8/3/2019 S. V. Astashkin, L. Maligranda and E. M. Semenov- On Complementability of Subspaces Generated by Contractions and Shifts of Functions

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390

Doklady Mathematics, Vol. 66, No. 3, 2002, pp. 390392. Translated from Doklady Akademii Nauk, Vol. 387, No. 5, 2002, pp. 583585.Original Russian Text Copyright 2002 by Astashkin, Maligranda, Semenov.English Translation Copyright 2002 by MAIK Nauka /Interperiodica (Russia).

Suppose thatE

is an r.i. space on [0, 1] (the defini-tion is given below), a

E

, 1

k

2

n

,

and n

= 0, 1,

.We set

and Q

n

, a

= span{

a

n

, k

, 1

k

2

n

}

. We denote the set of

a

E

such that Q

n

,

a

is uniformly complemented inE

,i.e., there exist projectors P

n

from E

onto Q

n

,

a

with

<

, by

(

E

)

. This paper studies the set

(

E

)

and the class of r.i. spaces E

coinciding with

(

E

)

. Itturns out that

(

E

)

is closely related to the space of ten-sor multipliers acting inE

.

Below, we give the necessary definitions.

A Banach spaceE

of measurable functions on [0, 1]

is called symmetric or rearrangement invariant (r.i.) if(i) |

x

(

t

)

|

|

y

(

t

)

|

and y

E

imply x

E

and

||

x

||

E

||

y

||

E

;

(ii) the equimeasurability of functions x

and y

andthe inclusiony

E

implyx

E

and ||

x

||

E

= ||

y

||

E

.

Following [1], we assume thatE

is separable or dualto a separable space.

Examples of r.i. spaces areL

p

with 1

p

and theOrlicz, Lorentz, and Marcinkiewicz spaces. Let

denote the set of increasing concave functions

(

t

)

on[0, 1] such that

(0) =

(+0) = 0

and

(1)

= 1. Each

an k, t( )a 2

nt k 1+( ) if k 1

2n

----------- tk

2n

-----

0 for other t 0 1,[ ]

=

Pnn

sup

function generates a Lorentz space () and aMarcinkiewicz spaceM() with norms

where x*(t) is the permutation of |x(t)| in decreasingorder. The space () is separable, and (())* =M().

If(t) is a convex increasing function on [0, ) and

then the Orlitz spaceL is, by definition, the set of func-tions such that (x) L1 for some > 0 with norm

The condition 2 means that (2t) C(t) forsome constant C> 0 and all t> 1.

In any r.i. spaceE, the operator family

acts continuously. The numbers

are called the Boyd indices of the spaceE. We alwayshave 0 EE 1. We use the CalderonLozanovskiiconstruction [2]. IfE0 andE1 are r.i. spaces and 0 < < 1,then denotes the space of functions with the

norm

x ( ) x* t( ) t( ),d0

1

=

x M( ) 1

s( )--------- x t( ) t,d

0

s

0 s 1 < sup=

t( )t

-----------t 0lim

t t( )-----------

t

lim 0,= =

x L inf : 0, x t( )

------------ td0

1

1>

.

8/3/2019 S. V. Astashkin, L. Maligranda and E. M. Semenov- On Complementability of Subspaces Generated by Contractions and Shifts of Functions

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DOKLADY MATHEMATICS

Vol. 66

No. 3

2002

ON COMPLEMENTABILITY OF SUBSPACES GENERATED 391

For any r.i. space E

, the continuous embeddings

L

E

L

1

hold. For an r.i. spaceE

,E

0

denotes the clo-sure ofL

inE

. IfEis separable, thenE0 =E; ifEL,then E0 is separable. The equality of two r.i. spacesmeans that they coincide as sets. By the closed graphtheorem, their norms are equivalent. By E' we denotethe space of functions for which

The spaceE' is isometrically embedded inE*. IfEisseparable, thenE' =E* and the norms are equal.

There exists a measure-preserving one-to-one map-ping of [0, 1] onto the square [0, 1] [0, 1]; therefore,for any r.i. space Eon [0, 1], the measure-preservingmapping generates a space of functions on the squareisomeric to this space. We denote this function space onthe square by the same symbolE. This allows us to con-sider the tensor multiplier (xy)(t, s) = x(t)y(s) as abilinear operator inLp, where 1 p. Every r.i. spaceEgenerates a space(E) of tensor multipliers with thenorm

Obviously, the embeddingsL(E) Ehold. Ten-sor products of r.i. spaces are considered in [3, 4] andelsewhere, and the space of tensor multipliers are stud-ied in [5]. In more detail, the r.i. spaces are studied in[1, 6]. Subspaces generated by shifts of one functionfrom an r.i. space on [0, ) are examined in [7].

The central result of this paper is the following the-orem.

Theorem 1.Let E be an r.i. space. Then,

Theorem 1 solves the problem of finding (E) forseparable r.i. spaces. In this case, (E) coincides with(E). Using the results describing (E) [35], weobtain the following corollary.

Corollary 1. (i)If1 < p < and1 q < , then(Lp, q) =Lp, min(p, q)and( ) =Lp.

(ii)If and (t) = , then

The equality(()) =() holds if and only if(ts) C(t)(s)for some C> 0 and all t, s [0, 1].

(iii)If 2, then(L) = Lif and only ifissemimultiplicative, i.e., there exists a C> 0 such that(uv) C(u)(v)forallu, v 1.

(iv)If E is a separable r.i. space, then(E) =Lifand only ifE= 0.

x E' x t( )y t( ) td0

1

y E 1 sup . 0 suchthat

for all aEand allcn, k 1,wherek= 1, 2, , 2nandn = 0, 1, ; here, n,kis the characteristic functionof the interval ((k 1) 2n, k2n).

The spaces Lp have the following characteristicproperty in terms of the class .

Theorem 6.Let E be an r.i. space. The followingconditions are equivalent.

(i)(E) =Eand(E') =E';

(ii) The tensor product operators fromEEtoEand fromE' E'toE'are bounded;(iii) Thereexists ap [1, ]suchthatE=Lp.Thus, if an r.i. spaceEdoes not coincide withLp for

1 p, then the class cannot contain more thanone of the spacesEandE'.

The problem of describing (Lp,) (1 < p < )remains unsolved. It is clear only that (Lp, )

=Lp.

2t( ) t( )

-------------t 0lim

M ( )( ) L M ( )\M0 ( )( ).=

t( ) t2( )------------

0 t 1