evgeniy pustylnik- ultrasymmetric sequence spaces in approximation theory

8/3/2019 Evgeniy Pustylnik- Ultrasymmetric sequence spaces in approximation theory

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Collect. Math. 57 , 3 (2006), 257277c 2006 Universitat de Barcelona

Ultrasymmetric sequence spaces in approximation theory

Evgeniy Pustylnik

Department of Mathematics, Technion - Israel Institute of Technology

Haifa 32000, Israel

E-mail: [email protected]

Received March 10, 2005. Revised February 19, 2006

Abstract

Let (t) be a positive increasing function and let E be an arbitrary sequencespace, rearrangement-invariant with respect to the atomic measure (n) =1/n . Let {an}mean the decreasing rearrangement of a sequence {|an |}.A sequence space ,E with symmetric (quasi)norm { (n)an} E is calledultrasymmetric , because it is not only intermediate but also interpolation betweenthe corresponding Lorentz and Marcinkiewicz spaces and M . We studyproperties of the spaces ,E for all admissibleparameters , E and use them forthe denition of ultrasymmetric approximation spaces X ,E , which essentiallygeneralize most of classical approximation spaces. At the same time we showthat the spaces X ,E possess almost all properties of classical prototypes, suchas equivalent norms, representation, reiteration, embeddings, transformation etc.Special attention is paid to interpolation properties of these spaces. At last, weapply our results to ultrasymmetric operator ideals.

1. Introduction

The ultrasymmetric function spaces L ,E with the (quasi)norm f = (t) f (t) E were introduced in [14] and shown to form a large class of spaces, including L p, Lorentzspaces L pq, Lorentz-Zygmund spaces L pr (log L) and many other classical examples.In spite of their generality, these new spaces can be investigated deeply and in detailwith rather sharp results that open a simple way to various applications. All these pos-sibilities appear owing to the following main property: any space L ,E is interpolation

Keywords: Interpolation, approximation, ultrasymmetric spaces, operator ideals. MSC2000: 41A65, 46B70.

257

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect


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Ultrasymmetric sequence spaces in approximation theory 259

2. Sequence spaces

A rearrangement-invariant (another name symmetric) Banach sequence space E isan interpolation space 1 in the Banach couple ( 1, ), i.e., any sequence space whichcan be represented as E = F ( 1, ) for some (real) interpolation functor F (about thisand other basic concepts of interpolation theory see, e.g., [1, 4]). It can be describedalso as a Banach space of sequences a = {an}with the properties

i) |an | |bn | for all n = { an} E {bn} E ,ii) {an} E = {an} E , where {an}means the decreasing rearrangement of {|an |}.

(Formally speaking, these properties give something more, since they may be used alsofor the denition of quasi-Banach symmetric spaces which are outside of the couple( 1, ).)

Given an arbitrary E = F ( 1, ), we dene another sequence space E =F ( 1, ), where{an} 1 =

n =1

1n |an |.

As a simplest example, we obtain the spaces p, 1 p , with the norm

{an} p =

n =1

1n |an |

p 1/p ,

(which, of course, makes sense as a quasinorm for p < 1 too).

The spaces E also are symmetric but with respect to another (atomic) measure(n) = 1 /n . Such spaces will be used as one of the parameters in the denition of ultrasymmetric sequence spaces. As the second parameter we will use an arbitrarypositive increasing function (t) with positive and nite extension indices

0 < := lims 0

ln m (s)ln s := lims

ln m (s)ln s

< ,where m (s) = sup t E (ts ) E (t). In what follows, any given parameter function

(t) could be replaced by an equivalent one, thus we always may assume that it isstrictly monotone, smooth and such that (t) (t)/t .

Definition 2.1 A symmetric sequence space G is called ultrasymmetric if its (quasi)norm {an} G is equivalent to { (n)an} E for some function and space E describedabove. Further on we will use the notation G = ,E .Notice that ,E is a Banach space if < 1, independently of the parameter space

E . In the case of > 1 it is only quasi-Banach and in the remaining case = 1the character of ,E depends on additional conditions. As a well-studied example of such spaces we can mention the Lorentz-Zygmund spaces pr (log ) , appearing when

(t) = t1/p (1 + ln t) and E = r (see, e.g., [7]).1 Recall that a space E is called interpolation between spaces E 0 , E 1 if any linear operator, bounded

on E 0 , E 1 , is also bounded on E .


