A Coanalytic Rank on Super-Ergodic Operators

A CO ANAL YTIC RANK ON SUPER-ERGODIC OPERA TORS MOHAMMED Y AHDI Abstract. T echniques from Descr iptive Set Theory a re a pplied in order to study th e T op ologica l Complexit y o f f amilies of op erato rs naturally connected to ergodic oper ators in infinite dimensional Banach Spaces. The families of er g o dic, unifor m-ergo dic, Cesar o - bo unded and power-bounded operato rs are shown to be Bor el sets, while the family of supe r-ergo dic op era tors is shown to b e either coanalytic or Bo rel accor ding to specific structures o f the Space . Moreov er, trees and coanalytic ranks a re intro duced to characterize sup e r-ergo dic o p erators as well a s spaces wher e the ab ov e classe s of op erator s do no t coincide. 1. Introduction Let T b e a b ounded op erator on an infinite dimensional Banach space X , and let A n = 1 n P n − 1 k =0 T k b e t he n th -Cesar o-me an of T . Consider the follo wing definitions: • T is ergo dic if the s equence { A n } x ≥ 1 con v erges in the space of op erators L ( X ) equipp ed with the stro ng op erator t o p ology S op • T is uniformly ergo dic if the sequence { A n } x ≥ 1 con v erges in L ( X ) equipped w ith its natural norm. • T is weak ly e rgo dic if for an y x ∈ X , the sequence { A n ( x ) } x ≥ 1 w eakly con v erges in X . • T is Cesaro-b ounded if the norms of { A n } n are uniformly b ounded. • T is p o w er-b ounded if the norms of { T n } n are uniformly b ounded. The de finition of a sup er-ergo dic op erator is in tro duced in [11] as the sup er prop ert y asso ciated with ergo dic o p erators. Let ℓ ∞ ( X ) b e the Banac h space of b ounded sequences in X and let C U ( X ) b e the subspace of sequences { x n } n suc h that: lim U k x n k = 0. T is sup er-ergo dic if f o r an y ultrafilter U on N , t he ultrap ow er op erator T U is ergo dic on the 2000 Mathematics Subje ct Classific ation. Primar y 47 A35, 54H05; Secondary 46B08 , 03E 15. Key wor ds and phr ases. Sup er -ergo dic op era tor, Borel set, Co analytic rank, Ba - nach spa ce. 1 2 MOHAMMED Y AHD I ultrapro duct X U := ℓ ∞ ( X ) / C U ( X ), where T U ( x ) := { T x n } n + C U ( X ) for an y x := { x n } n + C U ( X ) . In [11] it was sho wn that uniformly ergo dic implies sup er- ergo dic, whic h in turn implies ergo dic. Examples w ere give n to sho w that these implications are strict. An in teresting question is to determine the structures of the Banach space that place stro ng limits on thes e rela- tionships that migh t b e p ossible instead of just pa rticular examples. T ranslations of certain mathematical concepts t o families of sets, then examining their p ositions in the descriptiv e set hierarc h y ha v e b een pro v en to be a pro ductiv e approac h (see [9] and [10]). F or example, [10] in v estigates the “family of sup erstable op erat ors” where its to p ologi- cal complexit y is sho wn to b e connected to the structure of the space; suc h as ha ving some kind of unc onditional basis or being hereditarily indecomp o sable. In particular, this pro duced families of spaces where stable, superstable and uniformly-stable operator s are either equiv alen t or strictly separated. The imp ortant results in [8], [4 ] and [7] sho w the p ow er of some applications of descriptiv e set theory . In this w ork, some tec hniques f r om [10] are applied to sets related to classes o f ergo dic op erators. F or the part icular class of sup er-ergo dic op erators, a c