ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVI (1986)
Leszek Jan Ciach (Lôdz)
Some remarks on the non-commutative Lorentz spaces
Abstract. The non-commutative Lebesgue spaces and the non-commutative Lorentz ones of unbounded operators associated with a von Neumann algebra (equipped with a faithful normal semifinite trace or weight) are studied by many authors: [2], [5], [6], [8], [9] [10], [12], [13], [16]—[19], [21]. Here, at first, we shall show that Haagerup’s Lf-space l f ( M) (0 < <5 < 1) is a quasi-Banach space, that is, a complete quasi-normed space. Then, continuous linear functionals on the non-commutative Lorentz spaces are discussed. In the last section, convolution operators are introduced.
1. Preliminaries. Throughout, sé is a semifinite von Neumann algebra on a Hilbert space Ж, {Ж, ,5/ , m) is the regular gage space in the sense of [18]. <£m{sé) = = Ж is the *-algebra of m-measurable operators (see [2], [10]) and f f m(Ж) = Жт — У* — {аеЖ'т(sé): m(ef) < 00, e > 0}, where
_________ 00
yfa*a = \a\ = J Mek is the spectral resolution of \a\, У m is the *-subalgebra of sem.
For any azS £ m{sé) we define the rearrangement of a by am(oc) = a(a) = inf{A > 0: m(ef) ^ a, a > 0},
00
where \a\ = j Xdex.
0
For basic results on the rearrangements of operators we refer the reader to [12], [21].
The functional: Qm: [0, 00) (see [2]);
gm(a, b) = Qm(a — b) = inf (e > 0: m(e£х) ^ e} = inf {e > 0: (a — b)(e) < e},
00
where \a — b\ = J Xdek, is a metric in 5£, <£m{,sé) in this metric is a Fréchet space. The convergence in this metric is called convergence in 0
measure m (m-convergence, in symbols an^> a). The following simple fact is of great practical value:
Proposition 1.1 ([2]). I f a„ ^ a (a, a„e then an(a) a(a) at each point of continuity o f the function a (a).
202 L. J. C iach
Recall that an operator a e !£ m(sé) belongs to SBbm{sé) = S£bm — <£b (the non-commutative Lebesgue space) if
5£b are quasi-Banach spaces for 0 < Ô < 1 (see [2], [17]) and Banach spaces for <5^1 ([9], [18], [21]) with quasi-norms (norms) ||*||л.
The lattice of orthogonal projections from is denoted by Proj(«s/).
Two projections p, qe P ro j(^ ) are called equivalent (in symbols q) when there exists some uejrf such that u*u = p, uu* = q; p < q means that p is equivalent to some subprojection of q belonging to sé .
For an arbitrary operator a e & m{s/): supp a (support of a) stands for the smallest projection p e s / such that pa = a. It is not hard to demonstrate that
Definition 1.1. An operator aestf is said to be m-elementary if m (supp|a|)< oo. The ideal of m-elementary operators is denoted by one of the three symbols 3Fm (s/) =
One can show that m-elementary operators lie densely in 3?bm№ \ 0 < <5
< oo (see [2], Corollary 4.2).
We denote by ,JiXoc the *-algebra of locally measurable operators in the sense of [20].
2. Topological properties. The non-commutative Lorentz space
(G. G. Lorentz) is a collection of all aeS£m{sé) such that ||a||m>(5<T = ||a||i<T
< oo, where
As in the commutative case, it is not hard to show (see [7], p. 254) that ||a||*»2 < IMbffj, ax ^ о2, ae5£m. We may assume that, in the definition of IMIsa, the integration of the function aff(a) is performed with respect to the measure da0,0. Thus without any essential changes one can carry a number of properties of the spaces Sâ6 over to the case of the spaces ^ ь^(л/). And so,
|| ’\\0<t is a quasi-norm on ££ba, in which 5£be is a quasi-Banach space (cf. [2]).
The case a — oo needs separate discussion. For <5 = oo, J27®00 =«я/, IHLao
= ||-||. So, assume that 0 < ô < oo. From ( ii) of Proposition 2.4, [21] it follows at once that S£b™ is a quasi-normed space with quasi-norm || ||Ла0.
Later, we shall prove that 5£ь™ is complete.
00
IMkm = IMU = {jV (a)d a} 1/a < oo, 0 < <5 < oo.
о
supp a ~ supp a* ~ supp |я*| ~ supp |a|.
.sup(а1/йа(а): a > 0}, 0 < < 5 < o o , cr = oo.
Proposition 2.1. The topology o f m-convergence on !£ b^ (s/) is weaker than the topology of |HU*> on & 0т № )-
P ro o f. The case <5 = 90 is self-evident (cf. [10], [19]). So assume that 0 < ô < 00. Let ||а||йа0 ^ £(1/<5+1). Then в1/йа(в) ^ ||а||3а0 ^ that is, a (e )^ £ . In other words, p (a )^ £ . Consequently, \\a — а„||гао -> 0 implies a„-* a.m
Proposition 2.2. Let us fix 0 < a < 5 < 00. For any ae
IMIto ^ s u p ^ a 1' 1“ a 'W ^ Y " : «><)}•< (ЗДЙ-<»))1"||а||*».