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260 Pustylnik

Most properties of the ultrasymmetric sequence spaces can be obtained analo-gously to the properties of ultrasymmetric function spaces considered in [14]. The

main assertion is thata space

Gis ultrasymmetric if and only if it is a (real) interpolation

space between the extreme spaces and M with the (quasi)norms

{an} =

n =1

(n)ann

, {an} M = supn (n)an ,

so that G = F ( , M ) = ,E , where E = F ( 1, ) with the same functor F . Moreover,a space ,E is interpolation between two other such spaces ,E 0 and ,E 1 if and only if thespace E is interpolation between E 0 and E 1.

Interpolation of spaces ,E with different parameter functions is more compli-cated. The following result can be readily obtained from some general theorems of real

interpolation (see, e.g., [4, Theorem 4.3.1]).

Proposition 2.2

If 0 < < 1 then the space ,E is (real) interpolation between0 ,E 0 , 1 ,E 1 , independently of the spaces E, E 0, E 1. In particular, the inequalities

1/p 0 < < 1/p 1 are sufficient for the space ,E to be interpolation between p0 and p1 .

Another problem, important for the applications, is comparing ultrasymmetricsequence spaces one with another; the corresponding results also are analogous tothose from [14].

Proposition 2.3

The embedding ,E ,F is independent of the spaces E, F , if and only if

1

(t)t (t)

dt < . (2.1)

In order for this embedding to be always valid when E F , it is necessary and sufficientthat (t) C(t), for some positive constant C and all t 1.

Proposition 2.4Let r (n) = (2n )/ (2n ) and let E d be the cone of nonnegative and nonincreasing

sequences from the space E . Let, as usual, r mean the nonincreasing rearrangementof the sequence r . If the linear operator Qa = {r (n)an}is bounded from E d to F ,then ,E ,F . Moreover, in the case of the function r (n) being (almost) decreasing (that is, when r (n) r (n)), the condition Q : E d F is necessary for such anembedding.

Let us illustrate the last proposition by the following example.


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Example 2.5 Let E = p, F = q, p > q and let (t) = (t)(1 + | ln t|) , with some < 0. Then Qa {n an}and by the Holder inequality we obtain thatQa q

n =1(n an )q

1/q

n =1a pn

1/pq

n=1n qp/ ( p q)

( p 1)/pq< ,

for any a E d , if pq/ ( p q) < 1, that is, < 1/p 1/q . For any such , weget an embedding ,E ,F . Considering the sequence a = {n 1/p ln n}with1/p < < 1/q , it is easy to ascertain that the operator Q does not act from p to qif = 1 /p 1/q . Hence the condition < 1/p 1/q is also necessary for the abovementioned embedding.

At last we present the following corollary from Proposition 2.4.

Proposition 2.6,E = ,F if and only if (t) (t), for t 1 and E = F .

Many properties of ultrasymmetric spaces ,E (e.g., Proposition 2.4 above) re-quire simultaneous consideration of the parameter spaces E and E . In the case of function spaces this is easy because of the simple connection between these spaces:f (t) E (1, ) if and only if g(u) = f (eu ) E (0, ); moreover, f E = g E . Thecorresponding relation does not exist in the case of sequence spaces, and the norms{an} E and {a2n } E may be even not equivalent. However the needed connectionexists in some important situations, and the following lemmas will be used for showing

this.

For the rst assertion, let us dene a linear operator T such that(T a)n = a2k , for n = 2

k , 2k + 1 , . . . , 2k+1 1, k = 0 , 1, . . .and the weight function (n) equal to 1 when n = 2 k , for some k = 0 , 1, . . . and equalto 0 otherwise. For any E , the weight space E () will be dened, as usual, as havingthe norm a E () = {(n)an} E .Lemma 2.7

T : E () E for any symmetric sequence space E .Proof. If E = then

T a = supk |a2k | = supn (n)|an | = a () ,

which implies that T : () . If E = 1 then

T a1

= |a1|+12

+13 |a2|+ +

2k +1 1

n =2 k

1n |a2k |+

But, for any k,2k +1 1

n =2 k

1n 2 2

k +1

2k

dxx

= 2ln 2 ,


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262 Pustylnik

so that

T a1 2ln2

k=0|a2k | = 2ln 2 {(n)an} 1 = 2ln 2 a 1 () ,

which means that T : 1() 1.Let now E = F ( 1, ) for some interpolation functor F . It is easy to show thatthe same relation is true for the corresponding weight spaces with arbitrary weight;in particular, E () = F ( 1(), ()). Since, by denition, also E = F ( 1, ), therequired boundedness of the operator T follows immediately; moreover, the norm of operator T can be estimated independently of E .

