in L (Ω). As usually, we will write 1 for χ Ω. For more details on Sobolev saces we refer the reader to [GT77, PW03, KJF77]. We equi W k, (Ω) with the a.e. order, i.e., the order inherited from L (Ω). Note that X + is norm-closed. Indeed, let (f n ) be a sequence in X + such that f n f in W k, (Ω). Then f n f L 0, so that f 0 because the ositive cone of L (Ω) is closed. Thus, W k, (Ω) is an ordered Banach sace. It is well known that as a Banach sace W k, (Ω) is isomorhic to L (0, 1) when 1 < <, see, e.g., [PW03, Theorem 11]. However, we will see that the order structure of W k, (Ω) is very different from that of L (Ω). Remark 3.1. Observe that the norm of Sobolev saces is generally not monotone. For examle, let f W 1,1 [0, 1] be defined as follows: f(t) = t for all t [0, 1]. Then 0 f 1, but f > 1.

8 8 H. E. GESSESSE AND V. G. TROITSKY The question of existence of invariant subsaces of ositive oerators on Sobolev saces was investigated in [IM04], and a variant of Theorem 0.1 for W 1, (Ω) was roved there. If = then 1 is an order unit in W k, (Ω). It follows that W k, (Ω) is a Krein sace, so that if Q is a ositive quasinilotent oerator on W k, (Ω) then Q] has an invariant non-dense ideal by Theorem 1.4. For this reason, from now on we assume that <. Let X = W k, (Ω) such that Ω is regular (i.e., Ω is of class C 0,1, see, e.g., [GT77] for the recise definition of regular domains) and > N/k or = N/k = 1. The classical Sobolev embedding theorem asserts that in this case there exists C > 0 such that f C f for every f X. It follows that the unit ball of X is contained in [ C1, C1]. Combining this observation with Lemma 1.1, we obtain the following result. Theorem 3.2. If Ω is regular and > N/k or = N/k = 1 then W k, (Ω) is a Krein sace. In articular, W 1, [0, 1] is a Krein sace for any 1 <. The next result now follows immediately from Theorems 0.2, 3.2, 1.2, and 1.4. Theorem 3.3. Suose that Ω is regular and > N/k or = N/k = 1. If Q is a ositive oerator on W k, (Ω) then Q is bounded and there is a closed subsace invariant under {Q}. If, in addition, Q is quasinilotent, then Q] has a non-dense invariant ideal. 4. A secial case: W 1, (Ω) In this section we consider the case k = 1. The following fact from [GT77] shows that W 1, (Ω) is a sublattice of L (Ω). Lemma 4.1 ([GT77, Lemma 7.6]). If f W 1, (Ω) then f +, f, and f are also in W 1, (Ω) and { { f + f x = i if f > 0 0 if f 0, f 0 if f 0 = f if f < 0, f = for each i = 1,..., N (the equalities are a.e.). f if f > 0 0 if f = 0 f if f < 0

9 INVARIANT SUBSPACES OF POSITIVE OPERATORS 9 As usually, f, f +, and f are defined ointwise. Hence, W 1, (Ω) is a vector lattice and f = f for every f W 1, (Ω) (recall that W 1, (Ω) is generally not a Banach lattice by Remark 3.1). In articular, the ositive cone of W 1, (Ω) is generating. Together with Theorem 0.2 this immediately yields the following. Corollary 4.2. Every ositive oerator on W 1, (Ω) is bounded. Theorem 4.3. The Sobolev sace W 1, (Ω) is a lattice-ordered Banach sace with continuous lattice oerations. Proof. Let X = W 1, (Ω). In view of the receding discussion, we only need to verify the continuity of the lattice oerations in X. Take (f n ) in X such that f n f for some f X. It follows that f n f L 0 and fn f L 0 for each i. Since L (Ω) is a Banach lattice, f + n f + L 0. It is left to show that f + n f + L 0 for every i. Consider the following subsets of Ω: A + = {f > 0}, A = {f < 0}, and A 0 = {f = 0}. Also, for every n define A + n = {f n > 0} and A 0 n = {f n 0}. In order to show that f n + f + 0, Ω we slit it into integrals over the following six sets: A + A + n, A + A 0 n, A A + n, A A 0 n on A A 0 n, A 0 A + n, and A 0 A 0 n and A 0 A 0 n. In fact, by Lemma 4.1, the integrand vanishes a.e., so that there is nothing to do about these two sets. Next, we consider A + A + n and A 0 A + n. Lemma 4.1 yields that f n + a.e. on A +. Further, on A 0 we have f A + n and f + Therefore, (A + A 0 ) A + n = f f n + f + = (A + A 0 ) A + n f n f = fn = 0, hence f = f + Ω a.e. on f n f 0. For the remaining two sets, we show first that the sequences m(a + A 0 n ) and m(a A + n ) tend to zero as n, where m stands for the Lebesgue measure on Ω. a.e. Indeed, since f n f L 0, by assing to a subsequence we may assume that f n f a.e. on Ω. In articular, f n f on A +. By Egoroff s theorem we can find B A + such that f n f uniformly on B and m(b) m(a + ) ε. Since A + = k=1 {f > 1 }, we k have m ( {f > 1}) > m(a + ) ε for some k. It follows that f k n > 0 on {f > 1} B for k all sufficiently large n, so that A 0 n Similarly, we show that m(a A + n ) 0. Now Lemma 4.1 yields A + A 0 n f n + a.e.. {f > 1 k } B =. Therefore, m(a+ A 0 n ) 0. f + = A + A 0 n f 0,

