On minimal projections generated by isometries of Banach spaces

ROCZNIKI POLSKIEGO T O W A R Z Y ST W A M A T E M A T Y C Z N E G O Séria I: PRACE M A T E M A T Y C ZN E XXVII (1988)

A. L. Ko l d o b s k i (Leningrad) and V. P. Od i n e c (Bydgoszcz)

On minimal projections generated by isometries of Banach spaces

Let В be a real Banach space. Let A be a linear isometry in В which has a fixed point. The present paper is inspiried on a fact that is known to some degree but not precisely discused in any source, i.e., if the space В is represented as a direct (topologie) sum o f Im(7 — A) and Ker(/ — A) [where 7 is the identity map of B], then the projection P onto Im(7 — A) along Ker(7 — A) is the minimal projection ( г) (cf. below, Theorem 1.1).

In connection with this fact two problems arise. The first problem concerns a possibility o f the representation of a space В as a direct topologie sum o f Im(7 — A) and Ker(7 — A).

In Section 1 we g iv e . a necessary and sufficient conditions o f this representation for any linear operation in В (cf. [8], [9]).

The second problem concerns the estimation o f a norm o f a minimal projection P. The answer is given in Section 2 with the help o f the concave function g (which has no name for the present) and a Chebyshev radius, which were before used by the estimation o f a norm of a minimal projection onto hyperspaces [6].

As the examples showing the obtained results we take the vector-valued Orlicz space B = L Ml ([0, 1]; L M?(0, 1)), the Banach space with symmetric norm (in particular, /l5c0) and the space C (S*) o f the all continuous functions on the circle S1.

Note that in the last two examples, our results can also be obtained in a more complicated way with the help o f the theory of operators acting on compact topological groups (cf. for example [1 1], [1 2]).

1. A condition of the representation of В as В = Im(7 — 4 )© K e r (7 — A).

As usually the Banach space В is called (topologie) direct sum o f the

l 1) The projection P (i.e., a linear bounded idempotent operator) from В onto a comple­

mented subspace D is called minimal if ||P|| > ||P|| for any projection P : B - > D . A subspace is always assumed to be closed. The terms and also the notation o f the classical spaces lp, U , C ([a , b ]), which are encountered in this paper follow the books [2 ], [5 ], [17].

subspaces D and К if each x e B can be uniquely expressed as x = y + z, where y eD, z e K , and the linear operator P: B - > D , P(x) = у is bounded.

We shall write B — D @ K . The relative projection constant of a comple­

mented subspace D in a Banach space В is the number A(B, D) = inf {||P||: P projects В onto D}.

Let Л be a linear bounded map: B - * B . Let BA = Kev(I — A), BA

= Im (I —A), let в be the zero element in B, let N be the set o f natural numbers. If a set D <= В and / is a map on B, then by f\D we denote the restriction of / to the set D.

Theorem 1.1. Let A be a linear isometric operator of the Banach space В onto itself, and В ~ Im(/ — Л )© К е г(/ — A). Let P be the projection from В onto Im (f — A) annihilated on Ker (I —A). Then P is a minimal projection and it can be defined by

1 " - 1

(1) P = lim - £ A~ kP Ak,

и-*оо ft k= 0

where P is a projection from В onto l m ( I — A); A 0 = I.

P r o o f. Let P be a projection on subspace BA. Let P be the map defined by (1).

W e shall show that P is defined correctly for each x e B and it is a projection from В onto BA along BA. Indeed, if z e B A, then there exists a x e B such that z = x — Ax. Then Ak(z) = (I — A) (Akz )e BA for all k e N . Hence P ( A k(z)) = Ak(z) for all k e N and P (z ) = z.

If z e B A, then Ak(z) = z for all k e N . Since P ( z ) e B A and P (z) = x' — Ax' for a x 'eB , we have

P ( z ) = l i m - X A - k( x ' - A x ' ) = lim ~ ( A ~ n+1 ( x ' ) - ^ M ) = 0,

л -*oo ft k= 0 n-*aoft

because ||Л-1 || = 1.

By triangle inequality, we obtain ||P|j ^ ||P||. Therefore, by linearity and boundedness o f the map P defined by (1), P is the minimal projection from В onto BA along BA.

R e m a rk 1.1. The proof of this theorem is essentially a proof of some ergodic statistic theorem (cf. [5]). Note that some ergodic statistic theorems were used in fact before, in connection with the investigation o f a minimal projection (cf. [1], [13], [16]).

