Preliminaries. On a non-commutative analogue of the Hôlder inequality

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVI (1986) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXVI (1986)

Leszek Jan Ciach (Kielce)

On a non-commutative analogue o f the Hôlder inequality Abstract. Given three measure spaces ( X h ft), i = 1, 2, 3 and let L p( X h f t ) be the usual Lebesgue spaces. Let P be a product operator (see [7 ] and Definition 2.1). In [7 ] (Theorem 3.5) R. O ’N eil proved that

IIW^IIi^H/Ult/ILp

where l/ p + l/ q = 1, f e U ( X u f t ) , д е Ь Ц Х 2, f t ) .

In Theorem 2.2 we deduce an analogous inequality for the spaces £^"(31) associated with a semifinite von Neumann algebra Л equipped with a regular gage m.

1. Preliminaries. Throughout, ^ is a von Neumann algebra on a Hilbert space Hh rrii is a regular gage on % . The triple (Я,-, 31,-, w,) is termed a regular gage space (see [9]).

Definition 1.1 (see [8], [11]). For any closed, densely defined operator аеЗД,-, we determine:

П\а\(Л) = Щ( ^) , Л^ О (the m,-distribution of a) and

am (a) = a (a) = inf {X e [0, oo]: т{(е£) ^ a, a > 0}

00

(the rearrangement of a) where a* a = \a\ = J Xdek is the spectral о

decomposition of \a\.

Definition 1.2 (cf. [6], [11]). The *-algebra Q{%, m,) of m,-measurable operators is defined by

W = £mi = a(oc) < oo, a > 0}.

Definition 1.3 (cf. [10], [6], [11]). A sequence of m,-measurable operators {a„} is said to be mr convergent (convergent in measure wf) to a mr measurable operator a (a„^*a) if and only if (a — a„)(a)->0 as n-> oo for any a > 0.

The following proposition is of great practical importance.

Proposition 1.1 (see [1]). I f а (а, апе 2 т.{%), n = 1,2,...), then a„(a) T+ a(ix) at each point of continuity of the function a (a).

6 L. J. Ci ас h

The non-commutative Lorentz space is the collection of all mr measurable operators a such that \\a\\ôa < oo, where (cf. [4], [1])

J I (&110 am1(a))<T“ | > 0 < Ô, a < 0o, IM L =

I (

sup \a1/ôam. (a): a > 0}, 0 < <5 ^ oo, <x = oo.

is seen to be a quasi-Banach space and in some cases a Banach space

(cf. [4], [11], [2]). Note that (the non-commutative

Lebesgue space introduced in [9], [11])

2дт.{Ц ) = {ae2m.: \\a\\0 = {] aôm.{a)da}l/ô < oo}.

t)<5<Tl cz 2°П2 for ^ tr2.

We define an analogue of am.(a), a e 2 m.

For 0 < (j1^ ( T 2 ^go we have (cf. [4]) ||а||^2 ^ IM I^ . Hence,

ft

Q(и) = Qmi (p) = ^ J aM. (a) da, ц > 0.

It is clear that ^ am.(//), A* > 0 and Qm.(fi) < oo for ц > 0 if and only if ае'Ц. + й ^ ) . '

The following equality is obvious geometrically 00

Wifi) = 1Ш(Р)+ f *1\а\(Л)<М.

а (ц )

Proposition 1.2 (cf. [4], [2]). For any a e % F 2 l l. and \ < ô ^ c o , 1 ^ O’ ^ oo

Halite < ~ Halite < x^T Halite _{d — 1} (~ ~ r = 1),_{oo — l}

where

• lla||6(T

о

0 < <5 < 00, 0 < <7 < 00,

0 < f) ^ o o , er = go .

sup [a1/(5g (a): a > 0},

For any a, fteSIf + fii.: (a + b)(/i) < g(/i) + M^) (cf. [8], [12], [2]). In consequence, £^(91,) are Banach spaces for Ô > 1, a ^ 1 with norms <5(7 *

Non-commutative analogue of the Holder inequality 1 Note that i^.(9l,) = ^т,(чД) are Banach spaces with norms ||-||й (see [9], [5], [11]) for <5^1.

The set of all orthogonal projections from % is denoted by Proj(9l,).

2. Product operators.

Definition 2.1 (cf. [7], Definition 3.2). Given three regular gage spaces (Hh 91,, m,), i = 1, 2, 3. A bilinear operator P which maps measurable operators from iimi(9li) and (9l2) into measurable operators from i?m3(9l3) is called a product operator if

(i) ||P(a, b)|| ^ N I P H , (ii) ||P(a, Ь)\и ^ INK IN , (iii) Н^(я, bill! ^ INHIbll!.

