Math. Nachr. 113 (1983) 69-64

On the Type of Interpolation Spaces and Sp,p

By FERNANDO COBOS of Santiago de Compostela

(Received March 1, 1982)

1. Introduction

The type of real interpolation spaces has not been enough studied in spite of the importance of this notion in the study of the geometry of BANACH spaces. The known results concerning the type of (A,, A1)e,q in terms of the type of A, and A,, do not cover all values of the parameter q.

In this paper we study the interpolation properties of the operators of RADE- MACHER (p,s)-type (according the terminology of PIETSCH [lo]), which allows us to know the type of (A,, A1)e,q for the other values of q in [l, 00) not convered by BEAV- ZAMY [l]. We also give counter-examples which show that the results cannot be im- proved.

Finally, we derive the type and cotype of the LORENTZ operator spaces AS',,^. We obtain that the type (resp. cotype) of Sp#q is the same as the type (resp. cotype) of the LORENTZ functions space Lp,p, which has been studied by CREEKMORE in [3] by means of its st,ructure of BANACH lattice.

2. Definitions and Notations

Let X and Y be (real or complex) BANACH spaces. We denote by L ( X , Y) the col-

Definition 1 (see [lo], p. 293). Let 0 < p 5 2 and 0 < s < 00. An operator T E L ( X , Y) is said to be of RADEMACHER (p,s)-type if there is a constant M < 00

such that for every finite set

lection of all bounded linear operators from X into Y.

in X ,

Definition 1 can be formulated using the RADEMACIIER functions (r,,), defined by r,(t) = sign (sin (2"nt)) for 0 5 t 5 1, because

Remark 1. If T is of RADEMACHER (p,s)-type, then T is of RADEMACHER (w,s)-type

Remark 2. The parameter s is in fact superfluous in Definition 1 [lo, Prop. 21.1.91,

for every 0 < 17 < p .

furthermore every T E L ( X , Y ) is of RADEMACHER (1,l)-type.

60 Cobos, On the Type of Interpolation Spaces and i!3p,q

Definition 2 (see [7], l.e.12). A BANACH space X is said to be of type p for some 1 p s 2 if the identity map of X is of RADEMACHER (p,l)-type. The space X is said to be of cotype q for some 2 5 q < 00 if there exists a constant M < 00 such that for every finite set { x ~ } ; = ~ in X ,

We denote by lp,q the LORENTZ sequence space, formed by all sequences of scalars (l,,) converging to zero and having a finite quasinorm

if 1 < p < m q = m

if 1 < p < m 1 < q < m P 00

Il(tn)llp,q = ( z I E Z I ~ ~ ( ~ / P ) - ~ n = l

tl(5fl)llp,q = SUP (153 n19 n

where (5:) designates the rearrangement of the elements of (5,) by magnitude of the absolute values: 2 152+1 2 . . a . Of course, for p = q we get the classical space of p-summable sequences which is denoted by (Z,, I[ 11,).

Let 9 = (0, 00) and let p be the LEBESGIJE measure. We denote by Lp,q the collec- tion of (classes of) real-valued p-measurable functions f on 9 having a finite quasinorm

if l < p < m q = m

if r 00

11/11;,, = q/p J ( t l / ~ j * ( t ) ) q dt/t

ll/ll;,q = SUP ( t ” P / * ( t ) )

1 < p < m 1 5 q < 00

( 0

1>0

where f*(t) is the non-increasing rearrangement o f f on (0, 00) (see [2, section 1.31 and [3, 5 31). As before, we have L,,, = L,.

All these spaces are complete, moreover i t is possible to replace the quasinorm with a norm, which makes

Let H be a separable HILBERT space over the field of complex numbers. For 1 < p < 00 and 1 < q 5 00, we denote by Spoq the collection of all compact operators T E L(H, H ) having a finite norm

(resp. Lp,q) a BANACH space (see, e.g., [2, Thm. 5.3.1.1).