10 10 H. E. GESSESSE AND V. G. TROITSKY and A A + n f n + f + = 2 A A + n A A + n It follows that f + n f + L 0. f n f n f + 2 f A A + x n i 2 f n f + 2 Ω A A + n f 0. Combining Theorem 4.3 with Proosition 0.6, we immediately obtain the following result. Corollary 4.4. The closure of every ideal in W 1, (Ω) is again an ideal. Recall that Theorem 0.1 asserts that an oerator on a Banach lattice satisfying a certain condition has an invariant closed ideal. As we mentioned before, a variant of Theorem 0.1 for W 1, (Ω) roved in [IM04] states that an oerator on W 1, (Ω) satisfying the same condition has a closed invariant subsace. In fact, this subsace is roduced in [IM04] as the closure of a certain ideal. In view of Corollary 4.4, this closure is itself an ideal. Hence, the following result holds. Corollary 4.5. Let S and T be two ositive commuting oerators on a W 1, (Ω) such that S is quasinilotent and dominates a non-zero ositive comact oerator. Then T has a closed invariant ideal. As in [IM04], instead of S being quasinilotent, it is sufficient that S be locally quasinilotent at a ositive vector. Theorem 3.3 and Corollary 4.4 together yield the following result. Theorem 4.6. Suose that Ω is regular and > N or = N = 1. If Q is a ositive oerator on W 1, (Ω) then Q is bounded and there is a closed subsace invariant under {Q}. If, in addition, Q is quasinilotent, then Q] has a closed invariant ideal. 5. Minimal vectors in saces with a generating cone The method of minimal vectors was originally develoed in [AE98] to rove the existence of invariant subsaces for certain classes of oerators on Hilbert saces. The method was later extended to Banach sace in [JKP03, And03, CPS04, Tro04, LRT]. Following [LRT], we say that a collection F of linear bounded oerators on a Banach sace X localizes a subset A X if for every sequence (x n ) in A there is a subsequence

11 INVARIANT SUBSPACES OF POSITIVE OPERATORS 11 (x ni ) and a sequence (K i ) in F such that the sequence (K i x ni ) converges to a non-zero vector. It was roved in [Tro04] that if T is a quasinilotent oerator on a Banach sace X such that the unit ball of the commutant of T localizes a closed ball in X, then T has a hyerinvariant subsace. This imlies Lomonosov s theorem [Lom73] for quasinilotent oerators. In [AT05], the method of minimal vectors was used to roduce several generalizations of Theorem 0.1 and of other related results. In this section we show that the technique develoed in [AT05] extends naturally to ordered Banach saces with generating cones. In fact, it is even simler and more transarent in this more general setting, and the roofs are shorter. The crucial tool in extending the method is the following well-known theorem (see, e.g., [AAB92]). Theorem 5.1 (Krein and Šmulian). Let X be a Banach sace with a generating closed cone X +. There exists a constant > 0 such that for each x X there exist x 1, x 2 X + such that x = x 1 x 2 and x i x, i = 1, 2. The smallest such will be denoted (X). We will write B(u, r) for the closed ball of radius r centered at u X. We will also write B u = B(u, 1). In articular, B 0 stands for the unit ball of X. Suose that Q is a ositive bounded oerator on an ordered Banach sace X, u X +, and C R +. Following [AT05], we say that y X is a C-minimal vector for Q and u if y 0, y Q 1 (B u + X + ) and y C dist { 0, Q 1 (B u + X + ) }. Lemma 5.2. Suose that X is a Banach sace with a generating closed cone with = (X) and Q is a ositive oerator on X such that I 0 (Range Q) is dense in X. Suose that u X + such that 0 / B u + X +. Then there exists a (2 )-minimal vector for Q and u. Proof. Note first that Q is bounded by Theorem 0.2. Since I 0 (Range Q) is dense, it meets B u. Let x B u I 0 (Range Q). Then x Qh for some h, so that Qh B u +X +, hence Range Q (B u + X + ) is non-emty. Put D = Q 1 (B u + X + ), then D is nonemty. Since Q is continuous, 0 / D. It follows that d := dist{d, 0} > 0. Find z D such that z 2d. By Theorem 5.1 there exists y X + such that z y and y 2d. It follows from Qz B u + X + and Q 0 that Qz Qy, so that Qy B u + X +, hence y is a (2 )-minimal vector. Proosition 5.3. Suose that Q is a ositive bounded oerator on an ordered Banach sace X, u X + such that 0 / B u + X +, and C R +. Suose that y is a C-minimal

