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Владикавказский математический журнал 2018, Том 20, Выпуск 1, С. 30-37
УДК 517.98
THE UNIQUENESS OF THE SYMMETRIC STRUCTURE IN IDEALS OF COMPACT OPERATORS
B. R. Aminov, V. I. Chilin
This paper is dedicated to the memory of Professor Inomjon Gulomjonovich Ganiev
Let H be a separable infinite-dimensional complex Hilbert space, let L(H) be the C*-algebra of bounded linear operators acting in H, and let K(H) be the two-sided ideal of compact linear operators in L(H). Let (E, || • ||e) be a symmetric sequence space, and let Ce := {x € K(H) : {sn(x)}^' € E} be the proper two-sided ideal in L(H), where {sn(x)}JJ=1 are the singular values of a compact operator x. It is known that Ce fa a Banach symmetric ideal with respect to the norm ||x||cE = ||{sn(x)}£=1 ||e. A symmetric ideal Ce is said to have a unique symmetric structure if CE = Cf, that is E = F, modulo norm equivalence, whenever (Ce , || • ||ce ) is isomorphic to another symmetric ideal (CF, || • ||cf ). At the Kent international conference on Banach space theory and its applications (Kent, Ohio, August 1979), A. Pelczynsky posted the following problem:
(P) Does every symmetric ideal have a unique symmetric structure?
This problem has positive solution for Schatten ideals Cp, 1 < p < то (J. Arazy and J. Lindenstrauss, 1975). For arbitrary symmetric ideals problem (P) has not yet been solved. We consider a version of problem (P) replacing an isomorphism U : (Ce, || • ||ce ) ^ (Cf, || • ||cf ) by a positive linear surjective F
isometry U : (Ce, || • ||ce ) ^ (Cf, ||- ||cf ) is of the form U(x) = u*xu, x € Ce, where u € L(H) is a unitary or antiunitary operator. Using this description of positive linear surjective isometries, it is established that existence of such an isometry U : Ce ^ Cf implies that (E, || • ||e) = (F, || • ||f)•
DOI: 10.23671/VNC. 2018.1.11394.
Mathematical Subject Classification (2010): 46L52, 46B04.
Key words: symmetric ideal of compact operators, uniqueness of a symmetric structure, positive isometry.
1. Introduction
Let (co, || ■ ) ^e a Banach lattice of all sequences a = of real numbers convergent
to zero, where ||a||^ = supneN l£n| (N ^s ^te set of ^^taral numbers). If a = G c0, then
a non-increasing rearrangement a* = } of a sequenee a ^s a sequenee |} in decreasing order.
A non-zero linear subspaee E С c0 with a Banach norm || ■ ||e is called symmetric sequence space, if the conditions b G E, a G c0, a* ^ b*, that a G E and ||a||g ^ ||6|e-
Let H be a complex separable infinite-dimensional Hilbert space and let L(H) be an C*-algebra of all bounded linear operators acting in H. We denote by K(H) (respectively, F(H)) the two-sided ideal in L(H) of all compact (respectively, finite rank) linear operators. It is
© 2018 Aminov B. R., Chilin V. I.
well known that F(H) C I C K(H) for any proper two-sided ideal I in L(H) (see, for example, [10, Proposition 2.1]).
If (E, || ■ ||e) is a symmetric sequence space, then the set CE := {xG K(H) : |sn(x)}^=1 G E} is a proper two-sided ideal in L(H), where {sn(x)}^=1 are the singular values of x (i.e. the eigenvalues of (x*x)1/2 in decreasing order) [10, Theorem 2.5]. In addition, (CE, || ■ ||ce) is a Banach space with respect to the norm ||x||ce = ||{sn(x)}£=1 ||e [14, Ch. 3, § 3.5]. In this case we say that (CE, || ■ ||ce) is a symmetric ideal (cf. [12, Ch. III]).