T , , ~ ( T ) = SUP ( n( l /P ) - - l 2 a m ) if l < p < m q = o o k = l

where ( OL,,( T)) is the monotone non-increasing (non-negative) sequence converging to zero formed by the eigenvalues of the positive operator [T*T]lI2, each one repeated a number of times equal to its multiplicity. If [T*T]1/2 possesses less than n eigenvalues, we put a,(T) = 0. The norm T,.~ is equivalent t o the quasinorm u,,,(T) = ~ ~ ( L X ~ ( T ) ) ~ / , ~ ~ (see [lo, Prop. 13.9.51 and [12, Thm. 1.18.3/1]), whence (see [9, Thm. 11) (Sp,q, T,,~) is a BANACH space. Moreover, the SCHATTEN p-class S, (see [lo, section 15.51) is equal to S,,,, both spaces having equivalent norms.

For the notions relating to interpolation spaces we follow the terminology of [2] and [12].

Cobos, On the Type of Interpolation Spaces and fJp,q 61

3. On the Type of Interpolation Spaces

BEAUUY proved (see [ l , pp. 77-78]) that (Ao, Al)e,q is of type q whenever Ai

isof t y p e p i ( 1 5 p i 5 2 ) f o r i = 0 , l a n d 1 S q s p ( O ) , where-=-+-.In the proof i t is used the LIONS-PEETRE formula

1 1 - e e P(@ Po PI

( W A o ) , Lp,(Ai))e.q = Lq((Ao, Ai)e,*) valid for q = p(0) . Though there is no reasonable generalization of this formula to other values of q (see [4]), we have:

Lemma 1. Let (Ao, A , ) be a cmmpata%le couple of BANACH qam. Assume that 1 5 po,

p1 < CQ, o < e < I, -= - + - a d p 4 q < 00. Then Zp((Ao, A i h q ) zk c ~ t - 2, PO PI

tinuously embedded in (lpn(Ao), Zpl(Al))e.g.

coordinates a, $: 0 and a, E A, n Al for n = 1, 2, . . . Clearly

1 I - e e

Proof. Consider the set A of all sequences a = (a,) having only a finite number of

A = &((A09 A1)e.q) " ( ~ p o ( A o ) , Jp1(A1)e,q).

By Thm. 1.4.2 of [12], the inequality of MINKOWSKI and again Thm. 1.4.2, we have for every a E A with q = eplpl

5 4 0 2 I ~ ~ I ~ I ~ ( ~ A ~ . A ~ ) ~ . ~ ) where N is a natural number so that a, = 0 for n > N and C,, C2 are positive numbers which do not depend on a E A. This proves the lemma because A is dense in Zp((Ao,A1)e,q) [2, Thm. 3.4.21.

Remark 3. If p < q we have not, in general, the converse inclusion and if 1 5 q < p then Lemma 1 is not true, in general: Take 1 < po, pl < 00, po =I= p , and A. = AI = R (set of the real numbers) then we obtain [2, Thm. 5.3.11

( l P , V o ) , lpl(Ai))e,q = Z p , q GL lp = l p ( U o , A1)e.q)

zp((Ao, Ai)e,q) = lp o l p , q = (lPJAo), lP lVi ) ) e ,q

if P < q

if 1 5 q < P.

62 Cobos, On the Type of Interpolation Spaces and 8p,q

Lemma 1 and the technique used by BEAUZAMY in [l, pp. 77-78] allow us to prove :

1 i - e e Proposition 1. Assume that 1 I; po, p , 5 2, 0 < 0 < 1, - = - + -, 1 sq < 00

r, Po P1 and v = min(p, q). If (Ao, A,), (Bo, B,) are compatible couples of BANACH spaces and T E L(A0 Bi) is a n operator of RADEMACHER (pi,q)-type for i = 0, 1 then T E L((Ao, A1)o,q, (Bo,