P ro o f. The first inequality follows immediately, since a(fi) is non
increasing. We prove the second inequality.
J V o S ) ^ ) ' = a llô- ilo^ P alôaa{P)p-alôdP^ '
\1/<T
« « ‘"-«"IWIto /Г*/4<Ш = MM<V0i - a ) Y " .
0
Making use of Propositions 2.1 and 2.2, we can give a simple proof of the completeness of
Theorem 2.1. is a quasi-Banach space.
P ro o f. Assume that {an} is a Cauchy sequence in By Proposition 2.1, an^> a for some ae£Pm(st). Hence and from Proposition 1.1, (ak — an)(a)-j+ (a — a„)(a) nearly everywhere on (0, 00) with respect to the Lebesgue measure. This being so,
а1/*(а ^ limfcinf ^(ак-а„)а{Р)йр^
о о
< (ô/iô-o))11” lim inS\\ak — a„\\ô(X).
к In consequence,
l|e~flJI«oo < (à/{à-<r)fla lim inf||ak —aJU*.
к
Finally, a = (a — an) + ane 5£b™ and ||a —aH\\ô<x> -* 0, which ends the proof of the theorem.
An important application of the above theorem is contained in the following
204 L. J. C iach
Theorem 2.2. Haagerup's I?-space L?(M) is a quasi-Banach space, 0 < Ô
< 1.
P ro o f. We use the notation and terminology of [5]. The non- commutative Ü(M)~spaces, l^ < 5 < o o , associated with an arbitrary von Neumann algebra M, are Banach spaces [5], [6]. We shall show that 0 < ô < 1, is a quasi-Banach space. Indeed, for any a eL d(M): ae££ r°°(M o) and \\a\\ô = trflalV ^ = \\a\\ôaD, since 0S(M*) = \0s(a)\ô = |exp(-s/(5)a|^
= exp( —s)|a|5 and az(a.) = l/a 1/<5 t r ()<з)5)1/й (cf. [5]). Assume that {an}
is a Cauchy sequence in L?(M) (0 < ô < oo !). By Theorem 2.1, an-+a in
<eb? (M0) for some ae (M0). Moreover, anЛ a (Proposition 2.1). In con
sequence, exp( —s/<5)a„ = 0s(an)-^> в8(а) = exp{~s/ô)a and a is т-measurable.
Finally, a e lf(M ) and \\a -a n\\0 = \\а -а п\\0ж -> 0.
R em ark 2.1. Note that by the equality: ar(oc) = \/a}lô\\a\\ô we have gz(a)
— IMIa/(1 + <5)> that is, for a, aneÜ {M )\ a„-^ a if and only if a„~* a in LÔ(M) (cf. [8], Theorem 3.2).
Suppose that a eL â(M), ô > 1, and let for peProj(A /0) , f*(p) if т ( р К 1,
u if T ( p ) > 1.
l
Then ||fl||ltf = Jat (a)da = \\а\\л-д1{0 — 1). Hence I?{M) c ^ f|(M 0) and ||*||j is о
a norm (cf. [2], Proposition 6.6 and Lemma 3.2).
Similarly as in the commutative case (see e.g. [7]), the quasi-norm Ц-Ц*, can be replaced by equivalent quasi-norms and, in special cases, by a norm (see Corollary 2.1).
Definition 2.1. For fixed 6, a, ae <£m and for у such that 0 < у < со, y ^ a, y < ô, we define
âOT(a) = а (ос) = à(a, y)
_ [sup {m(p)~l,y\\pa\\y>m: a < m{p) < со}, 0 < a
" ( O , a > m(l).
In the above definition we assume that the supremum over an empty set is equal to zero.
Definition 2.2. For any a e £ P m№ ) and 0 < y ^ 1, we define
Д«(ос, У) = <?(<*, у) = а(а) = j~ , a > 0.
о
Note that gm(a, y) < oo, a > 0 if and only if a e s / + £?!„(<$/).
Proposition 2.3. (i) a( a) < a (a), 5(a) ^ a (a), a > 0.
(ii) a (a) ^ 5(a) ^ a (a), a > 0, ae (,c/).
P roof, (i) a{a) is a non-increasing function, hence, the first of the inequalities follows at once. To prove the second inequality, let us note that
) i/v
т(р)~11у\\ра\\у = m (p )_ 1 /7 <| | ^ m{p)~lly
о
m(p)
a’ W dffj
1/V
(ii) It suffices to prove that, in this case, a (a) ^ 5(a). If a {a) = a(oo), then a(a) = 0 ^ 5 ( a ) on account of the fact that а e ■9?m(,</). Let us notice that, for any а е & т{<$#), a (a) > a (со), т(е£а] + еШ)]) = m(lim e£a)- l/n), where
П
GO
|a| = f Mex. If a(oo) = lim a (a) = sup {A > 0: m(ef) = oo} < a (a), then,
0 a-* *
for n sufficiently large, m(e£a)- l/n) < oo and, in consequence, oo > m(lim е£а)- 1/я) = lim т(е£а)- 1/я) ^ a.
n n
00
In that case, if a = u\a\, \a*\ = J U fx, then m(e£a) + e[a(0l)}) = т ( / ^ а)+ /{в(я)}) ^ а
о
and, in consequence,
5(a) ^ m{fafa) + /{e(a)}) 1/7 \\(faU +/{«<«)}) a\\y
> -Ь/{«(а)>)_ l/v ||a (a) M* (/^ ) 4-/{e(a)}) w||y = a (a).