hara cterization in t erms of “tr e es” is in tro duced and mor e descriptiv e set theory to ols are applied. It is sho wn that the heigh t of these trees is a coanalytic r a nk, thus pro ving that the set of sup er- ergo dic op erators is coanalytic for the strong op erator top ology . All the other f amilies of o p erators are sho wn to be Borel. In particular, this giv es a general class of Banac h spaces for whic h sup er-ergo dic is strictly stro ng er than ergo dic and strictly w eak er than uniformly ergo dic. Mor eov er, examples ar e giv en to sho w the existence of suc h spaces. Throughout this work, X denotes an infinite dimensional Banac h space with a norm k . k . 2. Applica tion of descriptive s et theor y In a P olish space P , w e consider the natural hierarch y Π 0 ξ and Σ 0 ξ co v ering all Bor el subsets starting fro m the op en and closed sets for ξ = 1 to more complex ones defined b y induction on the countable ordinals ξ ; where a Σ 0 ξ is a countable union of Π 0 ξ − 1 sets, a Π 0 ξ is a coun table in tersection of Σ 0 ξ − 1 sets, and fo r α a limit ordinal: Σ 0 α = [ ξ <α Π 0 ξ and Π 0 α = \ ξ <α Σ 0 ξ A COANAL YTIC RANK ON SUPER-ERGODIC OPERA TORS 3 Borel sets are not the only subsets in a P olish space that are con- structible from op en and closed sets (see [2], [3] or [5]): • A ⊂ P is analytic if it is a contin uous image of a P olish space. • A ⊂ P is coanalytic if P \ A is analytic. The space L ( X ) of b ounded op erators equipp ed with the strong op- erator top ology S op is not a Polish space since it is not a Baire space. Ho w ev er, it is also p ossible to w or k in a standard Borel space; i.e., a space Bo r el isomorphic to a Borel set of a P olish space. Prop osition 2.1. F or a ny sep ar able B anach sp ac e X , L ( X ) of b o und e d op er ators is a s tand ar d Bor el sp ac e when e quipp e d with the σ -algebr a σ ( S op  gener ate d by the str on g o p er ator top olo gy. Pr o of. Let X b e a separable Banac h space. It is w ell-kno wn that with the strong op erator top olo g y the spaces L n ( X ) o f o p erators of norm ≤ n is a Polis h space in whic h T 7− → T n is clearly con tin uous (see, e.g., page 14 in [5] or lemma 3. in [10]). So L ( X ) = S n L n +1 ( X ) \ L n ( X ) is clearly standard Borel when equipp ed with the σ -algebra generated b y the strong op erator top olo g y .  The follo wing lemma is a mo dified result from [1]. Lemma 2.2. L e t T b e a Cesar o- b ounde d op e r ator on a Banach sp ac e X , and let { A n } n ∈ N b e the c orr esp onding se quenc e of Cesar o-me ans. Then T is er go dic if an d only if the se quenc e { A n x } n c onver ges in norm for al l x in a dense (in n orm) subset of X . Pr o of. Put E := { x ∈ X : { A n x } n ∈ N norm-con v erges in X } . Supp ose that E is dense in X and that M := sup n ∈ N k A n k < ∞ . Let y ∈ X b e fixed. F or ε > 0, consider x ∈ X suc h that k x − y k < ε . Then, there exits n 0 ∈ N suc h that k A n x − A m x k < ε, ∀ n, m ≥ n 0 . Th us, for all n, m ≥ n 0 w e hav e k A n y − A m y k ≤ k A n ( y − x ) k + k A n x − A m x k + k A m ( x − y ) k ≤ (2 M + 1) ε. Therefore the sequence { A n y } n ∈ N is Cauc hy , a nd hence y ∈ E .  