For these notions it can be seen

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STUDTA TJNIV. MB A BEŞ-B O L Y A I ” , M A T H E M A TI C A , Volume XLIV , Number 3, September 1999

S O M E A P P R O X I M A T I O N ID E A L S

N I C O L A E T I T A

A b s tr a c t. We consider some approximation ideals of operators on oper

ator spaces. The method used is similar to that from [8], [10], [11], in the case of the classical Banach spaces, or [2], [7] for the case of Hilbert spaces.

1. In tr o d u ctio n

The theory of the approximation ideals is well known for the case of linear and bounded operators on Hilbert, or Banach, spaces [2], [6], [7], [8], [11].

Here we consider the special case of the completely bounded operators on operator spaces. For these notions it can be seen [1], [3], [5].

We begin by recalling some definitions.

An operator space

E

, in short O.S, is a Banach space, or a normed space before completion, given with an isometric embedding

J

:

E

—►

L(H),

where

L(H)

is the space of all linear and bounded operators

T : H FI, H

being a Hilbert space.

We shall identify often

E

with

J(E)

and so we shall say that an O.S is a (closed) snbspace of

L(FF).

If

E

C

L(H)

is an operator space then

Mn

O

E

can be identified with the space of all

n

x

n

matrices having entries in

E,

that it will be denoted by

Mn (E) .

Clearly

Mn (E)

can be seen as an o.s. embedded in

L(Hn),

where

Hn = H

0 ... ©

H

(number of

H

is n ) .

Let us denote by || ||n the norm induced by

L(Ifn)

on

Mn (E)

, in the par

ticular case n — 1 we get the norm of

E.

Taking the natural embedding

Mn (E)

—»

A/n + i (i?) we can consider

Mn {E)

included in Mn+i

(E) ,

and || ||ri induced by |H|n+1 . Thus we may consider

(E)

a normed space equipped with it’s natural

n n orn l ll'llco •

109

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NICOLAE TI TA

We denote by

K [E]

the completion of

\jMn

(

E

). If we denote by

n

A'o =

\jMn,

the case

E =

C, then the completion of

K

o coincides isometri-

n

cally with the C*-algebra,

C(l2),

of all compact operators on the space

l2.

It is easy to check that (jA /n

(E)

can be identified isometrically with

K

q

Ç)E.

n

The basic idea of o.s. is that the norm of the Banach space

E

is replaced by a sequence of norms {|| ||ril} n on

{ Mn ( E) } n

or by a single norm H-H^ on the space A'

[E],

D efin ition 1. Let

E\

C

L( H

i) and

En

C

L(H

2) be operator spaces,

u

:

E

i

—> E2

be a linear map and

Un

: (xij) G Mn

(E\) —y

( u( xj j) ) G

Mn (E2) .

We say that, u is com pletely b o u n d e d , c.b ., if sup ||w„|| < oo and we define

\\u\\c b

sup||nn||.

n n

D efin ition 2. (equ ivalen t)

u

is co m p le te ly b o u n d e d if the maps

un

can be ex

tended to a single bounded map

Ur<>

:

K

[AJ

K [E2]

and we have ||t/.||c

b

= . D efin ition 3.

c.b

.

(E\, E2)

{?/ :

E\

—)•

E2

:

u

is c.b.} . We shall consider the

c.b.(E\<E2)

equipped with IHL.6. •

Remark L

The similar definition of the uniform norm for the bounded operators can be written as follows:

IMIc.6. = sup {||«oo|| ■ x e K [A ], 11*11 <

1

} .

Remark 2.

Likewise the case of an isomorphism between two Banach spaces, we say that two o.s.

E

i ,

E2

are c o m p le te ly isom orph ic, co m p le te ly isom etric, if there is an c.b. isomorphism

u

:

E\ —y E2

with c.b. inverse and in addition |M|r.& —

||u- 1 |«(El)l|c.fr. = 1

Let.