De f i n i t i o n 2.3. For a n y a e we define 00
11^11<5<г,т a da
I j 5w(a)a<T/<5_15 a | ' ,
о
sup {a1/<55(a): a > 0},
0 < <5 < oo, 0 < <7 < 00, 0 < <5 ^ oo, c — oo,
and
) I / o
ccalô~ i gff(a) dix > ,
sup {a1/<5a(a): a > 0},
0 < <5 < oo, 0 < c r < o o , 0 < <5 ^ oo, cr = oo.
206 L. J. C iach
Proposition 2.4. For any 1 < <5 < oo, 1 < <r < oo, M L < INI* < ~IML < (à/{à-y)yh \\a\\ôa, ae
The proof of Proposition 2.4 follows easily from Proposition 2.3 and Hardy’s inequality (see [7]).
Corollary 2.1. JFô£{<ç/), 1 < <5 ^ oo, 1 ^ <x < oo, are Banach spaces.
P ro o f. For fixed 1 < ô < oo, 1 ^ о < oo, we have FFbZ{sf) <=■ У*w(-<&) and (a + b)(txy 1) ^ а (а, l) + £(a, 1). Consequently, II ' WôG is a norm.
Analogously for _|| -Ц*, (see Lemma 3.2). In order to conclude the proof, it is enough to use Proposition 2.4.
To close this section, we shall consider continuous functionals on Lorentz spaces.
Proposition 2.5. Let pe Proj(,c/), m(p) < oo, hne J*m(stf) and \bn\ ^ p, m(|bJ) = ll&Jli->0. Then | | h J L - 0.
P ro o f. C ase I. 0 < о < oo:
lim|\bn\\ô<T = lim{ J b°(a.)docff/ô}ila = {f lim ban{<x)d<xa,â}1/a = 0,
n n 0 0
since 6„(a)->0 almost everywhere on (0, oo) and b„(oc) ^ x(0,w(p)>(a).
C ase II. <T = oo, 0 < <5 < oo:
• Fix e > 0 and 0 < y < ô. It is evident that
'*<o,m<p)>(a), 0 < c t^ m(p) ,
bn(x, У) < l/a 1/y |jb„||y, a > 0, and
sup {all0bn(a, у): а > 0] < max [sup Ja1/<5 bn(a, y): 0 < a < eô < m(p)};
sup {oL1/ôbn{a, y): a > e*5}} < max {e, 0Lllâ~lly\\bn\\y: а > e*5}
< max [e, е(у_г)/уй||6и||у} ^ e for n ^ Ne.
From the arbitrariness of £ > 0 and Proposition 2.4 follows the proposition.
By making use of Propositions 2.1 and 2.5 and Theorem 4.3, the proof of Theorem 4.4 in [21], it is not hard to prove the following fact.
Proposition 2.6. Let h be a continuous functional on {sd). Then h(b)
= m(ba), be3*m{s/) (the closure in whereas a e J ? ]oc, 0 < <5 < oo.
Theorem 2.3. If.srf has no minimal, non-zero projections, then any linear, continuous functional on vanishes on the closure of ,#'m (s/) in
for 0 < <5 < 1.
The proof of this theorem is based on the following
Lemma 2.1 ([2], Lemma 6.3). For 0 ^ a es0 + and 0 ^ к ^ m(l), the equality
я
j’fl(a)c?a = sup(m(ap): m(p) = 2, pe Proj (,*/)}
о
holds, if does not have any non-zero minimal projections.
P ro o f of T h e o re m 2.3. Assume that h is a continuous, linear functional on 0 < <5 < 1. In virtue of Proposition 2.6, h(b) = m(ba), b e , ^ m, where ae,90 + JFlm (see [21]). What is more, for any p e P ro j(^ ), m(p) < oo:
l/m(p)1/<5-m(p|a|) = l/m(p)1/<5 • m(pu* a) = h(pu*/m(p)l/ô) ^ \\h\\ < oo, where a — u\a\ is the polar decomposition of a. From this and Lemma 2.1 we obtain
||/z|| ^ l/m(p)llô-m(p\a\) = l/m(p)l/ô m(\a\ p)
and
\\h\\^ su p \l/X 1/ôm(\a\p): peProj(j^), m(p) = Я}
я
= \/Àl/ô§a((x)doL ^ X/X110 a(<x0), X ^ a 0.
о
From the arbitrariness of a0 and from Xl ~ i,0-> oo as 2 -» 0 it follows that а(а) = 0, а > 0, that is a = 0, and thus h(b) = m(ba) = 0, b e ^ m{sé).