Denote b y E ( X ), S E ( X ), U E ( X ) , L cb ( X ) a nd L pb ( X ), resp ectiv ely the subsets of L ( X ) of ergo dic, sup er-ergo dic, uniformly ergo dic, Cesaro- b ounded and p ow er b ounded op erators on X . Prop osition 2.3. F o r a n y sep ar able B anach sp ac e X , the s ets L pb ( X ) of p ower- b ounde d op er ators, L cb ( X ) of Cesar o-b ounde d op er ators, E ( X ) 4 MOHAMMED Y AHD I of er go dic op er ators and U E ( X ) of unifo rm ly er go di c op er a tors ar e a l l Bor el in  L ( X ) , σ ( S op )  . Pr o of. Let { x n } n ∈ N b e a dense sequence in the unit closed ball B X of X . It is not difficult to show t ha t L pb ( X ) = [ k ∈ N \ m ∈ N \ n ∈ N  T ∈ L ( X ); k T m x n k ≤ k  . It follow s from the con tin uit y of T 7− → T n that L pb ( X ) is F σ in  L ( X ) , σ ( S op )  . Similarly for the set L cb ( X ) using the S op -con tin uit y on b ounded subsets of L ( X ) of the maps R 7− → 1 n P n − 1 i =0 R i . Let T b e a b ounded op erat o r on X . By lemma 2.2, T is ergo dic if and only if T is Cesaro-b ounded and the sequence { ( 1 n P n − 1 i =0 T i ) x k } n is norm-Cauc h y for all k ∈ N ; i.e. ∀ ε > 0 , ∃ N ∈ N : ∀ n, m ≥ N      1 n n − 1 X i =0 T i x k − 1 m m − 1 X i =0 T i x k      < ε. In other terms, T ∈ \ k ∈ N \ p ∈ N [ N ∈ N \ n,m ≥ N n R ∈ L ( X ) :      1 n n − 1 X i =0 R i x k − 1 m m − 1 X i =0 R i x k      < 1 p o . This prov es the result f or E ( X ) using the contin uity of T 7− → T n . The same arg umen ts prov e the result for U E ( X ) since T is uniformly ergo dic if and only if T is Cesaro-b ounded and the sequence { A n } n ∈ N of its Cesaro-means is Cauc h y for the norm of L ( X ).  3. Set of sup er-ergodic op era tors First, a n en tropy -tree will b e defined to help c hara cterize t he super- ergo dicit y of an op erato r. F or an ultrafilter U on N , denote b y A U n the n th Cesaro-mean of the ultrap o w er T U : A U n = 1 n n − 1 X k =0 T k U By definition, an op erator T is no t sup er-ergo dic if a nd only if there exist an ultrafilter U and ¯ x ∈ X U suc h that { A U n ¯ x } n do es not con v erge, i.e., ∃ ¯ x ∈ B X U , ∃ ε > 0 , ∃ J = { j p } p ∈ N ↑ N : ∀ p ∈ N ,    A U j p ¯ x − A U j p +1 ¯ x    X U > ε where N ↑ N is the set of infinite a nd strictly increasing sequenc es o f N . A COANAL YTIC RANK ON SUPER-ERGODIC OPERA TORS 5 Let ( x n ) n ∈ N ∈ ¯ x b e c hosen in the unit ball B X . The condition ∀ p ∈ N ,    A U j p ¯ x − A U j p +1 ¯ x    X U > ε is then equiv alen t to ∀ p ∈ N , ∃ E p ∈ U : ∀ n ∈ E p ,   A j p x n − A j p +1 x n   > ε, or again, by using E m = \ p ≤ m E p , ∀ m ∈ N , ∃ E m ∈ U : ∀ n ∈ E m ,   A j p x n − A j p +1 x n   > ε ∀ p ≤ m. This implies in particular tha t ∀ m ∈ N , ∃ x m ∈ B X :   A j p x m − A j p +1 x m   > ε ∀ p ≤ m. Therefore, this prov es that if T is not sup er-erg o dic, then T satisfies the follo wing conditio n, noted ( N S E ); ∃ ε > 0 , ∃ J = { j p } p ∈ N ∈ N ↑ N , ∀ m ∈ N , ∃ x m ∈ B X :   A j p x m − A j p +1 x m   > ε ∀ p ≤ m. Lemma 3.1. T is not sup er-er g o dic if and o n ly if T satisfies ( N S E ) . Pr o of. One direction w as prov ed ab o v e. Supp ose no w that T satisfies ( N S E ). Let U b e the ultrafilter on N that con tains all sets E n := { n, n + 1 , . . . } . Put ¯ x = ( x m ) m ∈ N + C U ( X ) ∈ X U . The conditio n ( N S E ) implies that ∀ m ∈ E p ,   A j p x m − A j p +1 x m   > ε. So, U − lim m   A j p x m − A j p +1 x m   > ε 2 , a nd thus   A j p ¯ x − A j p +1 ¯ x   U > ε. Since this is true f or an y p ositiv e in teger p , it follows that the sequence  A U n ( ¯ x )  n ∈ N is not Cauch y a nd hence T is not sup er-ergo dic.  