For the second assertion, we dene a linear operator S such that ( Sa )1 =a1, (Sa )2 = a2/ 2 and

(Sa )n+1 =2n

k=2 n 1 +1

akk , for n = 2 , 3, . . .

Lemma 2.8

S : E E for any symmetric sequence space E .Proof. Again we start by showing that S : 1 1 and S : . For 1, we haveimmediately that

Sa 1

n =1

1n |an | = a 1 .

For the second space, we have

Sa = sup |a1|,12 |a2|, . . . ,

2n

k=2 n 1 +1

akk

, . . .

sup |a1|,12 |a2|, . . . , max2n 1 +1 k 2n |ak|

2n

k=2 n 1 +1

1k

, . . . .

As before,2n

k=2 n 1 +1

1k 2

n

2n 1

dxx

= ln 2 , n = 2 , 3, . . . ,

which gives

Sa 2 ln 2 sup {|a1|, |a2|, . . .}= 2ln 2 a ,so that the required assertion follows from interpolation properties of the spaces E and E .

Theorem 2.9

Let {an}be a nonincreasing sequence of nonnegative numbers. Then a E {a2n } E for any symmetric sequence space E , with equivalence constant independentof E .

Proof. We already know from Lemma 2.7 that T a E C a E () . The given proper-ties of the sequence {an}imply that ( T a)n an for each n, thus the lattice properties of


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any space E give that a E C a E () = C {a2n } E . On the other hand, Lemma 2.8asserts that Sa E C a E and, for our sequences,

(Sa )n +1 =2n

k=2 n 1 +1

akk a2n

2n

k=2 n 1 +1

1k

12

(ln2) a2n +1 , n 2,

so that Sa E 12 ln 2 {a2n } E and thus {a2n } E C a E .

3. Approximation spaces

Let X be a quasi-Banach space and let {Gn}, n = 1 , 2, . . . , be an approximation familyof subsets from X , that is, arbitrary sets having the propertiesa) G1 G2 . . . X, G 0 = {0},b) G n Gn for all R and all n ,c) Gm + Gn Gm + n for all m, n .

The approximation numbers of arbitrary f X are dened as

an (f, X ) = inf g Gn 1

f g X ,

so that a1(f, X ) = f X . Evidently, {an (f, X )}is a nonincreasing sequence of non-negative numbers.The main idea of approximation theory is to classify the elements f X in

accordance with the properties of their approximation numbers. In particular, we canconstruct various approximation spaces of f , requiring that {an (f, X )} A for specialtypes of sequence spaces A. A general description of such an approach was given in [2];the papers [3, 12] contain effective realization and applications for the case of weight p-spaces taken as A. In the present paper we consider as A all ultrasymmetric spaces.

Definition 3.1 The ultrasymmetric approximation space X ,E is dened as the setof all f X such that {an (f, X )} ,E . We put

f X ,E := {an (f, X )} ,E = { (n)an (f, X )} E , (3.1)

so that the space X ,E with this quasinorm becomes a quasi-Banach space.

Evidently, any embedding ,E ,F entails an analogous embeddingX ,E X ,F , thus we may use the conditions of Propositions 2.3 and 2.4 (and alsoExample 2.5) as sufficient for the last embedding. The necessity of these conditionsdepends on completeness of the considered interpolation scheme, that is, the exis-tence of functions f X with prescribed sequences of approximation numbers (up toequivalence).

As a rst step for studying ultrasymmetric approximation spaces, let us passto another form of the quasinorm (3.1), containing the parameter space E insteadof E .


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264 Pustylnik

Theorem 3.2

The quasinorm (3.1) is equivalent to

f expX ,E := { (2n )a2n (f, X )} E , (3.2)with equivalence constant dependent only on .

