ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX0000-0 ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS MARCH T. BOEDIHARDJO AND WILLIAM B. JOHNSON 2 (Commuicated by Abstract. I this paper, we give a geometric characterizatio of mea ergodic covergece i the Calki algebras for Baach spaces that have the bouded compact approximatio property.. Itroductio We begi by recallig that i 974, M. Li [4] proved the followig uiform ergodic theorem, where, as usual, B(X deotes the algebra of all bouded liear operators o the Baach space X. Theorem.. Suppose that X is a ( Baach space ad that T B(X satisfies T lim = 0. The the sequece T k coverges i orm to a + elemet i B(X if ad oly if I T has closed rage. To the authors kowledge, there has bee o attempt ( to make a geometric charaterizatio of the orm covergece of the sequece T k i the + Calki algebra B(X/K(X, where K(X is the closed ideal of compact operators i B(X ad T is the image of T i B(X/K(X. The mai difficulty ivolved here is to fid a aalog of the property that I T has closed rage i the settig of B(X/K(X. I spite of the theory of semi-fredholm operators, the fact that a operator S B(X has closed rage does ot, i geeral, imply that S + K has closed rage for every K K(X. To obtai the desired characterizatio, we itroduce the followig cocept. Let X be a Baach space ad let (P be a property that a subspace M of X may or may ot have. We say that a subspace M X is a essetially maximal subspace of X satisfyig (P if M itself has property (P ad if every subspace M 0 M havig property (P satisfies dim M 0 /M <. Usig this cocept, we ca state 99 Mathematics Subject Classificatio. Primary 46B08, 47A35, 47B07. Key words ad phrases. Mea ergodic covergece, Calki algebra, essetial maximality, essetial orm, compact approximatio property. Supported i part by the N. W. Naugle Fellowship ad the A. G. & M. E. Owe Chair i the Departmet of Mathematics. 2 Supported i part by NSF DMS-0032. c XXXX America Mathematical Society

2 MARCH T. BOEDIHARDJO AND WILLIAM B. JOHNSON 2 the mai result of this paper. For T B(X, the essetial orm T e is the orm of T i B(X/K(X. Theorem.2. Suppose that X is a Baach space havig the bouded compact T e approximatio property. If T B(X satisfies lim = 0, the the followig coditios are equivalet. ( ( The sequece T k coverges i orm to a elemet i B(X/K(X. + (2 There is a essetially maximal subspace of X o which I T is compact. The authors thak C. Foias ad C. M. Pearcy for helpful discussios. 2. The Calki represetatio for Baach spaces I this sectio, X is a fixed ifiite dimesioal Baach space. Let Λ be the set of all fiite dimesioal subspaces of X directed by iclusio. The {{α Λ : α α 0 } : α 0 Λ} is a filter base o Λ so it is cotaied i a ultrafilter U o Λ. Let Y be a arbitary ifiite dimesioal Baach space ad let (Y U be the ultrapower (see e.g., Chapter 8 i [2] of Y with respect to U. (The ultrafilter U ad the directed set Λ do ot deped o Y. If (yα is a bouded et i Y, the its image i (Y U is deoted by (yα. Cosider the (complemeted subspace { } Ŷ := (yα (Y U : w - lim yα = 0 of (Y U. Here, w - lim yα is the w -limit of (y α through U which exists by the Baach-Alaoglu Theorem. Wheever T B(X, Y, we ca defie a operator T B(Ŷ, X by sedig (yα to (T yα. Note that if K K(X, Y the K = 0. Recall that a Baach space Z has the bouded compact approximatio property (b.c.a.p. if there is a uiformly bouded et (S α 0 i K(Z covergig strogly to the idetity operator I B(Z. It is always possible to choose Λ 0 to be the set of all fiite dimesioal subspaces of Z directed by iclusio. If the et (S α 0 ca be chose so that sup S α λ, the Z is said to have the λ-b.c.a.p. It is 0 kow that if a reflexive space has the b.c.a.p., the the space has the -b.c.a.p. Theorem 2.. Suppose that X has the λ-b.c.a.p.. The the operator f : B(X/K(X B( X, T T, is a orm oe (λ + -embeddig ito B( X satisfyig f( I = I ad f( T T 2 = f( T 2 f( T, T, T 2 B(X. Proof. It is easy to verify that f is a liear map, f( I = I ad f( T T 2 = f( T 2 f( T, T, T 2 B(X. If T B(X the clearly f( T T ad thus we also have f( T T e. Hece f ad f is cotiuous. It remais to show that f is a (λ + -embeddig (i.e., if f( T (λ +. T e> To do this, let T B(X satisfy T e >. Sice X has the λ-b.c.a.p., we ca fid a et of operators (S α K(X (where Λ is defied at the begiig of this sectio covergig strogly to I such that sup S α λ. The T (I S α =

ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS 3 (I S α T T e >, α Λ. Thus, there exists (x α X such that x α = ad T (I S α x α > for α Λ. Note that for every x X, lim sup [(I S α x α]x sup x α[(i S α x] lim sup (I S α x = 0, ad so the et ((I S α x α coverges i the w -topology to 0. By the costructio of U, this implies that w - lim (I S α x α = 0. Therefore, due to the defiitio f( T = T, we obtai ( + λ f( T f( T lim (I S α x α = f( T ((I S α x α f( T ((I S α x α T (I S α x α. It follows that f( T ( + λ wheever T e >. Remark. We do ot kow whether Theorem 2. is true without the hypothesis that X has the b.c.a.p. Remark 2. The embeddig i Theorem 2. is a isometry if the approximatig et ca be chose so that I S α = for every α. This is the case if, for example, the space X has a mootoely ucoditioal basis. However, we do ot kow whether the embeddig is a isometry if X = L p (0, with p 2. If N is a subset of Y, the we ca defie a subset N of Ŷ by N := { } (yα Ŷ : lim d(y α, N = 0. Lemma 2.2. If N is a w -closed subspace of Y, the for every (y α Ŷ, d((y α, N 2 lim d(y α, N. Proof. Let a d(y α, N. Let δ > 0. The A := {α Λ : d(y α, N < a + δ} U. Wheever α A, yα zα < a + δ for some zα N. If we take zα = 0 for α / A, the, sice sup yα <, sup z α = sup α A zα (a + δ + sup yα <. α A

4 MARCH T. BOEDIHARDJO AND WILLIAM B. JOHNSON 2 ( As a cosequece, zα w - lim zβ N, sice N is w -closed. Therefore, ( ( d ((yα, N d (yα, zα w - lim zβ y α zα + w - lim zβ lim yα z α + w - lim zβ (a + δ + w - lim(zβ y β (a + δ + lim z β y β 2(a + δ. But δ ca be arbitarily close to 0 so d ((y α, N 2a = 2 lim d(y α, N. Propositio 2.3. If X ad Y are ifiite dimesioal Baach spaces ad if T B(X, Y has closed rage the T B(Ŷ, X also has closed rage. Proof. The operator T has closed rage so T also has closed rage. Let c = if{ T y : y Y, d(y, Ker T = } > 0. The by Lemma 2.2, for every (y α Ŷ, T (y α T y α c lim d(y α, Ker T c 2 d((y α, (Ker T. But obviously, (Ker T Ker T ad so T (y α c 2 d((y α, Ker T, (y α Ŷ. I other words, T has closed rage. Lemma 2.4. Suppose that X Y ad that T B(X. Let T 0 B(X, Y, x T x. The T 0 Ŷ = T X. Proof. If (yα Ŷ the for each α Λ, T 0 yα = T (yα X ad (y α X X. Thus we have T 0 (yα = (T0 yα = (T (yα X = T (yα X T X. Hece, T 0 Ŷ T X. Coversely, if (x α X the we ( ca exted each x α to a elemet yα Y such that yα = x α. Thus we have yα w - lim yβ Ŷ. Note that ( ( T0 w - lim yβ = w - lim T0 yβ = w - lim T x β = T w - lim x β = 0. This implies that T (x α = (T x α = (T0 yα ( ( = T0 yα w - lim yβ = T 0 (yα w - lim yβ T 0 Ŷ.

ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS 5 Therefore T X T 0 Ŷ. lim Proof of Theorem.2: ( (2 : Let P : T = 0, (2. ( I T P ( I I T + T +... + + Thus, T P = P ad so T I + T +... + + T I T + = 0. +. Sice P 2 P + T P +... + T P ( + P = P + +. Hece P is a idempotet i B(X/K(X. From Fredholm theory (see e.g. [3], we kow that sice σ( P {0, }, the oly possible cluster poits of σ(p are 0 ad. Thus, there exists 0 < r < such that {z C : z = r} σ(p =. The P = 2πi z =r (z I P dz ad so P 2πi z =r (zi P dz K(X. Therefore, by replacig P with P = 2πi z =r (zi P dz, we ca assume without loss of geerality that P is a idempotet i B(X (see e.g., Theorem 2.7 i [7]. Equatio (2. also implies that (I T P K(X which meas that I T is compact o P X. We ow show that P X is a essetially maximal subspace of X o which I T is compact. Suppose that I T is compact o a subspace M 0 of X cotaiig P X. Note that lim I + T +... + T P + K = 0, + for some K, K 2,... K(X, which gives [ ] lim I + T +... + T I + K (P I + = 0. I+T +...+T Sice I T is compact o M 0, + I is also compact o M 0 ad so is I+T +...+T + I + K o M 0. Thus, (P I M0 is the orm limit of a sequece i K(M 0, X, ad so P I is compact o M 0. Sice P X M 0, (I P M 0 M 0. Therefore, (P I (I P M0 = I (I P M0 is compact ad so (I P M 0 is fiite dimesioal. I other words, dim M 0 /P X <. (2 ( : Without loss of geerality, we may assume that X is a subspace of Y = l (J for some set J. Defie T 0 B(X, l (J, x T x. The by assumptio, there is a essetially maximal subspace M of X o which I T 0 is compact. By Theorem 3.3 i [5], there exists K K(X, l (J such that K M = (I T 0 M. We ow show that I T 0 K B(X, l (J has closed rage. Sice M Ker (I T 0 K ad M is a essetially maximal subspace of X o which I T 0 K is compact, Ker (I T 0 K is a essetially maximal subspace of X o which I T 0 K is compact. Let π be the quotiet map from X oto X/Ker (I T 0 K. Defie the oe to oe operator R : X/Ker (I T 0 K l (J, πx (I T 0 Kx. If R does ot have closed rage, the by Propositio 2.c.4 i [6], R is compact o a ifiite dimesioal subspace V of X/Ker (I T 0 K. Hece, I T 0 K is compact o π V ad so by the essetial maximality of Ker (I T 0 K,

6 MARCH T. BOEDIHARDJO AND WILLIAM B. JOHNSON 2 dim π V/Ker (I T 0 K <. Thus, V = π V/Ker (I T 0 K is fiite dimesioal, which is a cotradictio. Therefore, R has closed rage ad so I T 0 K also has closed rage. By Propositio 2.3, I T 0 K has closed rage. But K = 0 so Î T 0 has closed rage ad by Lemma 2.4, Î T = I T has closed rage. T e Sice by assumptio lim = 0, lim T T = 0. By Li s ( Theorem. [4], the sequece T k coverges i orm to a elemet + i B( X. By Theorem 2., the result follows. Refereces. J. W. Calki, Two-sided ideals ad cogrueces i the rig of bouded operators i Hilbert space, A. of Math. 42 (94, 839-873. 2. J. Diestel, H. Jarchow ad A. Toge, Absolutely Summig Operators, Cambridge Uiversity Press, Cambridge, 995 3. I. Gohberg, S. Goldberg ad A. M. Kaashoek, Classes of liear operators: Vol. I, Birkäuser, Basel, 990. 4. M. Li, O the uiform ergodic theorem, Proc. Amer. Math. Soc. 43 (974 337-340. 5. J. Lidestrauss, Extesio of compact operators, Mem. Amer. Math. Soc. No. 48 (964 2 pp. 6. J. Lidestrauss ad L. Tzafriri, Classical Baach Spaces I, Spriger-Verlag, Berli, 977, Sequece spaces. 7. H. Radjavi ad P. Rosethal, Ivariat Subspaces, secod editio, Dover Publicatios, New York, 2003. DEPARTMENT OF MATHEMATICS, TEXAS A&M UNIVERSITY, COLLEGE STA- TION, TEXAS 77843 E-mail address: march@math.tamu.edu DEPARTMENT OF MATHEMATICS, TEXAS A&M UNIVERSITY, COLLEGE STA- TION, TEXAS 77843 E-mail address: johso@math.tamu.edu

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