If lm(/ — A) is a reflexive subspace, then the existence of minimal projection follows from the Isbell-Semadeni results ([10]), i.e., if a comple­

mented subspace D of a Banach space В is isometrically isomorphic to a space Z *, then there exists a minimal projection from В onto D.

For using o f Theorem 1.1 we ought to have a condition for a decompo­

sition o f В as a direct sum В = BA@ B A. The next proposition gives the conditions o f this decomposition (if A is a linear continuous map) in terms of a convergence o f averaging operators A(n):

Example 1.1 (Fiirstenberg [7]). Let C^S1) be the space o f all continuous functions on the circle S1. Then for every linear isometric surjective operator A: CfS1) -> CfS1) and for every x e S 1 there exists lim (A(n))x, where A(n) is defined by (2).

Proposition 1.1. Suppose that a linear continuous operator A in a Banach space В is such that BA is closed and

In order that В be a direct sum BA® B A it is necessary and sufficient that (4) there exists lim (Л (п ))(х ) for any x e B .

P r o o f. Necessity. Notice that if x e B A, then (A (n ))(x ) = x, and therefore lim (Л (и ))(х ) = x. If x e B A, then there exists ye В such that x = y — Ay, i.e., (Л (п ))(х) = (\/n)(y — A n(y)). Hence, by condition (3), lim {A(n))x = 9. Thus, for each x e B , (4) is true.

The proof o f sufficiency follows directly from [5 ] (Chapter V III, §5.2).

R e m a rk 1.2. (a) If A is an isometry, i.e., a linear isometric operator in B, then from the proof o f necessity we have ВА слВл = {9}. In general case the last equation is not true. For example, if В = span {ex, xex] < = C ([ 0, 1]) and A is the differentiation operator in B, then BA — BA = span {ex} .

(b) If A is an isometry o f В onto itself and В is a reflexive space, then В

= Ba@ Ba (cf. [5]), where BA is the closure o f BA = (I — A)B.

(c) For the convergence o f the sequence (v4(n))J° it is not sufficient that A can be an isometry of В onto itself. Indeed, let 21 = (аи), 1 ^ j, i ^ 2, be a matrix o f order two, a^eN, 1 < i , j < 2, det$I = 1 , |tr 2I| > 2. (For example,

A ( n ) = -1 I Ak (n = 1 ,2 ,...), A 0 = I.

(2)

(3)

Let T be a standard automorphism of a torus V = S1 x S 1 corresponding to ЗД, i.e., for a point v — (z, w)eV, z = e2nx, w = e2my, x, y e R, we have

Tv = (е2Ща11Х + а12У) е2я*<«21х + 022У)).

Let A be an isometry of the space C ( V ) o f all real continuous functions on V, generated by T (i.e., A<p(v) = (p(Tv), where (peC(V), v eV). Then by a result of H. Fiirstenberg from [7 ] (Theorem 3.3) in C (V) there exists a function / for which the sequence {(Л (п ))/ }® is not convergent, though A is a surjection. By Proposition 1.1,

C ( V) ф ( C(V) )A@ { C ( V ) y .

Proposition 1.2. Let А : В -* В be a linear continuous operator in B.

Then the following statements are equivalent : (i) В = ВА® В Л;

(И) В л is a subspace of В and (1 — А)\Вл is a one-to-one operator of BA onto BA;

(iii) for each x eB there exists x' eB such that (I — A ) x = (I — A)2x' and for each sequence (xk) f such that lim ( I — A)2xk = в, there holds lim (I — A ) x k

к ->oo к -*oo

= в.

P r o o f. (ii)=>(i). W e shall prove that for every element x e B there exists y e B A and z eBa such that x = y + z.

Let ^{x e}B . Since (I —A)\Ba is a surjective map onto BA, there exists ^{z e}B^a such that x — A x = z — A z . Let у = x — z ; then y — A y = (x — A x ) — (z — A z )

— в. Hence у еВа. Next, assume that x = y + z and x = y ' + z ', where y , ÿ e B A, z , z'eB a .Then z — z ' = y — ÿ eB a . Hence, (I — A ) ( z — z ') = 6. Since the operator (I —A )\B is one-to-one, we have z = z ' and y = ÿ .

