Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces
aa r X i v : . [ m a t h . F A ] J u l GEOMETRIC, SPECTRAL AND ASYMPTOTICPROPERTIES OF AVERAGED PRODUCTS OFPROJECTIONS IN BANACH SPACES
CATALIN BADEA AND YURI I. LYUBICH
Abstract.
According to the von Neumann-Halperin and Lapidus the-orems, in a Hilbert space the iterates of products or, respectively, ofconvex combinations of orthoprojections are strongly convergent. Weextend these results to the iterates of convex combinations of productsof some projections in a complex Banach space. The latter is assumeduniformly convex or uniformly smooth for the orthoprojections, or re-flexive for more special projections, in particular, for the hermitian ones.In all cases the proof of convergence is based on a known criterion interms of the boundary spectrum. Introduction and background . What this paper is about.
Let H be a Hilbert space, and let M , . . . , M N be closed subspaces of H . Denote by P k the orthoprojectiononto M k , and let T = P P · · · P N . It was proved by von Neumann [29] for N = 2 and by Halperin [16] for any N that T n with n → ∞ convergesstrongly to the orthoprojection onto M ∩ M ∩ · · · ∩ M N . The same wasproved by Lapidus [21] for T = P Nk =1 α k P k with α k >
0, 1 ≤ k ≤ N , and P Nk =1 α k = 1. Some different proofs of these results were recently given in[20]. The von Neumann-Halperin and Lapidus theorems were generalizedto uniformly convex Banach spaces by Bruck and Reich [7] and Reich [30],respectively. For a survey see [10, Chapter 9].In the present paper we consider the situation when T is a convexcombination of products of some projections in a complex Banach space.With some concordance between its geometry (uniform convexity or uniformsmoothness, or reflexivity) and a class of projections (orthoprojections, her-mitian projections, etc.) we establish a spectral property of T which impliesthe strong convergence of T n as n → ∞ . The necessary background is pre-sented below. Mathematics Subject Classification.
Primary 47A05; Secondary 46B20 , 47A10.
Key words and phrases. orthoprojections, Apostol modulus, boundary spectrum. . Spaces and operators.
From now on we denote by X a complexBanach space and by B ( X ) the Banach algebra of linear bounded operatorson X . The identity operator will be denoted by I .Recall that a space X is said to be uniformly convex if for every ε ∈ (0 , δ ∈ (0 ,
1) such that for any two vectors x and y with k x k ≤ k y k ≤ k x + y k / > − δ implies k x − y k < ε . Accordingly,the nondecreasing function δ X ( ε ) = inf (cid:26) − k x + y k k x k ≤ , k y k ≤ , k x − y k ≥ ε (cid:27) is called the modulus of convexity of the space X . This classical definition,due to Clarkson [8], can be formally applied to all Banach spaces, so theuniformly convex spaces are just those which satisfy δ X ( ε ) > ε .Every Hilbert space H is uniformly convex, its modulus of convexity is δ H ( ε ) = 1 − r − ε . For more information on the modulus of convexity see e.g. [5], [15] and thereferences therein.A space X is called uniformly smooth if for every ε > δ > k x + y k + k x − y k < ε k y k holds forany two vectors x and y with k x k = 1 and k y k ≤ δ . A relevant modulusof smoothness was introduced by Day [9]. However, for the purposes of thispaper we only need to know that all uniformly convex and all uniformlysmooth spaces are reflexive and a space X is uniformly smooth if and onlyif its dual X ∗ is uniformly convex, see e.g. [23].Let H be a Hilbert space. An operator T ∈ B ( H ) is hermitian ( ≡ self-adjoint) if and only if k exp( itT ) k = 1 for all real t . In any Banach space X the latter property is a definition of a hermitian operator. (In [26] suchoperators were called conservative . This is just the case when T and − T are dissipative , i.e. generate semigroups