Ei

C

L(FI

i) and

E2

C be operator spaces. There is an embedding J :

Ei

O

E2 —y L(H

i 0

Hn)

defined by

J

(x\0 x 2)

(h\

0

h2)

= xi

(h.\)

0 x 2

(h2) .

We denote by

E\

0 min

E2

the completion of

E\

0

E2

equipped with the norm

1 1 0

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SOME A P P R O X IM A TIO N IDEALS

O b viou sly

J

can be extend to an isom etric em bedding. So we can see

E\

® mjn En as an o.s. em bedded into L(H\ ® a Hn). T his space is called the m i n i m a l, s p a t ia l, t e n s o r p r o d u c t o f E\and En. (Hi ®a Hn is the hilbertian tensor p rod u ct, [7], [] 1].)

If E C L(H ) is an o.s. then Mn 0 min E can be identified with the space Mn (E) and K [E] can be identified isom etrically with K ©min F. Thus, for any linear m ap u : E\ —►En we have ||t/.||c6 = ||/® u : K ® min E\ —ï K © mjn En\\ = Il/ O V ' K Omin E\ -> K ® min En\\c b . M ore generaly it can be shown that, for any o.s. F C L (H), we have ||7> ® u : F ® mjn E\ -> F ®min En\\ < ||u||c 6 . Further on, if v : F\ Fn is another c.b . m ap, we obtain

Il

U

0

v

:

Fl

® min

El

- >

F2

©min

En\\cb

< IMIc.6. * ll^llc.6. • T h is relation will be very useful in the sequel.

For others properties o f the m inim al tensor product it can be seen the papers [1]. [!)], etc.

2. A p p r o x i m a t i o n n u m b e r s o f c o m p l e t e l y b o u n d e d o p e r a t o r s

D e f i n i t i o n 4 . Let u : E -» F be a com pletely bounded m ap, u E c.b. (E , F ). T he a p p r o x i m a t i o n n u m b e r s , acnb( « ) will be defined as follow s

acnb (u) := in f {\\u— a.||c b : a E c.b. (E , F ), rank (a) < n) , n — 1 ,2 ,....

Remark 3. From this definition it results that ||w||c 6 = b (u) > a ţ b (u ) > ... > 0.

P r o p o s i t i o n 1. The approximation numbers a £ b (u) verify the following inequali

ties:

k k

1. acnb- (ui + u2) < 2 • ( rt" fc ( U l) + a" b (U2)) ,fo r k = l ’ 2 ’ •••

n = l n = l

k k

2.

^ 2 anb

(wi 0 ^2) < 2 •

^ 2

(ani> (wi) * anb (ul)) i for fc = F 2, ...

77 = ¹ 77 = ¹

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NICO LAE T IT A

Proof. 1) Let. e > 0. T here are a ,, i = 1 ,2 , such that rank (a*) < n and ll«i - rt>llc 6. < a n b ( « « • ) + I -

W e ob ta in :

n2b n -i

(«1 + «2) < ||(«1 + « 2) -

(ai

+ a2)||c.6. <

< ll« l - “ lllc.b. + llu 2 - 0 2 ||c.6. <

< < b ( « 1 ) + « n b ( U 2 ) + S.

Since e is arbitrary it follow s that:

a - ï n- 1( « 1 + «2) < < 6 ( «1) + ( «2) •

Further on it results:

E < 6 (Ml + «2) < E «2 n - l (Ml + «2) + E «2-n (« 1 + U2) <

ri = 1 n = 1 n = 1

< 2 • E « 2 ' n - l ( M l + M2 ) < 2 • E ( " n b ( « l ) + M *'6 ( l / 2 ) ) .

n = l n ~ 1

2) W e consider also a ,, ?’ = 1 ,2 , such that, ran/c (a,-) < n and

||«i - a i||c ,fc. < a n 6 ' ( M . ) + §•

W e obtain :

« n b ( « 1 ° M2 ) < 11 ( w 1 O m2 ) - [ « 1 O

a2

+ a y o ( u 2 - a 2 )]||c 6 =

= | | ( « i ~ M l ) O ( w 2 - a 2 )||c b < { a cn b ( « ! ) + § ) • ( a ^ b ( t * 2 ) + § ) .