Corollary 2.2. The dual space of JFÔ£(.90) is trivial for 0 < Ô < 1, 0 < cr
< oo if ,90 has no minimal, non-zero projections.
P ro o f. By :¥J,90) = 0< < 5< 1, 0 < <r < oo.
Problem. ££*£* — [0} if s0 has no, minimal non-zero projections, 0 < b
< 1?
Proposition 2.7.
— {ae {s0\. lim oti,ôa(ct) = 0 as ot-*0 or а -> oo}, where the closure is taken in (s0), 0 < Ô < со.
00
P ro o f. Let lim а 1/<5а(а) = 0 and an = a(e„ — e1/n), |a| = J XdeÀ. Fix
«-►O.oo 0
e > 0. Suppose that ос1/да(а) ^ e/(3*21/<5) for 0 < а ^ у and а 1/<5а(а) ^ e/(3• 21/й) for a ^ 1/y. For any и = 1 ,2 ,..., a — an = ae^ + ael/n, ane ¥ m. Hence
ot1,s(a-a„)(oL) ^ ocl/0(ae^)(ot/2) + cc1/6(aelf„)(ot/2)
= 21/ô(<x/2)l/ôa(ix/2)X(o,2m(e^yF 2l/ô{ot./2)110(aelln){ot/2)
< £/3 + 21/ô(;x/2)llô(l/n)x(0>2l7) + 2llô(<x/2)llôa(oc/2)x(2iy,ao) ^ e for n ^ (3/e)(2/y)1/<5, m(ei) ^ y.
208 L. J. Ci ас h
From the arbitrariness of e > 0 it follows that ae<Fm{,c/). Conversely, let a e $ 'm and \\a — an\\000 -* 0 as n-+ oo, where a „ e ^ m. Fix £ > 0 and let Ilfl-aJU® < e/21/ô, « ^ Ne. We have
a 1/0 a (a) ^ a 1/5 (a — a„)(a/2) + a 1/<5a„(a/2)
= 21 /<5 [(a/2)1 /<5 (a - a„) (a/2) 4- (a/2)1 / л a„ (a/2), and, in consequence,
lim a1,ôa(oc) < s + 2i,s lim (a/2)1/ôaNe{x/2) = £.
a - >0,o o a - + 0 , a o
From the arbitrariness of £ > 0 the proposition follows.
If h is a continuous linear functional on £? )%(.<&), о > 1, then h is a continuous linear functional on J^i,1 (,c /) = & xm(s4) (||ili® < IH In = l l , lli), so there exists an operator ае-я/ such that h(b) = m(ha) for all
(see [4], [18]) and \m(ba)\ ^ const ||h|j1(T.
Proposition 2.8. I f $0 has no minimal non-zero projections, then a — 0 and hence the trivial functional h = 0 is the only continuous linear functional on
1 < (T < oo.
00
P ro o f. Suppose that a = u\a\, \a\ — j Mex and m(eEl) = yx > у > 0 for
о
some 0 < £ < ||a||. Let 2n/y — ),k — 2n/(yk2k~2), к — 2, 3, ..., n, a0 = 0,
П
al = y/2n, a* = 2k~1y/2n and b„= £ *kpk, where p{ I p j, i Ф j, pt e Proj (s/),
k= 1
Pi ^ ef, m ( £ Pi) = ak. We have p j l i = m(b„) = £ (l/k);
i=1 Л= 1
IIMI, „=(L = i 1Д Т "
/с — 1 2
00
^ ( l + ( 2 ff- l ) £ 1//са)1/<7.
/с=2
Consequently, \m{bn/\\bn\\x ■ и*а)) < const||ЬМ|| 1<T/j|b„j|j -*■ 0 as n-* oo. But M M Ih J li -u*a)| = l/||h j|i • £ 2km(pk\a\pk) ^ l/II^JIj • X h ^ t ( p ke fp k)
f c = l f c = l
n
= # J l i • Z = e k= 1
yielding a contradiction.
Let us now assume that 1^< 5< оо,< т = 1. Then, for any 1/(5 + l/ô' = 1 and b e J ? s£{st) (see [21], Theorem 3.3)
oc
\m{ba)\ j b (a) a (a) dct < S \\a\\ô^ \\b\\ôu
о
that is, hae S e 6£ and P J K where h j - ) = m(- a).
Conversely, let ha(-) = m(- a)e£Fôtf ‘, where йе J ioc (see Proposition 2.6). From the proof of Theorem 6.3, [2] we immediately get that for peProj {s$\ m(p) < oo,
l/m{p)-m(\pa\) ^ \\ha\\ m(p)~1/0'.