of contractions [25]).Note that every real combination of pairwise commuting hermitian op-erators is hermitian as well. In particular, the operator T − αI is hermitianfor any hermitian T and any real α .For any operator T ∈ B ( X ) its spectrum is usually denoted by σ ( T ).If T is hermitian then σ ( T ) ⊂ R . If T is a contraction, i.e. k T k ≤ σ ( T ) ⊂ D , where D is the open unit disk in the complex plane. Theintersection of σ ( T ) with the unit circle ∂ D is called the boundary spectrum of the contraction T . Every point λ ∈ σ ( T ) ∩ ∂ D of the boundary spectrumis an approximate eigenvalue, i.e. there is a sequence of vectors x k of norm VERAGED PRODUCTS OF PROJECTIONS IN BANACH SPACES 3
T x k − λx k →
0. The boundary spectrum may be empty. Thishappens if and only if there is n ≥ T n is a strict contraction(i.e. k T n k <
1) or, equivalently, k T n k → n → ∞ .1.3 . Classes of contractions. A contraction is called primitive if itsboundary spectrum is at most the singleton { } . If the space X is reflexivethen the iterates T n of any primitive contraction T ∈ B ( X ) are stronglyconvergent. This fact is the key to all convergence problems studied in thepresent paper. Actually, it is a purely logical combination of two knowngeneral results:1) If the space X is reflexive then every contraction T with at mostcountable boundary spectrum is almost periodic, i.e. all orbits ( T n x ) n ≥ areprecompact [32].2) In any Banach space the iterates of any primitive almost periodiccontraction are strongly convergent [18]. (See also [27] for a general theoryof almost periodic operator semigroups.)An alternative proof (see Section 4 of the present paper) uses theKatznelson-Tzafriri theorem [19]: in any Banach space lim n →∞ k T n − T n +1 k = 0 for every primitive contraction T .Note that all the results stated above for contractions are automaticallytrue for any power bounded operator T ∈ B ( X ) since T is a contraction inan equivalent norm on X . On the other hand, even the weak convergenceof T n implies the power boundedness of T .The following geometric condition was introduced by Halperin in [16]:(H)there is K ≥ k x − T x k ≤ K (cid:0) k x k − k T x k (cid:1) ( x ∈ X ) . Under this condition (the same as ( K ) in [13]), T is a contraction, and allstrict contractions satisfy (H). We denote by K ( T ) the smallest value of K .In particular, K ( I ) = 0.Halperin proved that in a Hilbert space the iterates of every (H)-contraction are strongly convergent. In fact, this is true in any reflexiveBanach space. Indeed, from (H) it follows that(S) k x k k ≤ , k T x k k → ⇒ x k − T x k → . However, every (S) -contraction is primitive . Indeed, let k T x k − λx k k → λ ∈ ∂ D and a sequence of normalized vectors x k . Then k T x k k → k T x k − x k k → λ = 1. As a result, the CATALIN BADEA AND YURI I. LYUBICH iterates of every (S) -contraction in a reflexive Banach space are stronglyconvergent.
In Hilbert space this was proved in [3], where the condition (S) appearstogether with its weak version(W) k x k k ≤ , k T x k k → ⇒ x k − T x k → , and the correspondig convergence result. The latter was extended to thereflexive Banach space in [12].Obviosly, the condition (W) implies(W’) k T x k = k x k ⇒ T x = x. Conversely, (W’) implies (W) if the space is Hilbert (see [3]) or, more gener-ally, if it is a reflexive Banach space with a weakly sequentially continuousduality map (see [12]).Note that for the strict contractions the conditions (S) and (W) areformally fulfilled but empty in content.In [11] Dye proved that in a Hilbert space the condition (H) is equivalentto(D) there is r ∈ (0 ,
1) : k T − rI k ≤ − r. Obviously, under the condition (D) the operator T is a contraction. Hence, k T − rI k ≥ − r , so, finally, k T − rI k = 1 − r . Every (D) -contraction is primitive.