Lemma 2.1. I f P is a product operator and

c = P(a,b), ae L*mi (9li), b e £ m2(9l2), where ap1 = 0, mx (p) = Ô, \a\ ^ (11, then for a > 0

(i) ac(a) ^ /Ш>(<5), (ii) c(oe) ^ $&(a).

Proof. Let p > 0 and let Ьц = и(\Ь\е^ + pef), W = b — Ьц = u(\b\ ef — oo

— pe^), where b = u\b\ (the polar decomposition of b), \b\ = j Mek (the

о

spectral decomposition of |b|),

c = P(a, b) = P(a, b^ + Pia, bA) = cx + c2.

It is easy to see that |1У*| — (\b\ — p)e^ and

^ m2(ei+n) = Ч\ъ\(Л + Р)- In consequence

l|c,ll « Ml IIMI « fin, Ile,II, « N I, IIMI ^ fiSn,

IN I, < I W I M , < ^ b in w x . Let further p = b(0). Then

a a

a£(a) ^ a £ !(a ) + a£2(a) = j c l (y)dy + $c2(y)dy ^ \\cjh T l k ^

0 0

ao

« fiôb(»)+fi j qw o ) d x = pôbtS).

««>

L. J. C ia c h

To prove the second inequality of our lemma, we let p = b{a). We have

a a

ссф) < atQi {<*) + <*£2(а) ^ $ c l {y)dy + J c2(y)dy ^ « fl^ ill + l k 2|li

о

0

^ <xpb{ix) + p J щь\ (X)dX = (a)

b( a)

and (ii) follows, by dividing by a.

Lemma 2.2 (basic lemma on product operators). I f P is a product operator and c = P{a, b), where either Ъф s-?l2 + ^ 2 or be ЗД2 + £т2, ae ^ or be ЧЛ2, ae then for any p > 0

00

j ml (eï)k(mi(eï))dÀ + pk(p)a{p),

a(n)

00 where |a| — J Xdex.

о

Proof. Fix p > 0 . Consider a doubly infinite sequence {^n} ^ _ x such that

X0 = a(p), Xn^ X n+1, lim Xn = oo asn -*oo, lim Xn = 0 as n -> — oo.

CO

Let ип = м[(Яп- Я п_ 1) ^ и + (|а|-Яи_ 1)е(Яи_ 1дп>], where a = u\a\, \a\ = { АЖ?Я

о

and c(n) = P(a„, b). It is clear that

Fromm Lemma 2.1 (i) we obtain for n = 1, 2, ...

Wl(n) (ji) ^ (X„ - A„_ J my (ein_ t) k К (с|я_ t))

and by the second inequality of Lemma 2.1 we obtain for n = 0, —1, —2, - 3 , . . .

PQ{n){p) ^ { K - K - X ) h i v ) - p - Furthermore,

c = P(a, b) = P(u{\a\-X0)efQ, b)+P(u{\a\eXo +X0ej;0), b) = cx+ c 2,

/ о

Let a(,) = £ a„ and a<~° = Z fln, /= 0, 1, 2, ... It is easy to see that

n= 1 n= — l

a(l)eXl = a(l)e{x0tXl> = и (М -А 0)<?А0е(Яо1Я|> = u(|a| - А 0)<?|0еЯ1,

Non-commutative analogue of the Holder inequality ⁹

and so

u(\a\-À0) e iQ- a il) = (u(\a\-X0)ei Q- a {l))e^ = u(\a\ - l t)e£r

In consequence

Cl = P(u(\a\-^0)ej-0, b) = P(a('\ b) + P(u(\a\-Я ,)е ^ ^b) = c f + c$, P£i (p) ^ P£(i )(p) + p d )(p)

< E (^ n -^ «-i)mi ( eAn_ 1)è (w i(e in_ 1))4 -^ <i )(^).

w — 1

If a e i? ir ЬеШ2, then ы(|я| — Àt)e2l -> 0 in 2щ and

\\P(u(\a\-Ài)eir Ь)||, ^ ||м(|а|-Я|)^||1 -||Ь|| -► О as /-юо.