We shall demonstrate that the last inequality implies а е У В1. Assume that,
00
for some X0 > 0, m(e|0) = oo, where \a*\ = [ Xdek. Let p denote a projection
b
such that p ^ e £0, со > m(p) > Àôô’ \\h\\ô’. Then p\a*\p = pefQ\a*\p ^ A0p and X0m(p) ^ m(p\a*\) = w(|a*|p) — m(u* a* p) ^ m(\a* p\) = m(|pa|), where a* =u\a*\ is the polar decomposition of a*. Hence
l/m(p)1/ô■ m{\pa\) ^ \/m(p)lls■ X0m(p) = XQm{p)110' > \\ha\\
which leads to a contradiction. In other words, a* e Sfm (stf), which implies a e S fm{sé). In that case, by Proposition 2.4, ||a||*'œ < ||а||£® < ||ЛЯ||, i.e., ae F£6'ff (stf). To sum up
Theorem 2.4. Any linear, continuous functional h on 0 ^ 1 , is of the form h(b) = m(ba), beFFô^{s^), where ae F£âm°° (<c/), l/<5 + l/<5' = 1 and
М Ь ' а о < I M l £ * < P l l < < $ | М 1 й ' о о -
Corollary 2.3. F o r 0 < ct < 1, 1 ^ <5 < oo, & * % ( £ /) * = { s é \ l/<5 + l/<5'=--l.
P ro o f. This follows easily from the following facts:
ll'IU ^ II -iL; IIpIL = Midi = »*(р)1/л, P£ Proj (,*/), m(p) < oo ; the simple operators are dense in F£à^{sé) (cf. [2], Corollary 4.1).
Note that the space F£ô™{sî\ 1 < Ô < oo, is also called the non- commutative Marcinkiewicz space J i ô (,.c/).
For 1 < <5 < oo, 1 < a < oo, we have (cf. [7], [21], Theorem 4.4)
Theorem 2.5. Any linear, continuous functional h on is of the form: h(b) = m{ba), be F fô,°(sé), where ae (stf), 1/Ô + 1/Ô' = 1, 1/a + l/o'
= 1.
P ro o f. Assume that й е У ^ 'М . Then, for any be ^ ^ (.c /), M M I < Ï b(ct)a(ot)dcc ^ {S/<r)1 / a \\b\\ôa\\a\\ô-^ < oo,
о
210 L. J. C iach
that is, ha{-) = m ( • and \\ha\\ ^ {ô la Ÿ lo{ô'la'Ÿla' \\a\\y<r>. Let us now assume that h is a continuous, linear functional on S£ \ B y making use of Proposition 2.6, h{b) = m(ba), b e J ? ô£, where a e J i Xoc. Since ЦЬЦ^ ^ ||Ь||Л (see also the proof of Theorem 2.4), h is also a continuous functional on
00
jSf'îî and, in consequence, ae i? ^ 00 {sf). Let a = u\a\, |a| = f Xdek and
n b
K = Œ xf ~ X Pi)u* ‘> 0 = K+ 1 < K < K- i < .. • < , Pi = е(Я.+ 1>Ài>, oc0 = 0,
1i
ai = w ( Z ft)* We have
* = i
= M M ) I < £ 2?' ( a . - a , ^ ) i= 1
^ const • ||b„IL = const • ( £ I f (dfl0-a fL \)Y la.
i =1
The above inequality implies that ae£P6ma (srf).
3. Approximation and interpolation lemmas. In this section we shall apply the notations and terminology from [1] and [15]. We shall start by introducing a new space of m-measurable operators.
Definition 3.1 (cf. [1], [16]). For any а е ^ м(.с/), we define IM!o,* = IMIo = w(supp|a|)
and
— ^ ° m — — {aE &m ‘- Hallo < 00}.
Directly from the properties of the support of an operator and the regular gage m the following properties of ||*||0 result:
(i) Hello > 0* IWIo = 0 implies a = 0, ||0||0 = 0 {a eS fm)\
(ii) ||/?a||o = ||e||0 fie Q ; (hi) ||e + 6||o ^ IMIo + IMo (a, be (iv) ||e||o = IMIlo = l|e*||o ( a e ^ J ;
(v) llobflo < min(|Hlo, ||b||0) (a, be (vi) ||e||o ^ ll^llo (0 ^ a < b, a, beS£).
In the sequel, for the functional ||*||0 we stick to the term “norm”
although the condition of “homogeneity” is not satisfied. From properties (i)- (iii) of the norm |j-|lo it follows that is a linear space. If is considered as a group (with the addition of operators as a group operation), then the norm ||*||0 determines in i f 0 the topology in which J5f° is a topological group.
Proposition 3.1. The topology of m-convergence on ££° is weaker than the topology given by the norm || * ||0-
00
P ro o f. Let us fix г > 0. Let ||a||o ^ e and |a| = j Ыех. Since supp|a|
о
= c(0>oo), therefore m(ef) ^ rn(ei0<aD)) = ||a||o < e. In that case, д(а) ^ £.
Proposition 3.2. (i) an-> a in J*?0 if and only if т(е{еп)1)-+ 0 uniformly with
00
respect to e > 0, where \a — an\ = J Me(f . о
(ii) an-> a in ST° if and only if there exists a sequence \p„} of projections from с/, such that ||(u —a„)p„|| = 0 , m (pf)-> 0.
P roof, (i) su p p |a-a„| = ejg*», «!o.®) = lim e(£n)1.