Indeed, if λ ∈ σ ( T ), then λ − r ∈ σ ( T − rI ), so | λ − r | ≤ k T − rI k ≤ − r , whence λ = 1 for | λ | = 1.Thus, the iterates of every (D) -contraction in a reflexive Banach space arestrongly convergent. . Projections. Recall that a linear operator P ∈ B ( X ) is called a pro-jection if P = P or equivalently, Ker( P ) = Ran( I − P ). Obviosly, k P k ≥ P = 0. A projection P is called an orthoprojection if it is a contraction,i.e. k P k = 1 or P = 0. In Hilbert space this definition is equivalent to thestandard one: the subspaces Ker( P ) and Ran( P ) are mutually orhogonal.Equivalently, this means that P is hermitian. In any Banach space everyhermitian projection is an orthoprojection . Indeed, for any projection P wehave(1.1) exp( itP ) = ( I − P ) + e it P. Hence, P = 12 τ Z τ − τ exp( itP ) e − it d t VERAGED PRODUCTS OF PROJECTIONS IN BANACH SPACES 5 that yields k P k ≤ P is hermitian. However, if P is a hermitian pro-jection then so is I − P , while for the orthoprojections this is not true,in general. Another specific feature of the non-Hilbert situation is that forsome subspaces the orthoprojections do not exist. We refer the reader to [2]and [6] for more details and references.For our purposes it is important to note that all (D) -projections areorthoprojections . Also note that every hermitian projection P satisfies (D)with r = 1 /
2, i.e. it is a u -projection in the sense of [14]. This immediatelyfollows from (1.1) by taking t = π . Obviously, if P is a u -projection then sois I − P and both are orthoprojections. Main Theorem.
Let P , · · · , P N be some orthoprojections in a complexBanach space X , and let S = S ( P , · · · , P N ) be the convex multiplicativesemigroup generated by P , · · · , P N , i.e. the convex hull of the semigroupconsisting of all products with factors from { P , · · · , P N } . Assume that oneof the following conditions is satisfied: (i) the space X is uniformly convex; (ii) the space X is uniformly smooth; (iii) the space X is reflexive and all P k are of class (D) .Then for every operator T ∈ S ( P , · · · , P N ) the iterates T n converge stronglyto an orthoprojection T ∞ . In addition, if P k are of class (W’) then (1.2) Ran( T ∞ ) = ∩ k ∈ F T Ran( P k ) where F T is the set of all indices k occurring in the decomposition of T as a member of S ( P , ..., P N ) . The formula (1.2) is true in the class of allorthoprojections if the space X is uniformly convex or uniformly smooth andstrictly convex. Recall that a Banach space is called strictly convex if all points of itsunit sphere are extreme.In the case (i) the strong convergence of T n , where T is a product orconvex combination of orthoprojections, was proved in [7] and in [30], respec-tively. The space X in these papers is real, but the results are automaticallytrue for the complex unifomly convex spaces by realification. On the otherhand, there is an example of divergence in l ∞ , R , i.e in R endowed with themax-norm ([7], p.464). Another related example is in [28]. In fact, there isan example even in l ∞ , R , a fortiori, in l ∞ , C . Namely, let P = (cid:18) − (cid:19) and P = (cid:18) (cid:19) . CATALIN BADEA AND YURI I. LYUBICH