Since e is arbitrary it follow s that:

«2 n - l (Ml o Wo) < acnh (w i) • acnb- (w2) •

Likewise the 1) results 2). □

Remark f For the case o f the linear and bounded operators between Banach spaces the above inequalities are known, [8], [11].

In the sequel we deduce an inequality for the case o f the c.b . operator u\ Q mm Î/-2 using a sim ilar m eth od with that from [9], [10], used for the classical case o f the bou nded operators on B anach spaces.

P r o p o s i t i o n 2 . The approximation numbers acnb (iii ® mm w.o) verify the inequalities:

L

an

(ul

0min

Url)

< 6

k ncb

£ —

n = l

(

m

, H M I

c.b.+ «n 6' (M2)

I k I L

for fc = 1 , 2 , . . .

1 1 2

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SOME AP PR O XI MA TI ON IDEALS

Proof.

Let

€

> 0. There are a,-,

i =

1, 2, such that

rank

(a,) <

n

and

I K - a.ilc.6. < anb' (««O + §•

We obtain:

('M l © m i n d o) ^ ||m i © m i n w 2 ~ a l © m i n ^*21|c .6. =

= | | («1 - « l ) © m i n U2 ~ d\ © m i n ( « 2 - d 2 )\\c 6 <

< ll «l - « i l k • I k l k . + H«1 Hr.6. • ||«2 — «2IL.6. <

< («0 + §) • I k l k . + ||«1 ~ «1 + «iile

6.

• («n6 («s) + §) <

< ( « n 6 (m i) 4- § ) • I M L f t . + ( | k - m i||cb + | k l k . ) • ( < 6 (l<2) + I ) <

< ( < b ( «1) + § ) • I k l k + 2 • I k I k • ( < b (t»a) + 1 ) .

Since

e

is arbitrary we obtain:

aCJ (M ) ® m i n M2 ) < 2 • (acn b ( «1) • | k l k + acn b (u 2) • ||m i||cb ) .

T a k in g account th at the sequence o f the aproxim ation num bers is decreasing we can w ritte:

^2^{a n}

b'{"iv>m\n'U

(** 1 <8> m in'** 2 ) 2

)

< ÿ2 (2 •

n

+ 1)

n = 1 11

N ow we o b ta in : k

3

n = 1E 3 E n — I

a% (wl<8>ininW2)

, where j 2 < k < ( j + l ) 2

^

Y

2 (2 • n -J- 1 ) fl»2‘^tl®minU2^ < 3 .

Y l n '

°n2 ^tl®minti2^ <

n- 1 n = 1 n = l

0 ^2 a n h(U l) llU3||e.b + a n b (t<2)-||»l||c.fc: < 6 ‘ ^ fln * ' (Ui H IM Ic . b . + fln * ' (“ - H I ” dlc.b.

n = l H n = l

T h is finishes the proof. □

Remark 5. B y m ea n s o f these ap p roxim ation num bers we can define special approxi

m ation ideals in e.b. (E, F ) .

3. S p e c i a l a p p r o x i m a t i o n i d e a l s

D e f i n i t i o n 5 . Let x = {a?i, a ? o , b e a real sequence and let card (# ) be card {i ÇL N : Xi 0} .

Let K be the set o f all real sequences x G /no having the follow in g two properties:

1.

card

(j?) <

n

(x),

n

(a;) is a natural number

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N ÎCOLAE T IŢ A

2 . X \ > X 2 > . . . > æ n (ar) > 0 .