With lemma 3.1, the sup er-ergo dicity can b e describ ed in t erms of trees using the following nota t ions: • N ↑ < N denotes the set of finite and strictly increasing sequences in N as we ll as the empt y sequence. • F or s ∈ N ↑ < N , s p denotes the p th elemen t of s and | s | denotes the length of s . • F or s, s ′ ∈ N ↑ < N , s ≺ s ′ means that | s | < | s ′ | and hav e t he same first | s | elemen ts. 6 MOHAMMED Y AHD I Definition 3.2. Let X b e a Banac h space and T ∈ L ( X ) with A n its n th Cesaro-mean. F or all ε > 0, A e ( T , ε ) is the tree on N defined b y the set o f all elemen ts s ∈ N ↑ < N suc h that | s | ≤ 1 or ∃ x ∈ B X suc h that ∀ 1 ≤ p < | s | ,   A s p x − A s p +1 x   > ε. It fo llo ws from Lemma 3.1 that T is no t sup er-ergo dic if and only if ∃ ε > 0 and ∃ J ∈ N ↑ N suc h that ∀ s ≺ J , s ∈ A ( T , ε ) . In other terms, T is not sup er-ergo dic if and only if the tree A e ( T , ε ) is not we ll fo unded for certain ε > 0; i.e., with infinite branc hes. Theorem 3.3. L et X b e a B anach sp ac e and T ∈ L ( X ) . The fol low ing assertions ar e e quivalen t: (a) T is sup er-er go dic. (b) F or al l ε > 0 , the tr e e A e ( T , ε ) is wel l founde d. (c) η e ( T ) := s up ε> 0 h ( A e ( T , ε )  < ω 1 , w her e h gives the height of a tr e e. Pr o of. The equiv alence b etw een ( a ) and ( b ) was show n earlier. The equiv alence b et w een ( b ) and ( c ) is obvious b ecause ( b ) is equiv alent to ∀ n ∈ N , A e ( T , 1 n ) is w ell f o unded.  The index η e defined in theorem 3.3 extends t o all T ∈ L ( X ) by η e ( T ) = ω 1 , if T is no t sup er-ergo dic. Theorem 3.4. L et X b e a sep ar able Banac h sp ac e, and L ( X ) b e the sp ac e of b ounde d op er ators e quipp e d with the str ong op er ator top olo gy S op . L et η e b e the index on L ( X ) define d ab ove . Then: (a) T is sup er-er go dic if and only if η e ( T ) < ω 1 . (b) The set S E ( X ) of sup er-er go dic op er ators is c o analytic. (c) η e is a c o analytic r ank on S E ( X ) . (d) ∃ α < ω 1 such that the set of uniforml y er g o dic op er ators U E ( X ) ⊆ { T ∈ S E ( X ); η e ( T ) ≤ α } . (e) S E ( X ) is a Bor el s et if and only i f η e ( X ) := sup T ∈S E ( X ) η e ( T ) < ω 1 . Pr o of. The as sertion ( a ) is part of the the orem 3.3. Let T b e a b ounded op erator on X and A n its n th Cesaro-mean. W e can write t ha t η e ( T ) = s up n ∈ N h  A e ( T , 1 n )  . W e need to construct a tree on N that con tains all trees A e ( T , 1 n ) while k eeping the information on the index η e ( T ). Let A e ( T ) b e the tree A COANAL YTIC RANK ON SUPER-ERGODIC OPERA TORS 7 formed by the finite sequences s = ( s 0 , s 1 , s 2 , ... ) such tha t s 0 co v ers N and ( s 1 , s 2 , ... ) cov ers the trees A e ( T , 1 s 0 ); i.e., the tree n σ ∈ N ↑ < N : | σ | = 0 or σ = ( k , s ) ∈ N × N ↑ < N with s ∈ A e ( T , 1 k ) o . Consider the map on the S op -Borel set of Cesaro-b ounded L cb ( X ), A e : L cb ( X ) − → { T rees on N } T 7− → A e ( T ) . Claim 3.5. Let σ ∈ N < N . Put σ = { T ∈ L cb ( X ); σ ∈ A e ( T ) } . Then σ is a S op -Borel subset o f L cb ( X ). Indeed, this is clear if the length of | σ | ≤ 2 since in this case either σ = L cb ( X ) o r σ = ∅ . Let σ = ( k , s ) ∈ N × N < N with | s | > 2. If s / ∈ N ↑ < N then σ = ∅ . If s ∈ N ↑ < N , w e hav e σ = n T ∈ L cb ( X ) : s ∈ A e ( T , 1 k ) o = n T ∈ L cb ( X ) : ∃ x ∈ B X , ∀ 1 ≤ p < | s | ;   A s p x − A s p +1 x   > ε o . Since X is separable , let { x n } n ∈ N b e den se in the unit ball of X . Then, σ = n T ∈ L cb ( X ) : ∃ n ∈ N , ∀ 1 ≤ p < | s | ;   A s p x n − A s p +1 x n   > ε o = [ n ∈ N | s |− 1 \ p =1 n T ∈ L cb ( X ) :   A s