Proof. Suppose first p 5 q. Since T E L(Ai, Bi) is of RADEMACHER (pi,q)-type, i t follows from Definition 1 that the operator F: Zp,(Ai) + Lq(Bi) defined by

and i t is of RADEMACHER (v,q)-type.

is continuous for i = 0, 1, therefore, using the interpolation theorem [2, Thm. 3.1.21, we have that i t is continuous from (Zpp(Ao), Zp l (A~) )~ ,q into (Lq(Bo), .Lq(Bl))o,q. But now, by Lemma 1, Zp((A0, is continuous embedded in (Zpo(Ao), Zp,(A1))e,q, and by the LIONS-PEETRE formula (see [ll, Thm. 1.18.41)

Thus F is continuous from lp((Ao, again the interpolation theorem, that

into Lq((Bo, B,)o,q), whence i t follows, using

T E L((Ao9 Ai)e.q, P o , B1)e.q)

is of RADEMACHER (p,q)-type. Suppose now 1 < q < p (if q = 1 the result is trivial by Remark 2). We can choose

1 5 pi 5 pi (i = 0 , l ) such that - = - + - By Remark 1, the operator

T E L(Ai, Bi) is of RADEMACHER (p:,q)-type and then, by the first part of the proof, i t follows that

1 i--8 e q 2-4 24-

T E L((Ao9 Ai)e,q, (Bop B1)e.q)

From Proposition 1, we obtain immediately:

Corollary 1. Let (Ao, A , ) be a mmpatibb couple of BANACH spaces with Ai of type pi (1 5 pi 5 2) for i = 0, 1. Then (Ao, A,)e.o is of type min ( p , q) provided that 0 < 0 < 1,

-- -- + -and 1 5 q < 00. P Po P1

is of RADEMACHER (q,q)-type.

1 i - e e

Remark 4. For 1 5 q < p 5 2 given arbitrarily, let us take A , = L3+ and A , = which are spaces of type p [3, Thm. 3.51. It follows from [2, Thm. 6.3.11 that

(Ao, Al)4,5,q = L5,q (equivalent norms)

and this space contains a subspace which is isomorphic to lq [6, Thm. 5.11 therefore, by considering the unit vector basis of lq, it is verified that L5,q is not of type r for any r > q. So then, Corollary 1 (and also Proposition 1) gives a sharp result in this case.

Cobos, On the Type of Interpolation Spaces and LY,,~ 63

Remark 6. In general, a similar result to Corollary 1 is not true for q = 60: Let us take A. = L, and A1 = L,, these spaces are of type 2 [7, p. 731 however

(Ao, Al)d,s,, = L5,, (equivalent norms)

which is not of type r for any r > 1 [3, Thm. 3.71.

4. Type and Cotype of the Spaces Sp,q

DIXMIER and TOMCZAK-JAEQERMANN proved (see [ l l , Thm. 2.21) that the moduli of smoothness and convexity of the trace classes S, have the same order as the corres- ponding moduli of L, (1 < p < 60). Therefore S, is of type min (2, p) and of cotype max (2, p ) [7, Thm. l.e.161, moreover since there exists an isometric imbedding of 1, into S, [8, p. 2671, we have that Sp is not of type r for any r > min (2, p ) and not of cotype s for any s .< max (2, p).

The following proposition shows the type and cotype of the Sp,q spaces:

Proposition 2. a) Let 1 < q < p < do. Then the following holds.

i) Sp,q is of type min (2, q). ii) If p =i= 2, Sp,q is of cotype max (2, p ) .

iii) S2,q is of cotype ( 2 + E ) , for ull E > 0. b) Let 1 < p < q < 60. Then the following holds.

i) If p =k 2, Sp,q is of type min (2, p ) . ii) S2.q is of type (2 - E ) , for all E > 0. iii) Sp,q i s of mtype max (2, q) .

1 1--8 e Proof. If 1 < po, p , < 00, 0 < 6 < 1 and - = - + - then 19, Thm. 13

Sp,q = (S,,, S,,)B,,, (equivalent norms).