Then ( P P ) n = (cid:18) − n +1 − n (cid:19) , so the iterates ( P P ) n are divergent.The space in our example is not strictly convex. An open question isabout existence of an example of divergence in a strictly convex space. Forthe affirmative answer the space must be infinite-dimensional since everyfinite-dimensional strictly convex space is uniformly convex.For any Banach space X and its closed subspace M , we denote by P M ( x ), x ∈ X , the set of points in M whose distance to x is minimal. If X is reflexivethen the set P M ( x ) is not empty for every x . If, in addition, X is strictlyconvex, then P M ( x ) is a singleton. In this situation P M ( x ) can be consideredas a point in X and P M as a mapping X → X , a nearest point projection onto M . In general, this ’projection’ is nonlinear. However, in a Hilbertspace P M coincides with the orthoprojection onto M .For a strictly convex reflexive space X with dim X > P M P N ) n converges strongly to P M ∩ N for every pair ( M, N )of closed subspaces of X , then X is a Hilbert space. Thus, the von Neu-mann theorem cannot be extended to the nearest point projections in anon-Hilbert space. See however [30, Lemma 3.1] for a relation between lin-ear nearest point projections and orthoprojections. This makes it possibleto obtain a counterpart of the Main Theorem for linear nearest point pro-jections. This observation was kindly communicated to us by S. Reich.Note that the weak convergence of the iterates of a product or a convexcombination of orthoprojections in a uniformly smooth space follows from[7] and [30] by duality.1.5 . Organization of the paper. The next section contains some infor-mation on the Apostol modulus ϕ T ( ε ) and its modification e ϕ T ( ε ) for acontraction T in a Banach space. In Section 3 we apply it to prove that theclasses (H), (S) and (D) are multiplicative semigroups, furthermore, (S) and(D) are convex . This is an important ingredient of the proof of the MainTheorem. The latter is given in Section 4 after a proof of the convergence ofthe iterates of a primitive contraction in a reflexive Banach space. We con-clude with an Appendix (Section 5) where we study some relations betweenthe Apostol moduli and a geometric characteristic of the boundary spec-trum. This yields a new look at a generalization of the Katznelson-Tzafriritheorem obtained by Allan and Ransford [1]. VERAGED PRODUCTS OF PROJECTIONS IN BANACH SPACES 7 . Acknowledgments.
The first author was supported in part by theANR-Projet Blanc DYNOP. The work of the second author was carried outwithin the framework of the TODEQ program. We thank Eva Kopeck´a,Simeon Reich and the referee for helpful discussions, remarks and references.2.
The Apostol modulus . Definitions and basic facts.
For a contraction T ∈ B ( X ) we considerthe Apostol modulus ϕ T ( ε ) = sup {k x − T x k : k x k ≤ , k x k − k T x k ≤ ε } , < ε ≤ . This function was introduced and studied by Apostol [4] in the case ofHilbert space. For our purposes the following modification is convenient: e ϕ T ( ε ) = sup {k x − T x k : k x k ≤ , − k T x k ≤ ε } . This definition is correct if and only if k T k = 1 since this is the only casewhen the set { x : k x k ≤ , − k T x k ≤ ε } is nonempty for all ε . Thus, wewill assume k T k = 1 anytime when dealing with e ϕ T ( ε ). On the other hand,in all further applications the case k T k < ϕ T ( ε ) and e ϕ T ( ε ) are nondecreasing and(2.1) 0 ≤ e ϕ T ( ε ) ≤ ϕ T ( ε ) ≤ k I − T k ≤ . Actually, the most interesting information relates to their behavior as ε → ϕ T = lim ε → ϕ T ( ε ) = inf ε> ϕ T ( ε ) ≥ e ϕ T = lim ε → e ϕ T ( ε ) = inf ε> e ϕ T ( ε ) ≥ . It turns out that these limit values coincide. In this sense the differencebetween the two versions of the Apostol modulus is not essential.2.2.
Lemma. If T a contraction of norm 1 and T = I then ϕ T ( ε ) > forall ε and ≤ ϕ T ( ε ) ≤ e ϕ T (cid:18) k I − T k εϕ T ( ε ) + 0 (cid:19) . Proof.