12 12 H. E. GESSESSE AND V. G. TROITSKY vector for Q and u. Then there exists f X (called a minimal functional) such that (i) f is ositive, and f(u) 1 = f ; (ii) There exists c > 0 such that f Q(B(0,d)) c and f Bu+X + c, where d = dist { 0, Q 1 (B u + X + ) } ; (iii) 1 C Q f y (Q f)(y) Q f y ; (iv) f(qy) C u ; (v) Q f C2 u y. Proof. (i), (ii), and (iii) are roved exactly as the corresonding statements in [AT05, Lemma 6]. Note that C 1 y B(0, d), hence C 1 Qy Q ( B(0, d) ). Since u B u + X +, (ii) and (i) imly C 1 f(qy) c f(u) u, which roves (iv). combining (iii) and (iv) we get (v). For the rest of this section we will make the following assumtions. Finally, (1) (i) X is a Banach sace with a generating closed cone; (ii) u X + such that 0 / B u + X + ; (iii) A X + such that A minorizes (B u + X + ) X + ; (iv) Q is a ositive quasinilotent oerator on X. The goal of this section is to show that under these assumtions lus some extra conditions, Q] has a closed invariant subsace. Before we roceed, we would like to make a few comments on these assumtions. For A one can always take, e.g., the entire set (B u + X + ) X +. However, in what follows, we will be localizing A, so that we will be interested in having A as small as ossible. If X is a vector lattice, one can always take A = [ u ( )] B u + X + X+. In [AT05], where X was a Banach lattice, the set B u [0, u] served as A. Clearly, 0 / B u + X + imlies u > 1. The converse is also true if the norm is monotone on X +, i.e., when 0 x y imlies x y. In articular, this holds if X is a Banach lattice. The following examle shows that the converse fails in general. Examle. Let X = R 2 ordered so that X + is the first quadrant. Norm X so that the unit ball is the absolute convex hull of (0, 1) and (2, 2). Let u = (0, 2), then u = 2. However, ( 2, 0) B u and (2, 0) X +, so that (0, 0) B u + X +. Finally, note that under assumtions (1), Theorem 0.2 guarantees that Q is bounded. Proosition 5.4. Suose that X, u, and Q are as in (1) and I 0 (Range Q) is dense in X. Put = (X). Then for every n N there exists a ositive (2 )-minimal