p x n − A s p +1 x n   > ε o . It fo llo ws from the con tinuit y of T 7→ T n that σ is S op -Borel. Lemma 3.6 b elo w and claim 3.5 imply that the set C :=  T ∈ L cb ( X ); A e ( T ) is w ell b ounded  is S op -coanalytic in L cb ( X ) with a coana lytic rank h ◦ A e ; whic h maps T to the heigh t o f the tree A e ( T ). On the other hand, b y the definition of A e ( T ) and theorem 3.3, C = n T ∈ L cb ( X ); A ( T , 1 n ) is w ell f o unded ∀ n ∈ N ∗ o = n T ∈ L cb ( X ); A ( T , ε ) is w ell founded ∀ ε > 0 o = S E ( X ) ∩ L cb ( X ) = S E ( X ) . Therefore, the set S E ( X ) is S op -coanalytic in L cb ( X ) and th us in L ( X ) b ecause L cb ( X ) is a S op -Borel subset of L ( X ), whic h prov es ( b ). The index η e is then a coanalytic rank on S E ( X ) since η e = h ◦ A e , th us 8 MOHAMMED Y AHD I pro ving ( c ) . The a ssertions ( d ) and ( e ) of the theorem follo w f r o m the lemma 3.7 b elo w on coanalytic ranks.  Belo w are ada ptations, in particular to the top olog y on the set of all trees on N , o f classical results of descriptiv e set t heory used in the pro of ab o v e (see [6] and [12]). Lemma 3.6. L et P b e a Polish sp ac e and ψ b e a map fr om P into the set of al l tr e es on N . I f for al l s ∈ N < N , the se t ¯ s = { x ∈ P : s ∈ ψ ( x ) } is Bor el, then the set C = { x ∈ P ; ψ ( x ) we l l founde d } is c o analytic in P with h ◦ ψ a s c o a nalytic r ank. Lemma 3.7. L et δ b e a c o analytic r ank on a c o analytic subset C of a Polish sp ac e P . T h en: (a) ∀ α < ω 1 , B α := { x ∈ C ; δ ( x ) ≤ α } is a Bor el set. (b) If A ⊆ C is analytic, then ∃ α < ω 1 such that A ⊆ B α . (c) C is B or el if and only if δ is b ounde d on C by a c ountable or d i n al. The top ological hierarc hy o f these families of op erators mak es it p os- sible to put strong limits o n p ossible relationships among using the index η e ( X ) of a space X or the rank η e ( T ) of an op erator T . This generates families of Ba nac h spaces with a desired relationship instead of just individual examples. In particular, if η e ( X ) = ω 1 , then the set of sup er-ergo dic op erato r s strictly separates the sets o f ergo dic and uni- formly ergo dic op era t o rs. Moreo v er, not only is an o p erator T sup er- ergo dic when its rank η e ( T ) is coun table, w e also ha v e the in teresting dic ho t o m y b elo w. It is an application of the fa cts that η e is a coanalytic rank and the set of ergo dic op erator is Bo r el. Corollary 3.8. F or every sep ar abl e Banach sp ac e X exac tly one of the fol lowing ho l d s: • either ther e ex ists a n e r go dic but not sup er-er g o dic op er ator on X , • either ther e exi s ts a c ountable or dinal α such that for any er- go dic op er ator T on X , η e ( T ) 6 α . In particular, for an y Ba nac h space t ha t contains a complemen t ed ℓ 1 ( N ), there exists ergo dic op erators with a r bit r a ry large ordinals. In- deed, the example 3.3 in [11] sho ws that the left-shift op erator S on ℓ 1 ( N ) is ergo dic since lim n →∞ k S n x k = 0, but it is not super- erg o dic since the { A U n ( ¯ e ) } n ∈ N is not Cauc h y; where A U n is the n th Cesaro-mean of S U and ¯ e is the classical canonical basis ( e k ) k ∈ N . A COANAL YTIC RANK ON SUPER-ERGODIC OPERA TORS 9 Reference s [1] C. D. Aliprantis, K.C. Border , In finite dimensional analysis , Spr ing-V er lag, Berlin, 1994. [2] Argy r os, S. A.; Go defroy , G.; Rosenthal, H. P . 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