£|0
(ii) Let ||a„ —a||0-> 0. Put pn = e\$). Conversely, it is sufficient to notice that pn ^ e%, that is <?$>a0) ^ pi.
R em ark 3.1. By making use of property (v) of ||-||0, it is easy to show, as in the proof of Corollary 2.6 in [2], that the multiplication of operators is continuous in J?°m (,</).
By making use of Propositions 1.1 and 3.1, it is easy to see that (see [3], Proposition 2.1)
Proposition 3.3. is complete in the topology given by |И |0.
Finally, we prove the following fact:
Proposition 3.4. Assume that there exists o^efO, 1) such that Then lim||u||" = ||a||0 as o-+ 0 and for any a e i f m\<£0m\
lim\\a\\Z = oo as a —* 0.
oo IMIo Hello
P roof. ||a||£ = J aff(a)da = J aff(a)da->- J dot = ||a||0 as a-* 0, since
o o о
uCT(a)-> 1, 0 < a < ||a||0, as u -> 0 and aff(a) < m ax jl, a*1 (a)}. For any oc
aeJ?m \ M il — f aa(ot)dot ^ ав(п)чг > 0 for every n — 1 ,2 ,... so that
ô
lim inf||u||" ^ n, n = 1 ,2 ,...
a -»0
It is quite natural to estimate the error of this approximation.
Let ae and 0 < e < ||a||0- Then Hello-»
(3.1) INI" ^ j aff(ct)dtx ^ uff(||a||0-e)(||a||o-£).
о In consequence
(3.2) MIMMIo > -s -|W lo K (IM Io -e )-l|-
2 — Prace Matematyczne 26.2
212 L. J. C iach
INI о
Assume further that де Jz?°n j/ = J^m. Then ||a||" = J aff(a)da ^ ||a||0|ja|ja, о
that is,
(3.3) N i ; - I W I o < I N o |I N r - i |.
From relations (3.2), (3.3) we obtain for
|N I" “ Mlo| < е + 1М1ошах{|||д|Г-1|, |aff(||a o ll-e)-l|} . If fi = fl(||a||ô -e ) > д(||д||0) = 0 (e.g. (\a\ç, ç) ^ const ||ç||2), then
|!Mla~INIo| < or||a||0max [|1п||д||| • max Ц, !|a!f<T], |ln j?|max Jl, /Г] ]
^ o-||a||0max f|ln ||a|||, |ln /i|)max |1, ||д|Г) for
Now, we are in a position to give some interpolation lemmas. For symmetry — the algebra ,q? will be denoted by . To begin with, we shall consider the 1-normed abelian couple (cf. [15]) {£f°m, i?7®)
= (& °, ^ 00).
Lemma 3.1 (cf. [1], Lemma 7.2.1; [12], Theorem 1). For any aeS£m{si) a(a) = inf{||a — b\\: beFF°m{s^), ||6||0 ^ a, a > 0 } = L ( a , a)
= E(a, д; ( i f 0, J5?00)).
P ro o f. It is not hard to notice, by making use of the polar decomposition of д, that E(a, a) = E(x, \a\), a > 0. If so, without loss of
00
generality we may assume that a ^ 0. Let a = j Adek. For any a > 0, о
m(ea(a)) ^ a - So, let us put ha = aef(a) = ^a(a) ‘a- H is easily seen that ||ha|j0 < a and E(a, a) ^ ||д — ba\\ = ||деа(а)|| ^ a (a). We shall now prove the opposite inequality. Suppose that E{a, a) < A < a(a) for some a > 0. Hence, in virtue of'Ihe definition of the function a(a), m(ek) > a. Besides, there exists an operator he i f 0 such that \\a — b\\ < A, ||b||0 ^ a. If (see e.g. [2], Proposition
1.1 (iv)) (ejr л (supp|b|)x)£ = (, ||£|| = 1, then
l|a -* || > ||(a -6 )ç || » ( ( a - b ) i , f) = («{, i) » >AeR, ç) = À.
We have got a contradiction. Finally, a (a) = E(a, a), a > 0.
Corollary 3.1 (cf. [1], Theorem 7.2.2; [15], Theorem 6.1). For any 0 < Ô < oo, 0 < a ^ cc
(with equality of norms). This being so, if в = ô/(ô +1) and т — ва, then
K eJ ^ ° , F£™) = { F £ bxf .
Corollary 3.2 (cf. [1], Corollary 7.2.3). I f 6 — l/if+ 1), x = во and \/0
= (1—в)/00 + в/01, then
Ep<x( ^ ô°, 3 ?ôl) = {g’ô°Ÿie.
For the pair j£?*) = {££a, JS?00) of quasi-Banach spaces, 0 < a
< oo, the following well-known relation in the commutative case, holds (cf.
also [12], Theorem 3 and [8], [16]).
Lemma 3.2. Assume that ae £ feT + ^ >co, 0 < o < c c . Then (3.4) K(ot0, a; <£\ &*>) - { J aa{ot)da}1/<T.
For <7 = 1, (3.4) is an equality.