13 INVARIANT SUBSPACES OF POSITIVE OPERATORS 13 vector y n and a minimal functional f n for Q n and u. Furthermore, there exists a subsequence (n i ) such that y n i 1 w y ni 0 and f ni g for some non-zero g X+. Proof. The existence of sequences of minimal vectors (y n ) and minimal functionals (f n ) follows from Lemma 5.2 and Proosition 5.3. Suose that there exists δ > 0 such that y n 1 y n > δ for all n, so that y 1 δ y 2... δ n y n+1. Let again D = Q 1( ) B u + X +. Then Q n y n+1 D, so that Q n y n+1 dist ( 0, D) y 1 2 δn 2 y n+1. It follows that Q n δn, which contradicts the quasinilotence of Q. Hence, 2 y ni 1 y ni 0 for some subsequence (y ni ). Since f n = 1 for every n and the unit ball of X is weak*-comact, we can assume by assing to a further subsequence that w g for some g X. Clearly, g 0. Since f n (u) 1 for each n, it follows that f ni g(u) 1, hence g 0. Theorem 5.5. Suose that X, u, A, and Q are as in (1). If the set of all oerators dominated by Q localizes A then Q] has an invariant closed subsace. Furthermore, if [0, Q] localizes A then Q] has a non-dense invariant ideal. Proof. Let = (X). By Theorem 0.2, every oerator in Q] and [0, Q] is bounded. In view of Lemma 0.5 we may assume without loss of generality that I 0 (Range Q) is dense in X. Let (y ni ), (f ni ), and g be as in Proosition 5.4. Then Q n i 1 y ni 1 (B u +X + ) X + for each i. It follows that there exists a i A such that a i Q n i 1 y ni 1. By assing to a further subsequence, we can find a sequence (K i ) such that Q dominates K i for every i and K i a i converges to some w 0. Then ±K i a i Qa i Q n i y ni 1. For every T Q] we have ± f ni (T K i a i ) f ni (Q n i T y ni 1) = (Q n i f ni )(T y ni 1) Q n i f ni T y ni u y ni T y n i 1 by Proositions 5.3(v). It follows from y n i 1 y ni 0 that f ni (T K i a i ) 0. On the other hand, f ni (T K i a i ) g(t w). Thus, g(t w) = 0 for every T Q]. If T w = 0 for all T Q] then the one-dimensional subsace sanned by w is invariant under Q]. Suose T w 0 for some T Q]; let Y be the linear san of Q]w. Then Y is non-trivial and invariant under Q]. Finally, Y ker g imlies Y X, hence Y is a required subsace.

14 14 H. E. GESSESSE AND V. G. TROITSKY Suose now that [0, Q] localizes A. Then we can assume that the vector w constructed in the receding argument is ositive. Put E = I 0 ( Q]w ). Then E is an ideal by Lemma 0.3(i); E is invariant under Q], and E is non-trivial as w E. Finally, g 0 imlies E ker g, hence E X. Theorem 5.6. Suose that X, u, A, and Q are as in (1), and I 0 (Range Q) is dense. Suose also that F is a family of ositive contractive oerators on X such that F localizes A and S L(X) is a semigrou such that SF Q]. Then S has an invariant closed subsace. Furthermore, if S consists of ositive oerators then it has a non-dense invariant ideal. Proof. Again, let (y ni ), (f ni ), and g be as in Proosition 5.4 with = (X). Since Q n i 1 y ni 1 (B u +X + ) X + for every i, we can find a i A such that a i Q n i 1 y ni 1. By assing to a further subsequence, we can find a sequence (K i ) in F such that K i a i converges to some w 0. Let T S. Since T K i Q], Proositions 5.3(v) yields 0 f ni (QT K i a i ) f ni (Q n i T K i y ni 1) Q n i f ni T y ni u y ni T y n i 1 0 since y n i 1 y ni 0. It follows that g(qt w) = 0 for every T S. Let Y be the linear san of Sw; then Y is invariant under S. Since I 0 (Range Q) is dense and g 0 we have Q g 0. This yields Y X because T w ker Q g for every T S. Finally, if Y = {0}, then T w = 0 for each T S, so that san w is a one-dimensional subsace invariant under S. Suose now that S consists of ositive oerators. Then w can be chosen to be ositive. Let A be the convex hull of Sw and ut E = I 0 (A); then E is an ideal by Lemma 0.3(i) and E is invariant under S. It follows from Q g 0 that Q g vanishes on E, hence E X. If E {0}, we are done. Otherwise, T w = 0 for each T S. Put F = I 0 (w); then F is a non-trivial ideal; F is invariant under S since every T S vanishes on F. Also, F X as otherwise every oerator in S is zero. Corollary 5.7. Suose that X, u, A, and Q are as in (1). If the set of all contractions in Q] localizes A then Q] has a non-dense invariant ideal. Proof. By Lemma 0.5 we may assume that I 0 (Range Q) is dense. Now aly Theorem 5.6 with F = { K Q] : K 1 } and S = Q].