It is said that (CE, || ■ ||ce ) to have a unique symmetric structure, if whenever (CE, || ■ ||ce ) is isomorphic to another symmetric ideal (Cf, || ■ ||cf) then necessarily, CE = Cf, i.e. E = F, with equivalent norms.
In Kent Conference (International Conference on Banach Space Theory and its Applications, Kent, Ohio, August 1979), A. Pelczynsky raised the following problem:
(P): Does every symmetric ideal have a unique symmetric structure?
In [3] it is proved that symmetric ideals Cp = Cp, 1 ^ p < to, have unique symmetric structure. In addition, J. Arazy proved (see [4, Corollary 5.9]) the following
Theorem l.Ifa symmetric sequence space E does not contain a subspace isomorphic to c0 and du SpSlCG E does not contain a complemented subspace isomorphic to l2, then (CE, || ■ ||ce ) has a unique symmetric structure.
Using the Theorem 1 it is easy to prove that for the Lorentz ideals the problem (P) is solved positively for q = 2 (see Section 2 below). If q = 2, then answer is also positive (O. Sadovskaya and F. Sukochev (unpublished)). At the same time, for arbitrary ideals the problem (P) has not yet been solved.
In this paper we consider the version of problem (P) (we call the problem (P+)) in the case when isomorphism U : (CE, || ■ ||ce) ^ (Cf, || ■ ||cf) is replaced by positive linear bijeetive isometry. We solve the problem (P+) in the class of strongly symmetric ideals of compact operators.
2. Preliminaries
Let p, q G [1; to), and let
lp,q = |a = {£4n=i e Co : \\a\\P,q (tip - (n-l)pjj * <to
be the Lorentz sequence space. It is known that (lp,q, (1 ^ q ^ p < to), in addition,
iip,q
)
ipp = ip = {U^=1 c co, |{e
-,nj IIp,p
= |{^n }||p =
n=1
l^n|p
< TO
If 1 < p < q < to, then || ■ ||p,q ^s a ^^^^^^^e quasinorm on the vector lattice lp>q; moreover, on lp,q there exists a norm || ■ ||(p>q^, which equivalent to the quasinorm || ■ ||p>q, such that (lp,q, || ' ||(p,q)) is a symmetric sequence space [5, Ch. 4, § 4].
The following important property of the Lorentz sequence space is proved in the paper [7]. Theorem 2. If 1 < p < to 1 ^ q < to, then every infinite-dimensional closed subspace
m
lq
L(H)
Cp,q = {x G K(H) : {sn(x)}£=1 G lp,q},
1
equipped with the norm ||x||p,q = ||{sn(x)}||P;q, 1 ^ q ^ p < to (respectively, ||x||p,q = ll(sn(x)}||(p,q), if 1 < p < q < to).
Using Theorems 1 and 2, we can give a positive solution of the problem (P) for symmetric ideal (Cp,q, || ■ ||p,q), q = 2.
Theorem 3. Let 1 < p,q < to, q = 2. If (Cp,q, || ■ ||p,q) is isomorphic to symmetric ideal (Ce, || ■ ||ce), then Cp,q = Ce, i- e. E = lpq, with equivalent norms.
< It is known that the Lorentz sequences space (lp,q, || ■ ||p,q) (respectively, (lp,q, || ■ 11(p,q))) is reflexive if 1 < p,q < to. And the conjugate space (lp,q)* is isomorphic to lr,s, where 1 < r, s < to, ^ + 7 = 1)^ + 7 = 1 (see, for example, [5, Ch. 4, § 4, Theorem 4.7]). By Theorems 5.6, 5.11 [11] we have that the Lorentz symmetric ideal (Cp,q, || ■ ||p,q), 1 < p,q < to, is reflexive too, in addition, the conjugate space (Cp,q)* is isomorphic to Cr,s, where 1 < r,s < to, and i + ^ = i + i = 1. Consequently, (Cp>q, || • ||p,q), 1 < p,q < to, does not contain a subspace isomorphic to co. Using now Theorems 1, 2, and the inequality q = 2, we have a positive solution of the problem (P). >
Let Sp,q denote the closed subspace of Cp,q consisting of all block diagonal matrices x = diag{xfc}^=1 with Xk a k x k-matrix for all k. By Corollary 5.8 [4] in the case when Sp,q is not isomorphic to Cp,q the Banach symmetric ideal Cp,q has a unique symmetric
Sp,q
it is not embedded in Banach spaces Cp,q, in particu 1 ar, Sp,q is not isomorphic to Cp,q for all 1 < p,q < to. Therefore, in the case q = 2, the Theorem 3 is true too.