Assuming ϕ T ( ε ) = 0 for an ε , we obtain k x − T x k = 0 for all x with k x k ≤ ε , so T = I . Now let T = I . Take q ∈ (0 ,
1) and find a vector x suchthat k x k ≤ , k x k − k T x k ≤ ε, k x − T x k = qθ where θ = ϕ T ( ε ) >
0. Then for the normalized vector z = x/ k x k we have1 − k T z k ≤ ε k x k , k z − T z k = qθ k x k ≥ qθ, CATALIN BADEA AND YURI I. LYUBICH whence e ϕ T (cid:18) ε ) k x k (cid:19) ≥ qθ. On the other hand, e ϕ T (cid:18) ε k x k (cid:19) ≤ e ϕ T (cid:18) k I − T k εqθ (cid:19) since k x k ≥ qθ/ k I − T k . Thus, qθ ≤ e ϕ T (cid:18) k I − T k εqθ (cid:19) . It remains to subsitute θ by ϕ T ( ε ) and pass to the limit as q → (cid:3) Corollary. e ϕ T = ϕ T for all contractions T of norm 1.Proof. Since e ϕ I = ϕ I = 0, one can assume T = I and apply Lemma 2.2. As ε → ϕ T ≤ e ϕ T . The opposite inequality is trival. (cid:3) From now on we denote by ω T the common value of ϕ T and e ϕ T . Forinstance, ω I = 0. Accordingly, (2.1) can be extended to(2.2) 0 ≤ ω T ≤ e ϕ T ( ε ) ≤ ϕ T ( ε ) ≤ k I − T k ≤ . Theorem. ω T = 0 if and only if T is of class (S) .Proof. ”If”. There is a sequence of vectors x k such that k x k k ≤
1, 1 −k T x k k ≤ /k and e ϕ T (1 /k ) < k x k − T x k k . The latter norm tends to zero if T satisfies conditon (S).”Only if”. Let k x k k ≤ k T x k k →
1. Without loss of generality onecan assume k T x k k <
1, otherwise, we change x k to q k x k where all q k ∈ (0 , q k → k → ∞ . Since ω T = 0 we have e ϕ T (1 − k T x k k ) →
0, whence k x k − T x k k → k x k − k T x k ≤ e ϕ T (1 − k T x k ) ( k x k ≤ , k T x k < . (cid:3) Remark.
Theorem 2.4 remains in force for k T k < ω T = 0in this case. The latter definition is natural. Indeed, if k T k < ϕ T ( ε ) ≤ k I − T k ε − k T k , whence ϕ T = 0. (Recall that e ϕ T is not defined for k T k < Remark.
Let T be an isometry. Then ϕ T ( ε ) = k I − T k for all ε , hence ω T = k I − T k , therefore, ω T > T = I . VERAGED PRODUCTS OF PROJECTIONS IN BANACH SPACES 9 . The Apostol modulus for orthoprojections. If P is an orthopro-jection, so that k P k ≤
1, then(2.3) k P x k = 12 k P ( x + P x ) k ≤ k x + P x k ≤ k x k Now let k x k ≤
1, and let 1 − k
P x k ≤ ε . Then k P x k ≤ k x + P x k ≥ − ε . Hence, k x − P x k ≤ β X ( ε ) where β X ( ε ) = sup {k x − y k : k x k ≤ , k y k ≤ , k x + y k ≥ − ε } . This results in the inequality(2.4) e ϕ P ( ε ) ≤ β X ( ε ) . The function β X was introduced and investigated in [5]. It is closelyrelated to the modulus of convexity. In particular, lim ε → β X ( ε ) = 0 if thespace X is uniformly convex, otherwise, this limit is the supremum of those ε for which δ X ( ε ) = 0. The latter quantity (or 0 if X is uniformly convex)is called the characteristic of convexity of the space X , see [15].2.8. Proposition. If P is an orthoprojection in a uniformly convex spacethen ω P = 0 .Proof. This follows from the inequality (2.4) by passing to the limit as ε → (cid:3) Corollary.
Every orthoprojection in a uniformly convex space is ofclass (S) . Remark.
This corollary can be obtained directly from (2.3). In thisway Proposition 2.8 follows from Theorem 2.4.The uniform convexity of X is not necessary for the existence of (S)-orthoprojections. For instance, if a projection P in X is such that k x k = k P x k + k x − P x k for all x ∈ X (an L - projection [17]) then P is an ortho-projection and ω P = 0. Indeed, either P = I or ϕ P ( ε ) = ε for all ε . In thissituation X may not be uniformly convex. An example is X = ℓ where anycoordinate projection is an L -projection.2.11. Remark.