A function $ : K —> R is called a sy m m etric n orm in g fu n ctio n if:

1. $ (.r) > 0 if x G K and x 0;

2. $ ( o r • x ) = a • $ ( x ) , for every o > 0 and x G A";

3. O (.r 4- y) < $ (x ) + $ (y) , for every x, y G A ';

4 . $ ( { 1 , 0 , 0 . . . } ) = 1 ;

k k

5. If x ,y G A' and ]T]x* < f ° r everY ^ = 1 , 2 , then $ (:r) < $ (y) .

j = l i = l

Remark 6. T h e above definition can be extend on the whole space taking

$ ( . r ) := lim $ ({x ^ , ..., x* , 0, 0 ...}) , where x* — { x * } if;/v is the sequence {k ?'| }?€;v rearranged in decreasing order.

D e f i n i t i o n 6. In the sequel we shall consider a subclass o f c.b. (E, F) which is defined as follow s:

4> - c.b. (E, F) := { « € c.b. (E, F) : |M|c* fc := $ ( R " ( « ) } „ ) < .

Remark 7. W e prove that this class has similar properties with the sim ilar classes defined for linear and b ou nded operators. (For the case o f the Hilbert spaces it can be seen [2], [7] and for the case o f the Banach spaces it can be seen [8], [9], [10], [11].)

P r o p o s i t i o n 3. — c.b., ||.||^6 ^ is a quasi-normed operator ideal.

Proof. 1. Any unidim ensional operator , a G c.b. (F , F) , belongs to

$ —c. 6. (F , F) because, in this case, the sequence { a £ 6- ( « ) } = {||w||c6 , 0 , 0 , . . . } and hence $ ( R 6 ( « ) } „ ) = I M R <

2. If u1,^2 G c .b .(E ,F ) then ui + un G c .b .(E ,F ). This results from the p rop osition 5 (1). and from the properties o f 3>, as follow s:

* ( R * R + u 2 ) } J < 2 • $ ( R b R ) + < 0 R ) } „ ) <

< 2 - ( * ( R 6 ( « , ) } „ ) + * ( { < * • R ) } J ) .

3. If v G c.b. (E, E) , u G $ — c.b. (E, F) and w G c.b. (F, F ) then 114

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iv

o « o ţ) G $ -

c.b. (E

,

F )

.

From the proposition 5 (2) and from the definition of a£6

(u)

it follows that

0C nb' (<<’ ° « ° »') < ll« ’tle 6 • anb ( « ) ’ I M U . a n d h e n c e

$ ( { < 6' ( l | ,0 « 0 t ’) } „ ) < ll«ilc.6. ( « ) } „ ) ll'ilc.fc - D

Remark

8. We present now some properties similar to the properties of the classical approximation ideals L<j>, [8], [11].

Lem m a 1.

The approximation numbers acnh (u) verify the inequalities:

SOME A P P R O X IM A TIO N IDEALS

k Y A ' b n -l

n — 1

k k

(«) < («) < 2 • E « 5 - n - l (W) =

l ' 2’ -■

n = 1 n = 1

Proof.

The first, inequality is a consequence of the fact that the sequence

{a^b

(u ) } n is decreasing.

The second inequality results as follows:

E ° n b ( « ) < E anb ( « ) < E « 2 n - 1 ( « ) + E « 2 n ( « ) <

n ~ 1 n — 1 n = 1 n = 1

**5: 2 * E a 2 n - l ( îf ) •**

n = l

□

C orolla ry 1. ||u||^6' ( { é t - i (w )}n) a

quasi-norm equivalent

with i m i; 6- = $ ( { < 6 ( w ) } J -

k k

/

Remark 9.