Moreover,

/^£4

4

1

0

00

о

Assume that ае . It is clear that an = 0 for nsufficiently large and ail)

= u(\a\ — Я0)е|0 for / sufficiently large. Finally ас

n= 1

for a e fiij, ЬеЗД2 or ee$ Ii, Ь е 2 т2(Ш2).

The series on the right is an infinite Riemann sum tending (with proper choice of A„) to the integral

J mi(ei)k(mi(ei))dA.

a(fi) Therefore,

ao

P£i (p) < f Щ ( e f)h { щ (ei)) d l. a(u)

We use the second inequality of Lemma 2.1 to evaluate ç2(fi). For any Zeei0(H):

é l)Ç = (Я0 ^{— Я_,_} j) ^ and for any Çeelx_ k_ ltX_k>{H), 0 < к ^ I: é l) Ç

~ и{\а\ £ —A_j_! £), and for any Çeex_ l_ l (H): a(~l)Ç = 0. Hence c2 = Р(и{\а\еЛо + Л0е£0), b)

= P(a(~l), Ь) + Р(м(А_1_ 1ея_ |_ 1 + |а|-ед_1_ 1), b) = 4"'> + 4~ °

10 L. J. C i a c h

and from Lemma 2.1 (ii) о

№s~l) < p Z ( ^ , - V i)Mp) = fé(n)№о—я_,_ x).

n = - l

Let further b e s2l2(L^2). Then, ||c(6~I)||(||c(6~,)||1) -*• 0 as /->oo. Therefore, for any p > 0, fle fimi, b e £ m2

ААЫр) < A4&(A*Mo = А*я(А*)£(а*)- Finally

AA£ (A*) ^ PQl (A*) + P£i (p) < J "*! (c/) k (mi (^я1)) ^ + pa M b (//)

a ( u )

for either fle fiij, h e s2I2 or a e ^ , b e9 l2 + £ i2 or b<£s2l2 + £™2.

Corollary 2.1 (cf. [11], Theorem 3.3). If P is a product operator, c = P(a, b), where either a e sU{, b e sll2 + £m2 or be*&2 or а ф % + or Ь ф % + ^т2, then

/ic(p) ^ j’ a (a) b (a) da.

0

Proof. If 0 Ф a$ or 0 # b<£ $I2 + £^2, then ja(a)b(a)da =

00

0

A*

If a = 0 or b = 0, then c(p) = 0 and f a (a) b (a) dot = 0, p > 0. Assume that

0

oe^Ij, b g ЭД2 + P»w2 or яеР,^, be$l2. We have

OO fl

j mx (el)b(ml (ef))dÀ — (/ = a {et)) = — J ab (a) da (a)

а(ц) 0

= —ctb(oc)a(oc)\lo+ ja(a)b(a)da

0 V

= — pb (p) a(p)+ j a (a) b (a) da.

0

It is now clear that

PQ (p) ^ j a (a) b (a) da.

0

Remark 2.1. Assume that an”^L+a implies c„ = P(a„, b)^->P(a, b) and

Non-commutative analogue of the Hôlder inequality ^И

let further an = aene \a\ = j Mek. From Proposition 1.1 we obtain о

и Д

цс (ц) = jc{ot)dot ^ lim inf j c„ (a) dot = lim inf p£n (p)

о » o »

n и

^ lim inf f a„ (a) b (a) dot = fa (a) b (a) dot

n o b

(since a„(a)|a(a), a > 0) for any a e £ mi, bei?m2.

Theorem 2.1 (cf. [7], Theorem 3.3). A bilinear operator P is a product operator if and only if for c = P(a, b), where either а ф ^ - У or Ьф Ш2 + + i?m2 or а е Л 15 be W2 + 21m2 or a e 2 lmr />ечЛ2

и

PQ(p) ^ Ja(a)b(a)da.

о

Proof. The necessity of the condition follows from Corollary 2.1. To prove the sufficiency we may assume, without loss of generality, that ae ЧЛ 1?

beW2 + 2 i 2 or Ь е Л 2. We have

и

1И1 = lim q(p) ^ lim sup- \ a{ot)b{ot)dot ^ а(0 + )Ь(0 + ) = ||а|| ||Ь||,

ft ~*0 fi -*o p J 0

IkHi = lim pq(p) ^ lim Ja(a)b(a)da ^ ||a|| \\b\h (IMIj ||b||).

f i- * oc fi-*oo о

Corollary 2.2 A bilinear operator P is a product operator if and only if (i) ||Р(а,*)|К ||a|| ||*||,

(ii) ||P(a, b) Hi ^ j a (a)/? (a) dot,

о

where either + or Ь ф ^ + Я ^ or a e % , + or ae Ь е Л 2.