From (2.3) it follows that every orthoprojection in a strictlyconvex space is of class (W’).3.
Structure properties of classes (H) , (S) and (D)In this section we prove the following theorem. Theorem.
In any Banach space the sets of contractions of classes (H) , (S) and (D) are multiplicative semigroups. In addition, they are convex inthe cases (S) and (D) . This theorem is an immediate consequence of the lemmas proven below.3.2.
Lemma.
Let A and B be two contractions satisfying condition (H) .Then the product AB also satisfies (H) and K ( AB ) ≤ K ( A ) , K ( B )) . Proof.
We have k x − ABx k ≤ ( k x − Bx k + k Bx − ABx k ) ≤ k x − Bx k + k Bx − ABx k ) , whence k x − ABx k ≤ K ( B )( k x k − k Bx k ) + 2 K ( A )( k Bx k − k ABx k ) ≤ K ( A ) , K ( B ))( k x k − k ABx k ) . (cid:3) Thus, the set of (H) -contractions is a multiplicative semigroup.
Remark. If T is an (H)-contraction then ϕ T ( ε ) ≤ p K ( T ) ε. Indeed, if k x k ≤ k x k − k T x k ≤ ε , then k x − T x k ≤ K ( T ) (cid:0) k x k − k T x k (cid:1) ≤ K ( T )( k x k − k T x k ) ≤ K ( T ) ε. In particular, if P is an orthoprojection in a Hilbert space H then k x − P x k = k x k − k P x k . Thus, P satisfies (H) with constant K ( P ) = 1. Hence, ϕ P ( ε ) ≤ √ ε .3.4. Lemma. (i)
Let A and B be some contractions of norm 1. Theneither k AB k < or e ϕ AB ( ε ) ≤ e ϕ A ( e ϕ B ( ε ) + ε ) + e ϕ B ( ε ) . (ii) Let T = N X k =1 α k A k be a convex combination of contractions A k of norm 1, and let all α k > . Then either k T k < or e ϕ T ( ε ) ≤ N X k =1 α k e ϕ A k ( α − k ε ) . VERAGED PRODUCTS OF PROJECTIONS IN BANACH SPACES 11
Proof. (i). Let k AB k = 1. Then k A k = k B k = 1, so the functions e ϕ A , e ϕ B are well defined along with e ϕ AB . Take any vector x such that k x k ≤ − k ABx k ≤ ε . Then k x − ABx k ≤ k x − Ax k + k Ax − ABx k ≤ k x − Ax k + k x − Bx k . Thus, k x − ABx k ≤ e ϕ A (1 − k Ax k ) + e ϕ B (1 − k Bx k ) . Let us estimate 1 − k Ax k and 1 − k Bx k . We have1 − k Bx k ≤ − k ABx k ≤ ε and then1 − k Ax k ≤ k A ( x − Bx ) k − k ABx k ≤ k x − Bx k + (1 − k ABx k ) . Thus, 1 − k Ax k ≤ e ϕ B ( ε ) + ε. As a result, k x − ABx k ≤ e ϕ A ( e ϕ B ( ε ) + ε ) + e ϕ B ( ε ) . (ii). Let k T k = 1. Then all k A k k = 1, so the functions e ϕ A k are welldefined along with e ϕ T . Take x such that k x k ≤
1, 1 − k
T x k ≤ ε , i.e.1 − k N X k =1 α k A k x k ≤ ε. A fortiori, N X k =1 α k (1 − k A k x k ) ≤ ε, whence 1 − k A k x k ≤ α − k ε for every k . Hence, k x − T x k ≤ N X k =1 α k k x − A k x k ≤ X k α k e ϕ ( α − k ε ) . (cid:3) As a consequence, if ω A = ω B = 0 then ω AB = 0, and if all ω A k =0 then ω T = 0. By Theorem 2.4 the set of (S) -contractions is a convexmultiplicative semigroup .Now for a contraction T we consider the set R ( T ) = { r ∈ (0 ,
1) : k T − rI k ≤ − r } . By definition, T is a (D)-contraction if and only if R ( T ) = ∅ .3.5. Lemma.