P ro o f. The part of Lemma 3.2 can be proved identically as in the commutative case (see e.g. [1], p. 143). In order to prove the opposite inequality, let us assume that a = и \a\ (the polar decomposition) and \a\
00
= f Mek (the spectral decomposition). Besides, define: ax = u(\a\ -ea(aff) +
о ' °
+ а Ю еаы°)) and ao = a - a x = u(\a\e^ - d i a l ) e^ ). It is clear that a ^ i f 00, ||a1||00<a(aS). What is more, \a0\ = (|a| < ao- This being so, uo(a) = 0 for a ^ m{eL ) and a0(ot) — a{oc) — a(oto) for
e<V
0 < a < miea(a<n)- consequence, a0£ Ffa. Moreover, a “o
“o
K(aQ, a ; S£a, JS?00) ^ ||floll<r + a0||a1||e> < { { (a(a)-a(aS))ffda}1/ff + a0a(aS)
= { f (aict) — aiotl)Y dot}1/a+ { j aai<xa0) dot] l/tr
^ const { j aaiot)doiYla, 0
where const = 1 for <7 = 1.
4. Convolution operators.
Definition4.1 (cf. [11], Definition on 1.1; [19], Definition on p. 48). Given three regular gage spaces (Jf), mf), i = 1 ,2 ,3 , a bilinear operator, T, which maps measurable operators from F£mxiséf) and F£miisé2) into measurable operators from $£тъ i$43) is called a convolution operator if
(i) ||T(a, b)||i < {fa||! ||b||,, (ii) ||T(a, b )|K IM Ii I N , (iii) \\Tia, h)\\ ^ Hullllhili.
Now we shall prove two lemmas whose commutative equivalents can be
214 L. J. C ia ch
found in [11]. The method of proofs is based on that of [11], where convolution operators in ££a function spaces are characterized.
Lemma 4.1 (cf. [11], Lemma 1.4). I f T is a convolution operator and c = T(a, b), be & m2( s /2), where ap1 = 0, m ^p) = 3, \a\ ^ pi, for some pe Proj^c^), then for a > 0
(i) c{a, 1) = c(a) ^ /tôb(à), (ii) c(a, 1) = c(a) ^ PSb(<x).
P roof. Let p > 0 and let b^ = « (jbj ^ b* = Ь — Ьм = u(\b\e^~pe^),
00
where h = n|h|, |bj = f Àde2. It is easy to see that JbM| = (|b| — p)ef. In о
consequence, for cx = T(a, Ьц); c2 = T(n, 6") we have I N K N I I I b l l , | т 2(еях)с/Я,
A
Ikill ^ INIIIbJIi ^ pôp,
l|c2lli < IMIi IH li ^ Pô f m2(et)dA.
The next part of the proof is similar to the proof of Lemma 1.4, [11].
Lemma 4.2 (cf. [11], Lemma 1.5). I f T is a convolution operator and:
a e s # 1 + & lmi, b e ^ int2 or a e £ ?1mi, b estf2 or a $ .tfx + & xmx or Ьф.&2 + + S f 1ni2, then for c = T(a, h) and any p > 0
00
c{p) ^ pa{p)b{p)+ j a(ix)b(oc)doL.
ft
P ro o f. The inequality is clear if аф я/х + J?1mi or Ь ф ,^2 + ^ ?1m2- Fix p > 0. Consider a doubly infinite sequence such that:
Я0 —a{p), Яя ^ Я я+1, Н т Я я = оо, lim A„ = 0.
n oo n -+ — 00
Let a„ = м[(Яя- Я я_ 1)е ^ + (|а |-Я я_ 1)е(Аи_ 1,Аи>] and c{n> = T{an, b), where
00
a — и \a\, \a\ = § AdeÀ. It is clear that \an\ ^ (Яя — Я„_ x) I and аея = 0.
о
From Lemma 4.1 (ii) we obtain for n = 1, 2, ...,
£(п)(а) ^ (Яп —Я п -^т! (efn_ j)b(a) and, by the first inequality of Lemma 4.1,
c(n)(а) ^ (Яя- Я„_ i) wit ( ^ и_ t)b(m x (ein_ x)) for n = 0, - 1 , - 2 , ...
Furthermore
c = T(a, b) = T(n(|a| —Я0) ef0, 6)+ T(n(|n| еЯо + Я0ел10), h) = cx + c2, c(p) ^ c 1(p) + c2(p).
i о
Let a(l) — Z a„ and a{~l) = Z a„, / = 0, 1, 2, ...
n= 1 n= — l
It is easy to see that
= а(1)е(л0(Д/> = и{\а\-Л0) е ^ е Цо>Л1> = u(\a\-ÀQ)ei0eXl and so
и (|u| - Л0) eiQ - a{l) = и (|a\ - A0) e^Q - a(l) eX{ = и (\a\ - A,) . In consequence
C> = т(и (|a| - Л„) e^, b) = T ( J \ Ь)+Т(и (|a| - A0) - a<», b) = c f + сЦ> ;
£i(b) < £ !? М + £ * (м) « У £m (b-)+ÇiM
п= 1
I
< )m 1{ein_l)b{n) + c{l){fx), fi> 0.