Proof. We show that { (n)an} E { (2n )a2n } E for all nonincreasing sequencesof nonnegative numbers {an}and for any parameter function (t), with 0 < < . As is known, the last inequalities provide that (t) (2t), for all t 1.Let T be as in Lemma 2.7 and let b = T { (n)an}. Then bn = (2k )a2k , for n =2k , . . . , 2k+1 1, k = 0 , 1, . . . Therefore bn (2k )an (2k+1 )an (n)an . ButLemma 2.7 implies thatb E C { (2n )a2n } E = { (n)an} E C ( ) { (2n )a2n } E .

For the opposite inequality, set c = S { (n)an}with the same operator S as inLemma 2.8. Then for any n 2,

cn +1 =2n

k=2 n 1 +1

1k

(k)ak (2n 1) a2n2n

k=2 n 1 +1

1k C ( ) (2

n )a2n ,

(of course, cn +1 C ( ) (2n )a2n for n = 0 , 1 as well). As a consequence,

{ (2n )a2n } E C ( ) { (n)an} E ,and we are done.

For various applications of approximation spaces, it is important to have a spe-cial representation of any element f X ,E via elements from the subsets Gn (therepresentation theorem , see [12]).

Theorem 3.3

The space X ,E consists of those (and only those) elements f X which canbe represented as f = gn , for some elements gn G2n , n = 0 , 1, . . . , such that

{ (2n ) gn X } E . Moreover, the norm of f in X ,E is equivalent tof repX ,E := inf { (2n ) gn X } E ,

where the inmum is taken over all possible representations f = gn , gn G2n ,converging in X .

Proof. a) Let f X ,E . For each n = 0 , 1, . . . , dene f n G2n 1 such that f f n X 2 a2n (f, X ) (in particular, f 0 = 0). For any n 2, set gn = f n 1 f n 2, so thatgn G2n 1 1 + G2n 2 1 G2n ; for the remaining n, set g0 = g1 = 0. Then

f k

n =0gn X = f f k 1 X 2 a2k 1 0,


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as k , due to the denition of the space X ,E . Hence, the series gn convergesto f in X as needed.Furthermore, for any n 2,gn X C ( f f n 1 X + f f n 2 X ) 4C a 2n 2 (f, X ),

hence

(2n ) gn X 4C (2n ) a2n 2 (f, X ) C (X, ) (2n 2) a2n 2 (f, X ).Due to symmetricity of the space E , this implies that

{ (2n ) gn X } E C (X, ) { (2n ) a2n (f, X )} E f X ,E ,which proves the necessity part of the theorem.

b) Now let a representation f = n =0 gn , gn G2n be given so that

{ (2n ) gn X } E.For any n, we have that n 1k=0 gk G1 + + G2n 1 G2n 1. Thus

a2n (f, X ) f n 1

k=0gk X

k= ngk X . (3.3)

Therefore it is enough to prove the following assertion.

Lemma 3.4

Let bn 0, n N 0 and let { (2n )bn} E . If cn = k= n bk then { (2n )cn} E for any positive function (t) with low extension index > 0.Proof. We show rst that the series k=0 bk converges and all cn are dened. Indeed,any symmetric sequence space E , so that (2n )bn M , for some M and all n,that is, bn M/ (2n ). Since > 0, there exists a constant C = C ( ) such thatm (t) C t / 2 for all t > 0. This implies that (1) / (2n ) m (2 n ) C 2 n/ 2and thus

k=0bk

CM (1)

k=02 n/ 2 < .

Let us dene a linear operator Q{bn}= {cn}and show that it is bounded on theweight space E ( (2n )), for any symmetric sequence space E . If {bn} ( (2n )) withunit norm, then bn 1/ (2n ). At the same time,(2n )(2k) m (2

n k ) C 2 (k n )/ 2,so that

cn

k= n

1(2k )

C (2n )

k=02 k/ 2

C (2n )

.

This implies that { (2n )cn} with norm not exceeding C .


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Lemma 4.1

There exists a constant C = C ( ) such that

a2n +1 (f, X ) C (2n )

a2n (f, X ,E ). (4.3)

Proof. Let two elements g1, g2 G2n 1 be chosen such that, for some > 0,

f g1 X ,E a2n (f, X ,E ) + (f g1) g2 X a2n (f g1, X ) + .

Then g1 + g2 G2(2 n 1) G2n +1 1, and hence by (4.1)

a2n +1 (f, X ) f (g1 + g2) X a2n (f g1, X ) +

C (2n ) f g1 X ,E + C (2n ) (a2n (f, X ,E ) + ) + .Since is arbitrary, this leads to (4.3).