15 INVARIANT SUBSPACES OF POSITIVE OPERATORS 15 Remark 5.8. In view of Proosition 0.6, if X is a lattice-ordered Banach sace with continuous oerations then the invariant ideals in Theorems 5.5 and 5.6 and Corollary 5.7 can be taken to be closed. 6. Alications of minimal vector technique Non-unital uniform algebras. Consider A sa for a non-unital C*-algebra A. imortant secial case is the sace C 0 (Ω) of continuous functions on a locally comact Hausdorff sace which vanish at infinity; equied with su-norm. These saces are ordered Banach saces with closed generating cones (but not Krein saces). Hence, Theorems 5.5 and 5.6, as well as Corollary 5.7, guarantee that if Q is a ositive quasinilotent oerator on any of these saces satisfying the conditions described in the theorems, then Q] has a common invariant closed subsace. Observe that for every u A + with u > 1 we automatically have 0 / B u + A + (this is one of the assumtions in (1)). Indeed, suose that 0 B u + A +, then there exist sequences (x n ) in B 0 and h n A + such that u + x n + h n 0. It follows that u + h n x n 0. However, in a C*-algebra, 0 a b imlies a b ; see, e.g., [SW99, VI.3.2.ii]. Hence u + h n x n u 1 > 0, a contradiction. Sobolev saces. Finally, we resent an alication of the localization technique develoed in Section 5 to W 1, (Ω). In the following theorem, we do not require that Ω is regular or that and N are related. Being a lattice-ordered sace, W 1, (Ω) clearly has a generating cone. Combining Corollaries 4.2 and 4.4 with Theorem 5.5 and Corollary 5.7, we obtain the following result. Theorem 6.1. Let X = W 1, (Ω) and suose that Q is a ositive quasinilotent oerator on X, u X + with 0 / B u + X +, and A X + such that A minorizes (B u + X + ) X +. (i) If the set of all oerators dominated by Q localizes A then Q] has an invariant closed subsace. (ii) if either [0, Q] or the set of all contractions in Q] localizes A, then Q] has an invariant closed ideal. Next, we show that the assumtions in Theorem 6.1 are not as restrictive as they might seem. Lemma 6.2. If X = W k, (Ω) and u X + with u L > 1 then 0 / B u + X +. An

16 16 H. E. GESSESSE AND V. G. TROITSKY Proof. Suose not. Then there exist sequences (x n ) in the unit ball of X and (h n ) in X + such that u + x n + h n 0. This imlies u + x n + h n L 0, so that u + h n L x n L 0. However, u + h n L x n L u L 1 > 0, a contradiction. Remark 6.3. Let X be a lattice ordered Banach sace such that x = x for every x X (for examle, X = W 1, (Ω)). Let u X + such that 0 / B u + X +. Then, of course, u > 1. Let A = {y + : y B u }. We claim that then A is norm-bounded, A (B u + X + ) X + so that 0 / A, and A minorizes (B u + X + ) X +. Indeed, suose that y B u. We have y + = y+ y 2, hence y + y u + 1, so that A is bounded. Also, y + = y + y B u + X +, so that A (B u + X + ) X +. Now, let z (B u + X + ) X +. Then z = y + h for some y B u and h 0. It follows from z 0 and z y that z y +, hence A minorizes (B u + X + ) X +. We now show that Theorem 6.1 imlies an analogue of Theorem 0.1 for W 1, (Ω). Theorem 6.4. Suose that Ω is a regular domain and Q is a ositive quasinilotent oerator on X = W 1, (Ω) and 0 K Q for a non-zero comact oerator K. Then Q] has a closed invariant ideal. Proof. Since K 0 we have K1 0 as, otherwise, K would vanish on all bounded functions, but bounded functions are dense in X. It follows that there exists ε > 0 such that 0 / K ( B(1, ε) ). Let u = 1 ε 1, then 0 / K(B u). By scaling u even further we may also assume that u L > 1, hence 0 / B u + X + by Lemma 6.2. As in Remark 6.3, take A = {y + : y B u }; then A minorizes (B u + X + ) X +. Since u is a multile of 1, it follows from Lemma 4.1 that y + u y u 1 for every y B u ; hence A B u. Observe that [0, Q] localizes A. Indeed, let (x n ) be a sequence in A. Since K [0, Q] and K is comact, there is a subsequence (x ni ) such that Kx ni w for some w. Clearly, w K(A) K(B u ), hence w 0. Now aly Theorem 6.1. References [AA02] Y.A. Abramovich and C.D. Alirantis, An invitation to oerator theory, Graduate Studies in Mathematics, vol. 50, Amer. Math. Soc., Providence, RI, [AAB92] Y.A. Abramovich, C.D. Alirantis, and O. Burkinshaw, Positive oerators on Kreĭn saces, Acta Al. Math. 27 (1992), no. 1-2, 1 22, Positive oerators and semigrous on Banach lattices (Curaçao, 1990). [AAB93], Invariant subsaces of oerators on l -saces, J. Funct. Anal. 115 (1993), no. 2,

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