/»= 1
If a e S £ xm , be <y?lm2{-<rf2), then u(\a\ — Xt)eXl -* 0 in S£xm and, in consequence, il40|!i -*• 0 (llcÿii 0) as /->oo. Hence, с^(^) -> О, ц > 0, as / -*• oo.
Assume that a e st/l . It is clear that an = 0 for n sufficiently large and, in consequence, a{l} = u(\a\ — X0)eX() for l sufficiently large. Finally,
OO
C l i t i c )m 1(ei„_1)b{n), ц > 0.
n — 1
To evaluate c2(/d we use the first inequality of Lemma 4.1. For any (Lf), g C(2_^_ j д _^ (L/), 0 < Л < /, ^ j (Л) we have, respectively, а(~'Ч = (A o-A -f-iK ; а(_г)£ = м(|а|£-Я _,_ j ç); = 0. Hence,
с2 = 7 > (~г), Ь) + T(u(À-l- l ei_ l_ 1 + \a\ex_l_ l), b) = с(5~l) -F с(6~0 and, in consequence,
о
£ 2 (^ K z + / x>0.
n= - /
For any a e j / j - t - b e & xm2 we have, ||c(6_/)|| у 0 and so £б"0 (ju)-♦ 0, H >0, as 1-+OQ. Assume that b e sг/2, In that case, ||w(Я_f_ x ex_ +
+ lalÉ?2_/_ 1)lli -> 0 and so ||c(6-,)| | -> 0 that is, c(6~0(ju) -*■ 0, ju > 0, as /-> oo.
Finally,
о
Z (^« — 1) mi (eA„-i)^(mi (^я„_ A))-
n= — OO
The remaining part of the proof is similar to the proof of Lemma 1.5, [11].
216 L. J. C iach
Co r o l l a r y 4.1 (cf. [11], Lemma 1.6). Under the assumptions of Lemma
4.2
c(p) ^ j a (a) b (a) da.
From Lemma 4.2 we obtain (cf. [11], Theorem 1.7).
Theorem 4.1. A bilinear operator T is a convolution operator if and only if for c = T(a, b), аед/х + ЗР1^ , b e & lm2 (ae Se *mi, b e s / 2)
GO
c{p) ^ pa(p)b{p)+ J a (a) b (a) dtx.
n
Corollary 4.2 (cf. [11], Corollary 1.8). A bilinear operator T is a convolution operator if and only if
(i) ||T(e,fe)||1 < |N |1|!b||1, (ii) || T{a, b)|| J a (a) b (a) dct,
о
+ or a e ^ lmi, b e s# 2.
Making use of Lemma 4.2, Theorem 4.1 and Hardy’s inequality, we easily obtain the following theorems:
Theorem 4.2 (cf. [11], Theorem 2.6). I f T is a convolution operator and T(a, b) = c, a e , + b e ^ \ 2 ( a e i ? 1^ , Ь е д /2)> arul lf а е ^ Л * 1, b e * £ ôml2> where l/ô 1 + l/ô 2 = l + l/(5, <5, <5l5 ô2 > 1, and a ^ 1 is any number such that l/a 1 + l/a 2 ^ l/<7, cr1, cr2 ^ 1, then
Ikllfo ^ const ||а||а1в1 \\Ь\\02<,2-
Corollary 4.3 (Young’s inequality, cf. [11], Theorem 3.1). I f Ô > 1, о > 1, у> 1 satisfy the equation l/ô + ljo — 1-Fl/y, then
||c||y < const 1ИУЫ1,.
Theorem 4.3 (cf. [11], Theorem 3.6). I f T is a convolution operator and c = T(a, b), + beS£'mi { a e £ flmi, Ь е я /2) and if а е & Лщ, Ье£Р*Л°1> where \!ô + l/ô 1 = 1, \/u x + \/a ^ 1, then
||c|| ^ ym ax{c/ô, ojd ^} ||a|L||b|U1<ri, l/(ffy)+l/fo y) = 1.
R e m a r k 4.1. Theorem 4.2 and Corollary 4.3 are true for ô, ôlt ô2 = 1 (о, ô, y = 1) and _|| *||loo (cf. [11]). Note that
- N i x » =IM!i H M I n ! , ае££тЫ ).
Problem. Assume that Г is a convolution operator, J - d№ 2 - dtP 3 5
«я/ i = 2 — ^ ъ-, Щ = m2 = m 3. Is U(T{a, b)) = U(a)U(b), for some unitary operator U (“Fourier transformation”) from 5£2m{sf) into
C o m m e n t s on t he proofs. Theorem 2.1, Propositions 2.1, 2.2, 3.4, Lemmas 3.1, 3.2 are true for a subadditive measure m (cf. [2], [3]). Corollary 2.2 remains valid for the measure m0 (see [3]), 0 < 3 < 1/2, and a finite and a-finite algebra .
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INSTYTUT MATEMATYKI UNIWERSYTET LÔDZKI EÔDZ