Next note that by the Banach Open Mapping Principle (cf. [2], p. 33) the operator ((I — A)\Ba)~ 1 : BA -+BA is a continuous linear operator. There­

fore for each sequence (zk)^° c= BA such that (I — A)zk -> в, we get lim zk = в.

к ->co k~* oo

W e define here the operator P : В -> BA by the formula Px = z, where z eBa is such that x = y + z (у еВа).

Next, we prove that P is a continuous operator. Indeed, let (xk) f be a sequence such that xk -> в, and (yfc)r , (zk)® are such that xk = yk + zk (yt eBA, zk = BA, к = 1, 2, ...). Then

||(/-Л)х,||«:((1+ | И11) М 1) - 0.

k-*ao

Hence,

( I - A ) x k -> в.

k-*<x>

By

( I - A ) x k = yk + zk- y k- A z k = { I - A ) z k, we obtain as above zk = P ( xk) в.

k-> ao

Since P is evidently a linear operator, P is a projection from В onto BA.

(i)=>(iii). Let x e B , x = y + z, where y e B A, z e B A. Then z = (I — A)x', where x 'eB . Hence {I — A ) x = (I — A)\y + z) = (I — A)z = (I — A)2x'. Next it is easy to verify that (I — A)\Ba is a one-to-one surjective operator. Finally, we note that by the Banach Open Mapping Principle the operator ((/ — A)\Ba)~1 is a continuous linear operator on BA.

The implication (iii) => (ii) is obvious.

R e m a rk 1.3. Let А: В -> В be a linear continuous operator. It is easy to see that

(a) If A{B) = B and В = ВАфВл, then A( Ba) = BA and BA~ 1 = B A, BA = BA, where A ~ x is the inverse operator to A.

(b) If dimB^ < oo, then В = BA@ B A o B A r\BA = {0}.

Example 1.2. Let M f (/ = 1,2) be two Orlicz functions: [0, + o o )- >

[0, + oo), i.e., continuous convex non-decreasing functions with M t (0) = 0 and M t ф 0. Let L M l {0 ,1 ) be an Orlicz space o f equivalence classes of such measurable functions h: [0, 1] ->( — oo, + o o ) for which

i

p|| = inf (f > 0: fM 2(\h(x)\/t)dx ^ 1| < + oo.

b

Let B = L Ml ([0, 1]; L M l (0, 1)) be the Orlicz space o f the equivalence classes of strongly measurable functions /: [0, 1] - > L M2(0, 1) for which

lll/lll = inf {t > 0: \ м ^ М ( х ) Щ й х « 1} < + 00. 0

Now, we shall define for every n e N a map т„: [0, 1] -*■ [0, 1] as follows: if n = 1, then r ^ x ) = x for every x e [0, 1] ; if n ^ 2, then

x + - 1

n ^{if x e} 0,n—1 T„(x) =< n- 1

x — - — - if x e n—1 , 1 if x = 0.

An operator g (x ): L M2(0, 1 )-> Т М2(0, 1) defined for every x e (0 , 1] by (Q(x)h)y — h(T[ltx](yj) (where h e b M2(0, 1), [1/x] is the greatest integer of the number 1/x, y e [0 , 1]), and Q(0)h = h, for x = 0, is a linear isometry.

Then the operator A: B - > B , such that A f ( x ) = Q ( x ) ( f (z2(x))) for each f e B , x e [0 , 1] is a linear isometry o f В onto itself.

BA will be a subspace o f all functions f e B , satisfying for every n e N the property:

(/ (* )) Ы у)) = ( ДТ2 (*)))(>>) for //-almost every x e

and y e [0 , 1] relative to Lebesgue measure //.

BA will be a subspace o f all functions f e B satisfying for every n e N \ { 1}

the property

(5) "S ( f ( x ) + f ( x + i))(y + k/n) = 0

k= 0

for //-almost every x e(l/ (n + l), 1 /n\ and y e [0 , 1/n] relative to Lebesgue measure //.

It is easy to verify that the subspaces BA and BA are infinite-dimensional and that the operator A satisfies condition (iii) from Proposition 1.2.

Now, let P be a projection from В onto BA which can be defined in the following manner: P f ( x ) = f ( x ) if x e [0 , !•); if 2 and x — l e (l/(n+1), l /и], then

(/ (* )) У if y e 0,

“ Z ( / ( * - £ ) + / (* )) ( y - “ ) if У е 1 •

*=o \ nJ L n

By Theorem 1.1 and Proposition 1.2 the projection P: В -> BA defined by (1) is a minimal one. By virtue o f results o f Section 2, ||P|| ^ 2.