Theorem 2.2 (Holder’s inequality, cf. [7], Theorem 3.5, [3], [5], [11]). I f P is a product operator: c = P(a,b), where either ае^1ь Ье*&2 + 2n2 or ae*?^, he Л 2, and if b e fij^ 2, where l/ô1 + l/ô2 = 1, 1/ctj -h l/cr2

^ 1, then ceQl,3 and

(i) Hclli ^ max{ÔJau 02/а2} \\a\\ôl(fl \\b\\02<,2, ôh Ф oo, i = 1, 2. In the particular case: <Sj = <xl5 b2 = <r2

(kill < ||a||a i||

6

(H) Iklli ^ l|a||^aollh|b2ff2, <?1 = oo, (üi) IkHi ^ M ôl<Jl |№ 2<30, a2 = go.

12 L. J. C ia c h

Proof, (i) We have

lldlj = lim Jc(a)da< J a(a)b(a)(/a = j (а1/г1/а1/У1)а(а)(а1/Й2/а1/У2)Ь(а)^а

ц-*ао 0 0 0

^ { j а71Л51 a7l(a)(l/a)(/a}l/71 { J a72/<*2 b72(a)(l/a)(/a}1/72

о о

= (<51/y 1) 1/71( ^ / y 2)1/ï2l! d l, l ï l llb |b 2V2

^ max { à j a u Ô2/a2} ||а|Ь1«т1 ЦЬ|1а2«,2 where

1

1

2

1

0

1

2

(ii) If а, — oo, then

IWIi < ] a 1*1 a(cc){ot1/Ô2/(x)b{ot)d(x ^ \\a\\ôiao\\b\\Ô2l < IMIajoJbll^j- о

The case a2 — oo follows analogously.

Theorem 2.3 (cf. [7], Theorem 3.4). I f P is a product operator, c

= P(a, b), where either йе91ь ЬеШ2 + 2„2 or b ç$ l2, and if ае2щ l 9 b e 2 ôJ ° 2, where l/ôt + l/ô2 = 1/S < 1 and a^ 1 is any number such that I/ctj H- 1/<t2 ^ I/o-, r/ien cefij," aw/

Ikllto ^ 00, ^(7, Sx, G !, <52, I № 2„2 Moreover,

0 0, <7, ^{S i ,} (7i> <52, <72) —

(i) <5/0- l)0/<5)1/ff max { { b j o t f 1*, 0 2/<*2)1/,T}; G, G i , G 2 ф 00, (ii) <5/0- l)0l/<5)1/<T; (7, (7i ^ QO, G 2= 00,

(iii) <5/0- 1)020 ) 1/<T; (7, (72 # 00, (Tj = ⁰⁰,

(iv) (1M ) 1" i(a A 6 - l))1,' i (52/<T,)1/*i ; G, G 1 := 00, 1 < g2 <: oo, 1/<т2 + l/<72 = 1,

(v) (1M ) ^{G, G 2}= 00,

,1 < G1 < 00, 1/(7! + 1/(7; = 1, (vi) <52; <7, (71 = 00, (72 = 1,

(vii) <5i; ^G, G 2 = OO, G J — 1, (viii) <5/0- 1); (7, (7j, (72 = 00.

Proof, (i) Let us suppose that 1/yi + l h i = 1, l /у,- < g/g{, i = 1, 2. We

Non-commutative analogue of the Hô'lder inequality 13

have

00 00

IklL « J = |^ J

U«

(by Hardy’s inequality, see e.g. [4])

<S/(S-1) aajô 1 ao ba ba ^ fa 1/<T

= <$/(<$ —1)^ — J (aa,èl l & lyi) aa (tx)(&102/oilly2)ba {<x) dot}^

0

(by Holder’s inequality)

(a 1/02b{a))ay2- } J„')1,a'n f r d^ 1,<T72 1/^2 j d(X I i l / l Mi . . . \ffvi dCCI

a )

= S/(S - I M S ) 1" ( M l °) /na(S2/y2 2ff • ||a||,l(yi<r) ||Ь||,2(У2<Т)

^ 0/(6- 1 ) ( ф ) 1/а max { ( Ô M 1" , (S2/<t2) 1'*} -HelU^j \\Ь\\д2вг

'ji/ff

(ii) IkIL ^ <7 \ i f 1* xfL(n)dn\ = < 7 (fiQ(n))aца1д ff Ыц u<*

^ 3/(3 — 1) ^ j* <xal0 1 aa (a) ba (a) da j (by Hardy’s inequality) 0

00

= c5/(<5-1)|| j" ( a l>l a(a))<,(a1M26 ((i))'^ | I/o

< а д й -1 )(а 1/а)1"||в||41. 1цыи2„ .