Since the Banach spaces (lp,q, || ■ ||p,q) and (lr,s, || ■ ||r,s) are isomorphic if and only if p = r, q = s (see [10]), Theorem 3 implies the following isomorphic classification of Banach symmetric ideal (Cp,q, H ■ ^p,q).
Corollary 1. Let 1 < p,q,r,s < to. The Banach spaces (Cp,q, || ■ ||p,q) and (Cr,s, || ■ ||r,s)
p = r q = s
It should be noted that the Banach space C2,2 = C2 is a separable Hilbert space and it is isomorphic to l2 = l2;2, in particular, C2 has local unconditional structure. For all other variants of the values of the parameters p, q the Lorentz symmetric ideal (Cp,q, || ■ ||p,q) has not local unconditional structure [13]. Since a Banach lattice has a locally unconditional structure, it follows that SpctC6S lp,q cUld Cp,q are not isomorphic, if 1 ^ p,q < to, p = 1 and q = 2, p = 2
Define in K(H) (respectively, in co) the Hardy-Littlewood-Polya partial order x XX y (respectively, {£}«=1 XX M^U), if
k k / k k \ Y^ sn(x) Sn(y) (respectively, nn v k g N-
n=1 n=1 \ n=1 n=1 /
It is clear that x XX y ^ {sn(x)}^=1 XX {sn(y)}^=1.
The Hardy-Littlewood-Polya partial order has the following important property
Proposition 1 [9, Proposition 2.1]. If x,y G K(H) x = x*, y ^ 0 and —y ^ x ^ y, x XX y
A symmetric ideal (a symmetric sequence space) (Ce, || ■ ||ce) (respectively, (E, || ■ |e)) is called a strongly symmetric ideal (respectively, a strongly symmetric sequences space) if the condition x XX y, x,y G Ce (respectively, a XX b, a,b G E) implies that ||x||ce ^ ||y||ce (respectively, ||a||E ^ ||b||E)- It is clear that a symmetric ideal (Ce, || ■ ||ce) is a strongly symmetric ideal if and only if a symmetric sequence space (E, || ■ ||e)) is a strongly symmetric sequence space.
The Proposition 1 implies the following.
Corollary 2. Let (CE, || ■ ||ce) be a strongly symmetric ideal. If x, y G CE, y ^ 0 x* = x and -y ^ x ^ y, then ||x||ce ^ ||y|cE-
3. Description of positive isometries of symmetric ideals
In this section we give a description of positive linear bijeetive isometry U : (Ce, || ■ ||ce ) ^ (Cf, || ■ ||cf )j when Cf is a strongly symmetric ideal.
The following Proposition establishes positivity of the inverse mapping of positive surjec-tive isometry.
Proposition 2. Let (CE, || ■ ||ce) be a symmetric ideal and let (CF, || ■ ||cf) be a strongly symmetric ideal. Let U: CE ^ CF be a positive linear surjective isometry. Then an isometry U-1 is also positive.
< Let x G Ce and U(x) ^ 0. Since U is a positive surjective mapping it follows that U(y*) = U(y)* for all y G Ce. Hence x* = x. Let x+ and x- be a positive and negative parts of an operator x. It is clear that x+,x_ G Ce- If x+ = 0 then U(x) ^ 0. Consequently, U(x) = 0 which implies x = 0.
Let now x+ = 0. Set y1 = U(x+) and y2 = U(x_). We have that
y1 ^ 0, y2 ^ 0 and y = y1 - y2 = U(x) ^ 0.