Since ^ (a£6- (u ))P < E ( » E < ' 6' («) i=1

P

<

r ( p )

•

E

( < 6’ (W))P . 1 < P < = 1,2,..., n = l

see the Hardy inequality [11], it follows that

* ( { K ‘ < » » ' } „ ) < * ( { ( ; î > ‘ w ) } J < ' ( » ) • * ( { « * < « » ' } „ )

and

hence

||«||$bp) is equivalent, with |M|*bp) := $ (P) ^ j j j • E « f 6' ( « ) } ) -

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N ICOLAE T IŢA

where <ï>(r) ( { * ; } ) = ^ } ) p is a sym m etric norm ing fu n ction , [2], [6], [7], [11], for 1 < p < oo.

Because acnb (u ) < ( a f 6 (u) • ... •acnb (u )) n < £ • J2ai'b' (n) l{ follow s> also, n

2 = 1

that

i a n b ( « ) } „ € l * (r) if 9 n («) := { ( « î fc («) • ••• • < 6 («)) " } n €

{•^n} £ ^(p) ^ ^ ^(p) ^

Remark 10. If we consider the special case o f the function

K 6 N ) p p

^ ( p ) : ( K b ( « ) } ) - > *

n

where 1 < p < oo, the classes <£(p) — c.6. (F , F ) are tensor product stable.

P r o p o s itio n 4. If G $ ( p) — c.6. (F * , F^) , fc = 1,2., then

U l

®min ^2 £ ^(p) C.6. (F^ ®min F2, Fţ ®min ^2) •

P roof It is sim ilar to that for the classical approxim ation ideals [10], [11].

First we remark that, using the relation

for A* = 1 , 2 ,..., see P rop osition 6 for p = 1.

Now taking into account the properties o f the functions it follow s that we can obtain

c (/>) • $

« * • («0 • ll«allc.fr.)P | («;

n

,c.b.

n

( « 2 ) ' I M I C b)

n

r

116

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SOME A PPR O X IM A TIO N IDEALS

Hence

$(r) ( { ° n 6' (U1 ®min «2) } ) <

< O

(P) '

((p) ( R («!)}) • I H U + *(P) ( R 6 («2)}) • IKIU.) <

< 2 - 0 (p) . ? (p) ({o*-4- («!)}) R P) ( R fc M } ) ,

C i ( p ) = c ( p ) r and |KI|c 6 < $(P) ( R 6' («*)}).* = 1,2.

The proof is fulfiled.

□

Rem ark ÎL

The above result remains true if we consider the m axim al ten sor p r o d uct.

R eferen ces

[1] D.P. Blecher, V .I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991) 262-292.

[2] 1. Gohberg, M . Krein, Introduction on the theory of the linear, non-adjoint operators on Hilbert, spaces, Nauka, Moscow, 1965. (in Russian)

[3] V . I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics, Longmans, London, 1986.

[4] V . I . Paulsen, Representation of function algebras, abstract operator spaces and Banach space geometry, J. Funct. Anal. 109 (1992) 113-129.

[5] G . Pisier, Operator spaces and similarity problems, Documenta Math. (I.C .M . 1998) 1-429-452.

[6] N. Salinas, Symmetric norm ideals and relative conjugate ideals, Trans. A .M .S . 188 (1974) 213-240.

[7] R. Schatten, Norm ideals of completely continuous operators, Berlin-Gottingen- Heidelberg, 1960.

[8] N. T iţa, Operators of <rp—class on non-Hilbert spaces, Studii Cercet. M at. 23 (1971) 4 6 7 -4 8 7 .(in Romanian)

[9] N. Ti(,a, On a class of operators, Collect. M ath. 32 (1981) 257-279.

[10] N. T ita, Some inequalities for the approximation numbers of tensor product operators, Anal. Şt. Univ. ” A1. I. Cuza” Iaşi 40 (1994) 329-331.

[11] N. Ti^a, S-numbers operator ideals, Ed. Univ. Transilvania, Braşov, 1998.

De p a r t m e n to f M^{a t h}., F^{a c}. ^of Sc i e n c e, ” Transilvani a” U^{n i v}., 2200 B^{r a ş o v}, Ro m a n ia

117