Theorem 4.2

The ultrasymmetric approximation scheme is stable with respect to iteration,namely, (X ,E ),F = X ,F for any parameter functions , and any parameter spaces E, F .

Proof. If f (X ,E ),F , then by Lemma 4.1

f (X ,E ),F {

(2n ) a2

n (f, X ,E

)

} F C

{(2n ) (2n ) a

2n +1 (f, X )

} F C {(2n +1 ) (2n +1 ) a2n +1 (f, X )} F C f X ,F f X ,so that

f X ,F C f X + f (X ,E ),F C f (X ,E ),F .Conversely, let f X ,F . By Theorem 3.3, there exists a representation f =

gn , gn G2n , converging in X and such that

{(2n ) (2n ) gn X } F C f X ,F .In particular, due to the inequality (4.2), this implies that for any n = 0 , 1, . . .

gn X ,E C (2n ) gn X C (2n ) f X ,F .But

n =0

1(2n )

1(1)

n =0m (2 n ) C

n =02 n/ 2 < ,

and hence the series gn converges to f in the space X ,E as well. Moreover,

{(2n ) gn X ,E } C {(2n ) (2n ) gn X } F, (4.4)which means that f (X ,E ),F with inequality f (X ,E ),F C f X ,F .


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268 Pustylnik

Both one-sided embeddings in this theorem depend only on the relations (4.2)or (4.3), which may connect in the same manner arbitrary spaces and even different

approximation schemes. This allows us to compare approximation spaces of variousnature one with another (see, e.g., [12] for the case of p-spaces as parameters).

Theorem 4.3

Let (X, Y ) be a quasi-Banach couple 2 with common approximation family {Gn}X Y . Suppose that there exist a constant C and a nonnegative nonincreasing function(t), t > 0, such that

g Y C (n) g X , for all g Gn and n = 1 , 2, . . .Then X ,E Y ,E , for every parameters (t) and E .

Proof. The proof of this theorem is essentially the same as for the second part of thepreceding theorem, only the inequality (4.4) should now be changed to

{ (2n ) gn Y } C { (2n )(2n ) gn X } E,which gives the required embedding.

Another kind of comparing different approximation schemes was also proposedin [12] as transformation theorem.

Theorem 4.4Consider two approximation families {Gn} X and {H n} Y and let a linear operator T : X Y be such that T (Gm ) H n , whenever n (m), for some nonnegative nonincreasing function (t) with positive and nite extension indices.

Then T (X (),E ) Y ,E , for every parameters (t) and E .

Proof. For a given n, let m be such an integer that (m) n < (m + 1) (withoutloss of generality, we may assume that the function (m) is strictly monotone and thusthis number always exists and is unique). Then T (Gm ) H n and thus

an +1 (T f ,Y ) = inf h H n

T f

h Y

inf

g GmT f

T g Y

T am +1 (f, X ). (4.5)

At the same time, from the condition , < , it follows that (m +1) (m) and(Cn ) (n) for any constant C ; thus (n) (m+1) (m) . Consequently,(n) an (T f ,Y ) C (m) am (f, X ) for any n N .Before comparing the norms of the corresponding sequences, it should be men-tioned that any number m may correspond to several different numbers n entering intothe same interval (m), (m + 1) . Let us show that this multiplicity can be ignoredin our estimates, that is, any number m may be taken only once.

2 Two spaces X, Y form a couple if they are continuously embedded into some common lineartopological Hausdorff space.


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Lemma 4.5

For arbitrary positive sequence a =

{an

}, let bn = am whenever (m)

n max(1 , ) and (t) = (t1/p

). ThenX ,E = ( X, Y p)K , where = ,E .

Proof. By the denition of the function (t), we have that < 1, which allows us towrite the norm of ( X, Y )K in a discrete form like (5.1), namely,

f K,,E {(2np )K (2 np , f ,X,Y p)} E . (5.3)The theorem will be proved if we show that f K,,E f X ,E .

Using the rst inequality in (5.2), we obtain immediately that

f K,,E (1/C ) {(2np ) a2n (f, X )} E = (1 /C ) { (2n ) a2n (f, X )} E = (1 /C ) f X ,E .