2. Evaluation of norm of minimal projection, generated by isometry. In this section, Sx (resp. U x) will denote the unit sphere (resp. the unit ball) of a real Banach space X.

Let D, К be subspaces of a Banach space В such that В = D @ K . Let x e S K and Dx = D ® span { x j . Let f eS * be such that / -1(0) = Z).

For every a e [0 , 1], let

1 1

n + Г n

Wax = / 1(a), Cxa = UDx n Wax, qxD{a) = inf sup ||z-y||,

г е Ж * yeC *

Cx and Qo{a) will be called, respectively, the hypercircle in D and the Chebyshev radius of Cx (cf. [6]).

Next write CXD for sup Qo(a) and C£ for supC£. Consider now the

a e [ 0 , 1) х €$к

function g: [1, 2] —> [1, 2] (cf. [6] ) defined as

j l + i ( ( t - l ) + У ( г - 1 ) 2 + 8(г-1)) if 1 У Й - З ,

(6) g ( 0 - 28(t~ 1> if V Î 7 - 3 < m;2 .

i f2 -h 4 (r — 1) v

Note that g is strictly increasing and concave. Moreover, # (1) = 1, g (2)

= 2, g{t) ^ t for each t e l l , 2]. In terms o f the function g and the number Сд we can evaluate the norm o f the minimal projection P defined by (1).

Theorem 2.1. Let A be a linear isometry of Banach space В onto itself such that В = Im(/ — Л )© К е г (/ — A). Let D = Im(/ — A), К — Ker(/ — A).

Let P be a projection from В onto D along K. Then (7) 1 ^ C£ < ЦВ, D) = ||P|| ^ g ( C « ) ^ 2.

P r o o f. Let x e S K and Dx = D@span {x }. By Remark 1.3 the operator Ax = A\Dx is an isometry o f Dx onto itself with Im(7 — A) = D, K e r{I —A)

= sp a n {x }. By Theorem 1.1, the projection P£: Dx - * D along span Jx] is a minimal projection, i.e., ||P£|| = X(Dx, D). In view o f the fact that cod im ^ D

= 1 and by a result o f C. Franchetty ( [6], Theorem 3) we have 1 ^ Cxd^ X ( D x, D ) ^ g ( C xD) ^ 2.

Observe also that in view o f the inequality À(DX, D) ^ X{B, D), we obtain 1 ^ ^ A(B, D). Next we use the fact that the projection P is minimal and

||P|| = X(B, D) (cf. Theorem 1.1).

If ||P|| = 1, then the theorem follows from the identity g ( l ) = 1.

Next assume that ||P|| > 1. Then for any £ > 0 with e <||P|| — 1, there exists x 0 e S B such that ||P(x0)|| +e > ||P||. Let yD and yK be such that x0

= Уй + Ук with y D e D and y K e K . Evidently, у к Ф в .

Let z = y*/|M |. Then ||P(x0)|| = ||РЬ(*о)Н ^ ЦРЫ1 = A(DX, D), where P:D is the projection from Dz = £>®span [z] onto D along sp a n {z}. Clearly, À(B, D) < X ( D Z, D) + f,. Since the function g is strictly increasing, we get

À(B, D) ^ sup X(Dx, D) ^ sup^(C^) ^ g(sup Q ) =g(C%) < 2.

xeSf' xe^K xe^K

By Theorem 2.1 and a result o f Franchetty in [6] (Theorem 4), taking into account the form o f function g, we get directly:

Corollary 2.1. Let A be a linear isometry of a Banach space В onto itself and В = l m ( I — A ) @ K e r ( I — A). Let D = l m ( I - A ) , K = Ker (I —A). Then (I) ЦВ, D) = 1 o C l = l o V x e S K: CXD = l o V f l 6(0, 1)

and V x e S K: QB{a) ^ 1, (II) X(B, D) < 2 o C B < 2 <^> V x eSK 3 ax e(0, 1): (ax) < 1 + ax,

(III) ЦВ, D) = 2 o C $ = 2.

In the next example we give a realization o f cases (II) and (III) of Corollary 2.1. For case (I), see [11] (Proposition 3.a.4).