P Po P1

Distinguishing the situations 1 < p < 2, p = 2 and 2 < p < a, choosing appropri- ately po < p < p1 and using the Corollary 1, we obtain the type of Sp,q. The cotype of Sp,o is obtained by duality taking into account that the dual space of Sp,q is S,,,,o, - . ~~ . .

1 1 1 1 (; + 7 = p + 2 = 1 [9, Thm. 41 and Prop. l.e.17 of [7].

Remark 6. An analogous result to Proposition 2 for the LORENTZ function spaces Lp,q can be derived from Corollary 1. This result was before proved by CREEKMORE in [3] by means of other procedures.

Just like it happens with Lp,m [3, Thm. 3.71, we have for Sp,m the following reault:

Proposition 3. Let 1 < p < do. i) S,,m ie of no type r > 1.

ii) Sp,m is of no cdype s < 60.

tionals associated to it. For every sequence Proof. Let (zn) be any orthonormal basis for H and let (fn) be the coefficient func-

= (&,) E I,,,, let T, E L(H, H) be the

64

compact operator defined by

Cobos, On the Type of Interpolation Spaces and Sp,q

00

TE(4 = z tfltfl(4 5-

(T,+T,) (4 = 22 rEf l latn(4 Zfl

n=1 We have

00

n=1

thus (afl(T,)) = (Itfl). Whence there exists an isometric imbedding of l p , , into f lP , , .

But Zp,, is isomorphic to the dual space of lP, , , (: + [12, Thm. 1.11.2]andIp.,1 .- - contains a complemented subspace which is isomorphic to Il [6, Prop. 4.e.31. Conse- quently, Sp, , contains a subspace which is isomorphic to I!,. This proves the proposition because I, is neither of type r > 1 nor of cotype s < 00.

References

[ 13 B. BEAUZAMY, Espaces d'interpolation reels; topologie et gbombtrie. Springer Lecture Notes

[2] J. BERQE and J. LOFSTROM, Interpolation spaces, an introduction. Springer, Berlin- Heidel-

[3] J. CREEKMORE, Type and Cotype in Lorentz Lpq Spaces, Indag. Math 43 (1981) 145-152 [4] M. CWIKEL, On (Lpe((A,,), L P I ( A ~ ) ) ~ , ~ , Proc. Amer. Math. SOC. 44 (1974) 286-292 [5] T. FIQIEL, W. B. JOHNSON and L. TZAFRIRI, On Banach lattices and spaces having local

unconditional structure with applications to Lorentz function spaces, J. Approx. Theory 13

[6] J. LINDENSTRAUSS and L. TZAFRIRI, Classical Banach spaces, Vol. I, Sequence Spaces.

[7] -, -, Classical Banach spaces, Vol. 11, Function Spaces. Springer, Berlin-Heidelberg-

[8] C. MCCARTHY, cp , Israel J. of Math. 6 (1967) 249-271 [9] C. MERUCCI, Interpolation dens C"(H), Compt. Rend. Acad. Sci. Paris 274 (1972) A 1163

Math. 666, Berlin-Heidelberg-New York 1978

berg-New York 1976

(1975) 395-412

Springer, Berlin-Heidelberg-New York 1977

New York 1979

to A 1166 [lo] A. PIETSCH, Operators Ideals. North-Holland, Amsterdam-New York-Oxford 1980 [ll] N. TOMCZAK-JAEQERMANN, The moduli of smoothness and convexity and the Rademacher

[ 121 H. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators. North-Holland, averages of trace classes Np(l 5 p < ao), Studia Math. 60 (1974) 163-182

Amsterdam- New York- Oxford 1978

Univeraidad de Santiago de Compoatela Facdtad de Matemciticas Departamento de Teoria de Funciones Santiago de Compoatela Spain

Current address: UniveraidacE Autdnoma de Madrid Fffiultad de Ciencias Divisidn de MathemrUicas Madrid (34). Spain