The second inequality from (5.2) gives that

f K,,E C (2np )n

k=02 p(k n )a2k (f, X ) E

.

At the same time, for any k

n,

(2np ) (2kp ) m (2 p(n k)) C () 2 p(n k)( + )(2kp ).Taking = (1 )/ 2, we obtain that

(2np ) C 2 p(n k)(1 )(2kp ),whence

f K,,E C n

k=02p(k n ) (2k )a2k (f, X ) E

.

For nishing the proof, we need the following:


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272 Pustylnik

Lemma 5.3

The linear operator P a = b, dened by the equalities

bn =n

k=02p(k n )ak , n = 0 , 1, . . . ,

is bounded in any symmetric sequence space E .

Proof. As usual, it is enough to show that P is bounded in 1 and . We have that

P a 1

n =0

n

k=02p(k n ) |ak| =

k=0|ak|

n= k2 p(n k) C

k=0|ak|,

and also

P a supnn

k=02p(k n ) |ak| supn maxk n |ak|

n

k=02 p(n k) C supn |an |,

as required.

Returning to the proof of Theorem 5.2, we obtain that

f K,,E C { (2n )a2n (f, X )} E = C f X ,E ,and the theorem is proved.

Corollary 5.4

A space X ,E is interpolation between the spaces X 0 ,E 0 and X 1 ,E 1 if and only if the space ,E is interpolation between 0 ,E 0 and 1 ,E 1 . Moreover, for any real interpolation functor F and any number p > max( 0 , 1 ), one has

F (X 0 ,E 0 , X 1 ,E 1 ) = ( X, Y p)K ,where = F ( 0 ,E 0 , 1 ,E 1 ), i (t) = i (t1/p ), i = 0 , 1.

This assertion follows immediately from Theorem 5.2, using general reiterationtheorems of interpolation (see, e.g., [4, p. 357]).

For applying this corollary, we should return to the problem, already posed inSection 2: when is the ultrasymmetric space ,E interpolation between two otherspaces 0 ,E 0 and 1 ,E 1 ? The solution is simple for the case of interpolation withnumerical parameters (for the denition and properties of this method, see, e.g., [1,Section 3.1]). For example, under conditions of Proposition 2.2, we get that

(X 0 ,E 0 , X 1 ,E 1 ),q = X , q , (t) = 0(t)1

1(t),

whenever 0 < < 1, 1 q .Interpolation with functional parameters is much more difficult; apparently, themost general interpolation theorem of such a type for ultrasymmetric function spacesis Theorem 5.2 from [14]. An analogous result can be proved for sequence spaces aswell, leading to the following assertion for approximation spaces.


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Proposition 5.5

Let the parameter functions 0(t), 1(t) and (t) be such that 1 / 0 > 0, 0 0 such that there exist elements g1, . . . , g r , withr 2

n 1

, giving thatT (U A)

r

i=1{gi + U B },

where U A and U B are closed unit balls of A and B respectively. As shown in [5,Theorem 1], for any p > 0, there exists a constant C = C ( p) such that

n en (T ) pC

n

k=1ak (T )

p, n = 1 , 2, . . . (6.3)

Theorem 6.2

If T L ,E (A, B ), then { (n) en (T )} E .Proof. First of all, let us show that the operator P a = b, such that b pn = (1 /n )

nk=1 a

pk ,

with xed number p, is bounded in any space ,E with < 1/p . In the case of p = 1this operator is linear, namely, the classical Hardy operator, which is known to bebounded in any symmetric space with upper Boyd index less than 1. Otherwise, thisoperator is only quasilinear, but this property is also sufficient for application of realinterpolation. Due to Proposition 2.2, it is enough to show that P is bounded on some p1 with < 1/p 1 < 1/p (as the second space p0 we can always take , where

the boundedness of P is evident). Setting an = a pn and bn = b pn , we obtain that theoperator P 1a = b is again the classical Hardy operator, and thus it is bounded in rwith r = p1/p . This means that

{bn} p 1 = {bn}1/p

r C {an}1/p

r= C {an} p 1 ,

and we are done.If now the ideal L,E (A, B ) is given, we take arbitrary p < 1/ , obtaining that

1n

n

k=1an (T ) p

1/p

,E C {an (T )} ,E ,

and it remains only to use inequality (6.3).