For any contractions A and B if r ∈ R ( A ) and s ∈ R ( B ) then rs ∈ R ( AB ) and αr + βs ∈ R ( αA + βB ) with α > , β > and α + β = 1 . Proof.
First, we have k AB − rsI k = k A ( B − sI ) + s ( A − rI ) k≤ k B − sI k + s k A − rI k ≤ − rs. Secondly, k ( αA + βB ) − ( αr + βs ) I k ≤ α k A − rI k + β k B − sI k≤ α (1 − r ) + β (1 − s ) = 1 − ( αr + βs ) . (cid:3) Thus, the set of (D) -contractions is a convex multiplicative semigroup.
The proof of Theorem 3.1 is complete.4.
Proof of the Main Theorem
The following general result is a key lemma in the proof of our MainTheorem.4.1.
Theorem. If X is a reflexive space and T is a primitive contraction in X then the iterates T n converge strongly. The limit operator T ∞ coincideswith the orthoprojection E T onto the subspace L = Ker( I − T ) along theclosure M = Ran( I − T ) . The convergence is uniform if and only if Ran( I − T ) is closed.Proof. According to the classical ergodic theorem [24], the Ces`aro means of( T n ) n ≥ converge strongly to the projection E T onto L along M . A part ofthis statement is that X is the direct sum L ⊕ M . Let x = u + v where u ∈ L , i.e. T u = u , and v ∈ M , i.e. v = lim k →∞ ( z k − T z k ) for a sequence( z k ) k ≥ . Given ε >
0, we take and fix k such that k v − ( z k − T z k ) k < ε .Then k T n v − ( T n − T n +1 ) z k k < ε for all n . Hence, k T n v k < ε + k T n − T n +1 kk z k k < ε for large n by the Katznelson-Tzafriri theorem [19]. Thus,lim n →∞ T n v = 0. As a result, lim n →∞ T n x = u = E T x , i.e. T ∞ = E T . Thelatter is an orthoprojection since T is a contraction.Now suppose that Ran( I − T ) is closed, i.e. M = Ran( I − T ). Theoperator I − T acts bijectively on the invariant subspace M. Since M isclosed, the inverse operator S = (( I − T ) | M ) − is bounded. Since ( T | M ) n =( T n − T n +1 ) S , we obtain k ( T | M ) n k → T n converges uniformly then the same is true for theCes`aro means, and then Ran( I − T ) is closed ([22]). (cid:3) An alternative proof is merely a logical combination of two results provedin [32] and [18] as we indicated in the Introduction.
VERAGED PRODUCTS OF PROJECTIONS IN BANACH SPACES 13 . Proof of the Main Theorem.
Let T ∈ S ( P , · · · , P N ) where P , · · · , P N are some orthoprojections in a Banach space X . Obviously, T is a contrac-tion. By Theorem 4.1 it suffices to show that T is primitive in all cases(i)-(iii). Recall that all contractions of classes (S) and (D) are primitive.(See Section 1.)(i). The space X is uniformly convex. Then by Corollary 2.9 all P k areof class (S). By Theorem 3.1 so is T . Therefore, T is primitive.(ii). The space X is uniformly smooth. Then X ∗ is uniformly convex and T ∗ ∈ S ( P ∗ , · · · , P ∗ N ). All P ∗ k are orthoprojections since k A ∗ k = k A k for anyoperator A . Therefore, T ∗ is primitive like T in (i). Then T is also primitivesince σ ( A ) = σ ( A ∗ ) for any operator A and T = T ∗∗ by reflexivity of X .(iii). The space X is reflexive. Since all P k are of class (D), such is also T by Theorem 3.1. Thus, T is primitive again.To complete the proof of the Main Theorem we note that the subspaceRan( T ∞ ) coincides with the subspace Ker( I − T ) of fixed points of theoperator T . Thus, it suffices to refer to the following lemma and Remark2.11. (cid:3) Lemma. (i) Let A and B be some (W’) -contractions. Then Ker( I − AB ) = Ker( I − A ) ∩ Ker( I − B ) . (ii) Let T be a convex combination of N (W’) -contractions: T = P Nk =1 α k A k with all α k > . Then Ker( I − T ) = ∩ k Ker( I − A k ) . Proof.