Example 2.1 Let В be a Banach space with the symmetric norm ( 2) (relative to a normal basis

(e,)*

(;(0)i=i

—j (i ) ^ 3. Now, let А: В -* В be a linear isometry such that A(eJ = es + 1 for every s e N and j ( i ) ^ s < ; ( i + l ) - l and Aj(i + 1)- l = em (i = 1, 2, ...).

Let

j(i+ D -l

= \x = ( «1, a 2, . . . ) e B : £ a v = 0, a v = 0

v = J(0

if (v < j ( i ) ) or (v ^ j ( i + 1))}, Bk(i) ~ SPan ^{ej(i)> • • • » ^ej(i+ 1)- 1} » 1 = 1 ,2 ,...

It is easy to see that dim (B k{i)/D m ) = 1. From Theorem 1.1 it follows immediately that the isometry A generates in the subspace B k(i) the minimal projection P t from B k(i) onto D k(i) and ||Р,|| = À (B k(i), D k{i)).

Now, in view of Proposition 1.2 it is easy to check that its condition (ii) holds. Hence, by Theorem 1.1 the projection P from В onto D = l m ( I — A )

OO

= @ D k(i) along Ker(7 — A ) is a minimal projection.

i = 1

It is obvious that

(8) ||P|| = supUP.-ll

I

00 j(i + 1 ) — 1

(because for each x = a w e have ||x|| ^ || £ avev||, i = 1,2, . . . ,

*= 1 v = j(i)

[18]).

Now, let В = lx or В = c0. We prove that ||Pt|| = 2 — 2/k(i) (i = 1, 2, ...).

Indeed, if В = /l5 then there exists a linear isometry F 1: B k{i) яш> l\(i\ so F i ( D m ) = / й Ч О ) , where f u = ( 1 , . . . , 1 )е (1 \ {1))* . If В — c 0, then there exists a linear isometry F 2: B m so F 2(Dki:t) = / ,-> (0 ), where / 2 l =

(1A(0,-.., l/fc(t))e(W.

By the result o f [3 ] we get in both cases: À (B k(i), D k(i)) = 2 — 2/k(i). By (8), ||P|| = 2 — in f(2/fc(0)-

i

Therefore, ||P|| = 2 and C£ = 2, where К = Ker(7 — A ), if supk(i) = + o o . i

If sup (k(i)) < + o o , then for each x e S K there exists nx e(0, 1) such that i

QdM < 1 + « x -

( 2) Let E be the set all e = (eb e2. •••) with г,- e { —1, 1) (i = 1, 2, ...). Let П be the set o f all permutations a: N - * N . A Banach space В with a normal basis is said to be symmetric iff ||eerx|| = ||x|| for every x e B , а е П , e e E (cf. [18]).

R e m a rk 2.1. Note, if В = c0, then the projection P from Example 2.1 is the unique minimal projection onto D.

If В = f , then the uniqueness o f the minimal projection P onto D (from Example 2.1) fails, although the projections Pt are unique minimal projec­

tions from Bk(i) onto Dk{i), i = 1, 2, ... (Cf. [14], [15].) From Theorem 2.1 we get immediately:

Corollary 2.2. Suppose D be a complemented subspace of Banach space В such that X(B, D) ^ 2, Then there exists no linear isometny 4 of В onto itself such that D = Im(7 — A) and В = 7 )® K er(7 — A).

Example 2.2. Let 7? = C ( [ 0, 2xc]) be the space of all continuous 2n- periodic real-valued functions defined on [0, 2 л], and D„ (n ^ 1), be the subspace in В consisting o f all trigonometric polinomials of degree < n.

W e shall prove that for n ^ 8 there exists no linear isometry A o f В onto itself such that l m ( I — A) = Dn. Suppose that for some n ^ 8 there exists such a linear isometry A.

By Proposition 1.1 and Example 1.1 it follows that В = 7)„® K er(7 — A).

Hence, by Theorem 2.1, X(B, Dn) ^ 2. On the other hand, X(B, Dn) is equal to the Lebesgue Constant gn such that q„ = (4/л2) log n + 1.27033 + e„, where 0.166 > e„ { 0 (cf., for example, [4]). Taking into account that n ^ 8, we have gn > 2, a contradiction.

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FIN A N C E -E C O N O M IC S INSTITUTE, LE N IN G R A D and

INSTITUTE OF MATHEM ATICS, P E D A G O G IC A L U N IVERSITY OF BYD G O SZCZ