Proceeding to the characterization of eigenvalues of operators from ultrasymmetricideals, we should observe that the set of these numbers is at most countable, since allsuch operators are compact. Let a compact operator T act in a complex Banachspace A. Denote by {n (T )}the sequence of all eigenvalues of this operator, countedaccording to their multiplicities and such that |1(T )| |2(T )| . . . 0. If T hasonly a nite number n0 of eigenvalues, we put n (T ) = 0 for all n > n 0.

As shown in [5, Theorem 4], the eigenvalues of a compact operator T , actingin a complex Banach space, are connected with its entropy numbers by the inequal-ity |n (T )| 2 en (T ), n = 1 , 2, . . . Therefore Theorem 6.2 entails immediately ananalogous assertion for the eigenvalues.


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276 Pustylnik

Theorem 6.3

Let T

L,E (A, A). Then

{(n)

|n (T )

|}E .

To nish, let us consider a simple application of this result.

Example 6.4 Let h(t) be a locally integrable 2 -periodical function. Consider thecorresponding convolution operator

T f (t) = 20

h(t s) f (s) ds. (6.4)It is easy to see that T L(L1, L1) with norm equal to h L 1 (0 ,2 ) . Moreover, usingtrigonometric approximation of the function h(t), similarly to Jacksons theorem, onecan show that a2n +2 (T ) C L 1 (h, 1n ), for any n = 1 , 2, . . . , where

L 1 (h, ) = sup0 s 2

0 |h(t + s) h(t)|dt.

Let now some parameter function (t) and parameter space E be given. Byanalogy to usual Besov spaces, let us dene

B1, ,E = h : h B 1 , ,E = h L 1 + (n) L 1 h,1n E

< .Like (6.2) we can show that

{(n) an (T )

}E

C T L (L 1 ,L 1 ) +

{(n) a2n +2 (T )

}E .

Then Theorem 6.3 gives immediately, that the eigenvalues n (T ) of the operator (6.4),arranged in the descending order, satisfy the condition { (n)|n (T )|} E wheneverh B1, ,E .

And one more consequence can be stated here. Namely, we may use the result of T. Carleman (see [13, Proposition 6.5.5]), claiming that the eigenvalues of the convo-lution operator (6.4) coincide with the (complex) Fourier coefficients of the functionh(t), multiplied by 2 and arranged in the descending order. Hence these coefficientsalso form a sequence belonging to ,E .

References

1. J. Bergh and J. L ofstr om, Interpolation Spaces. An Introduction, Grundleheren der Mathematis-chen Wissenschaften 223 , Springer-Verlag, Berlin-New York, 1976.

2. Ju.A. Brudnyi, Approximation spaces, Geometry of linear spaces and operator theory (Russian) ,Jaroslavl (1977), 330.

3. Ju.A. Brudnyi and N.Ja. Krugljak, A family of approximation spaces, Theory of functions of several real variables (Russian) , Jaroslavl (1978), 1542.

4. Ju.A. Brudnyi and N.Ja. Krugljak, Interpolation Functors and Interpolation Spaces 1 , North-Holland, Amsterdam, 1991.

5. B. Carl, Entropy numbers, s-numbers, and eigenvalue problems, J. Funct. Anal. 41 (1981), 290306.


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6. F. Cobos and I. Resina, Representation theorems for some operator ideals, J. London Math. Soc.(2) 39 (1989), 324334.

7. R.A. DeVore, S.D. Riemenschneider, and R.C. Sharpley, Weak interpolation in Banach spaces, J.Funct. Anal. 33 (1979), 5894.

8. F. Feh er and G. Gr assler, On an extremal scale of approximation spaces, J. Comput. Anal. Appl. 3(2001), 95108.

9. M.A. Krasnoselskii and Ja.B. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd.,Groningen, 1961.

10. J. Peetre and G. Sparr, Interpolation of normed Abelian groups, Ann. Mat. Pura Appl. (4) 92(1972), 217262.

11. A. Pietsch, Operator Ideals, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.12. A. Pietsch, Approximation spaces, J. Approx. Theory 32 (1981), 115134.13. A. Pietsch, Eigenvalues and s-Numbers, Cambridge University Press, Cambridge, 1987.14. E. Pustylnik, Ultrasymmetric spaces, J. London Math. Soc. 68 (2003), 165182.

evgeniy pustylnik- ultrasymmetric sequence spaces in approximation theory

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