(iii) The proof of case (iii) is identical with that of case (ii).

14 L. J. C ia c h

(iv) \\c\\ôaD ^ sup {nilôc{^i): fi > 0} ^ sup {ц1/д 1 ] a(ct)b(a)doi: ц > 0}

IMbiaoSup W * *Ja 1/01 b (a) da: ц > 0}

= Ilell^ooSup 1 * Ja 1,0 + 1/02 (oc1102/a1 !°2) b (a) dct ]

A /ô- 1

1/^2 J,x.S U p <^

« (ô/(ô - i))," i (i/<7i

)1"'2

^ ) 1'"2

а »2/г2 ^ « 2 ( а ) ^ { l/ff2

(v) The proof of case (v) is the same as that of case (iv).

(vi) IklL* ^ sup {n1/ôc{fi): ju > 0} ^ sup j/x1/<5 1 J a(oc)b(ct)dGi: fj. > 0

о fi

< ll^lUioosup|/4i;^-

1

1

1

1

52

2

о

(vii) The proof of case (vii) is the same as that of case (vi).

(viii) ||с||Лоо ^ sup{//1/,5c(^): // > 0} < sup {ц1,6~1 j a (a) b (a) doc: // > 0}

о

= sup \nl/0~1 j(l/a1/<5)a 1/<51 a{ct)oi1102 b{oc)doi.: j i > 0}

о

^S/(ô-l)\\a\\ôiaD\\b\\Ô2ao

for ô < oo and S = oo,

Hell.. = IHI = sup {с(ц): fi > 0} < sup {l/n ja(a)b(a)da} ^ ||a||||b||.

о

Remark 2.2. Assume that аи^->а implies cn = P(an, b)’^-*P{a, b). It follows from Remark 2.1 that a e f ij, 1^ 1, b e f i ^ 2 т а У be assumed in place of either а е чЯь b e s2l2 + £,L2 or b e sH2 in Theorems 2.2 and 2.3.

Making use of Theorem 3 in [5] and Theorem 2.3 we easily obtain the following theorem:

Non-commutative analogue of the Holder inequality 15

Th e o r e m 2.4. Let 1 < ôx, b2,

04

2

00

1

1

(1

1/У2 = l/52 + 1/^2 < 1, I/o- = (1 — 0) M + 0/<r2, 1/У = (l-0)/7i +

0

2

Then

l|P(a, *)||r « (п/(У1 - 1))’-°(Уг/(У2- DflWUlblU-

3. Concrete examples. Let Ah i — 1, 2, be any two linear operators from into £m3(2l3). Let further

НЛИ1 ^ |M|, a e % , i = 1, 2, IH ;(a)||i ^ INIi, a efii., * = 1,2.

Ex a m p l e 3.1. H x = H2 = H3, mx = m2 = m3, = 9l2 = 9l3.

(i) P(a, b) = A x(a) A 2(b) or

(ii) P(a, b) = i [ A l {a)A2{b) + A2(b)Al {a)].

Ex a m p l e 3.2. H L — H2, = ЧЛ2, m1 — m2. Let A x be a gage-preserving

^-isomorphism from i?,ni(9li) into £W3(9l3). We define (i) P(a, b) = A x (ab) or

(ii) P(a, b) = ^ [ A X (ab) + A x (baj].

Ex a m p l e 3.3. Suppose that = 1, i = 1, 2, and let (H3, 9I3, m3) be the product gage space (see [9 ], [10]) of ( Hx, 9I

l5

(i) P(a, b) = a®b (the tensor product) or (ii) P(a, b) = A x (a)®A2(b),

where A-t are any two linear operators from £m.(9I,) into £mi.(9l,), * = 1,2, and

И Л И I ^ INI» ae%, i = l,2, IH iH li < M h ’ ûeÛi,, * = 2- It is clear that

II_P(a, b)||= INI ПЫ1-

From Corollary 8.3 in [10] it follows that

\\P(a, 0)1 Ii = l|a®0|li = IMIx ||0|li ^ INI ||0||i (IN Ii l|0||)•

16 L. J. Ci ас h

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