In both cases the inclusion of the right-hand side into the left-handside is trivial. The proofs of the converse inclusions are as follows.(i) For x ∈ Ker( I − AB ) we have k x k = k ABx k ≤ k Bx k ≤ k x k . Therefore, k Bx k = k x k , whence Bx = x and then Ax = x by condition(W’).(ii) For x ∈ Ker( I − T ) we have k x k ≤ N X k =1 α k k A k x k ≤ N X k =1 α k k x k = k x k . Thus, k A k x k = k x k , hence A k x = x for every k . (cid:3) Remark.
The same argument shows that the contractions of class (W’) constitute a convex multiplicative semigroup. Appendix: the amplitude of the boundary spectrum
Let T be a contraction in a Banach space X , and let the boundaryspectrum of T be nonempty. We call the quantity a T = max {| λ − | : λ ∈ σ ( T ) , | λ | = 1 } the amplitude of the boundary spectrum of T . Obviously, 0 ≤ a T ≤
2, and a T = 0 if and only if the contraction T is primitive. In view of Theorem 2.4,the fact of the primitivity of the (S)-contractions is a particular case of thefollowing inequality.5.1. Proposition. a T ≤ ω T .Proof. Let λ ∈ σ ( T ), | λ | = 1. Then for every ε > x of norm 1 such that k T x − λx k ≤ ε . Hence, 1 − k T x k ≤ ε and | λ − | ≤ k x − T x k + k T x − λx k ≤ e ϕ T ( ε ) + ε. The result follows as ε → (cid:3) Corollary. If a T = 2 then ω T = 2 and e ϕ T ( ε ) = ϕ T ( ε ) = 2 for all ε .Also, k I − T k = 2 in this case.Proof. We have ω T ≥
2. Now everything follows from (2.2). (cid:3)
Obviously, a T = 2 if and only if − ∈ σ ( T ). Therefore, if − ∈ σ ( T ) then ω T = 2.5.3. Proposition.
If the space X is uniformly convex and ω T = 2 then a T = 2 .Proof. We have e ϕ T ( ε ) = 2 for every ε ∈ (0 , x = x ( ε ) of norm 1 such that k x − T x k ≥ − ε . Hence, k x + T x k ≤ β X ( ε )where β X is the function defined in Section 2. Since the space X is uniformlyconvex, we have lim ε → β X ( ε ) = 0. A fortiori, lim ε → k x ( ε ) + T x ( ε ) k = 0.This means that − ∈ σ ( T ). (cid:3) The amplitude a T is the maximal deviation of the boundary spectrumof T from the point 1 in the metric of the complex plane. Alternatively, onecan use the metric of the unit circle. This ”intrinsic” amplitude is τ T = 2 arcsin (cid:16) a T (cid:17) . In [1] Allan and Ransford obtained the following quantitative version of theKatznelson-Tzafriri theorem:lim sup n →∞ k T n − T n +1 k ≤ (cid:16) τ T (cid:17) , τ T < π. VERAGED PRODUCTS OF PROJECTIONS IN BANACH SPACES 15
In terms of the amplitude a T this means thatlim sup n →∞ k T n − T n +1 k ≤ a T p − a T , a T < . Combining this result with Proposition 5.1 we obtain5.4.
Theorem.
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Laboratoire Paul Painlev´e, Universit´e Lille 1, CNRS UMR 8524, Bˆat.M2, F-59655 Villeneuve d’Ascq Cedex, France
E-mail address : [email protected] Department of Mathematics, Technion, 32000, Haifa, Israel
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