In addition,
y1 + y2 = U (|x|) and ||U (|x|)|cf = HMH^e = ||x|ce • Using mathematical induction, we show that
||x+ + kx_||cb = 11y 1 + ky2||cf ^ ||x|ce (!)
for all k G N. If k = 1, then the inequality (1) is obvious. Suppose that it is true for k = n. Then
— (y1 + ny2) < (y1 - y2) - ny2 = y1 - (n + 1)y2 < y1 + ny2.
By Proposition 1 we have that y1 - (n + 1)y2 XX y1 + ny2. Since (Cf, || ■ ||cf) is a strongly symmetric ideal it follows that (see Corollary 2)
|y - (n + 1)y2|cf < 11^1 + ny211cF < ||x|Ce•
Thus
||y1 + (n + 1)y2|cF = ||x+ + (n + 1)x_||ce = || |x+ - (n + 1)x_| ||ce
= |x+ - (n + 1)x_ ^Ce = |y - (n + 1)y2|cF < ||x|CE • Therefore the inequality (1) holds for all k G N. Since
k||x_||ce ^ ||x+ + kx_|ce ^ ||x|ce for all k G N,
it follows that ||x_||ce = 0, that is x ^ 0 >
Remark 1. The proof of Proposition 2 is analogous to the proof of Theorem 1 in fl], where the positivity of the inverse mapping for isometries of Banach lattices is established.
Theorem 4. Let (CE, || ■ ||ce ) be a symmetric ideal and let (CF, || ■ ||cf ) be a strongly symmetric ideal. Let U: CE ^ CF be a positive linear surjective isometry. Then U(p)
(respectively, U 1(e)) is an one-dimensional projection for any one-dimensional projection p G F(H) (respectively e G F(H)).
< Suppose that U(p) = y is not a rank 1 operator. Since y ^ 0 y G Cf it follows that there exist pairwise orthogonal one-dimensional projections q1;q2 G F(H) and positive numbers A1, A2 such that 0 < A1q1 + A2q2 ^ y. By Proposition 2 we have that
0 < U_1 (A1 q1 + A2q2) < U_1 (y) = p.
If U _1 (qi) = xi5 then 0 < A^ ^ pi = 1, 2. Sine e p is an one-dimensional projection, it follows that AiXi = ji'p for some 7j > 0. Consequently, q» = U(xj) = U(^pp) = ^y, i = 1,2, which is
q1 q2 = 0
Therefore, U(p) = Aq for some one-dimensional projection q and positive number A. Now using the inequalities
1 = ||p||ce = ||U (p)||cf = A|q|cF = A,
we have that A = 1. Consequently, U (p) = q.
Similarly U_1(e) is an one-dimensional projection for any one-dimensional projection e G F(H). >
Corollary 3. Let (CE, || ■ ||ce ) (Cf, || ■ ||cf ) and U: CE ^ CF be the same as in Theorem 4. Then U(F(H)) = F(H).
A linear bijeetive mapping p: L(H) ^ L(H) is called an Jordan isomorphism, if p(x2) = (p(x))2 and p(x*) = (p(x))* for all x G L(H). If p: L(H) ^ L(H) is an Jordan isomorphism, then there exists an unitary or an antiunitary operator u G L(H) such that p(x) = u*xu for all x G L(H) (see, for example, [6, Ch. 3, § 3.2.1]).
H k L(H) = K(H)
If (E, || ■ ||e) C c0 is a symmetric sequence space, then the set
Ce (k) := {x G K (H) : {s1 (x),...,sfc (x), 0, • • • } G E} = L (H)
k
||x||ce (k) = ||{s1(x),... ,sfc(x), 0, ••• }||E •
Using the description of all positive linear surjective isometries of strongly symmetric spaces E (M, t) in the case a finite von Neumann algebra M and a finite trace t [8, Theorem 3.1], we have the following
Theorem 5. Let (E, || ■ |E) C c0 be a symmetric sequence space with a strongly symmetric norm. Let U: (CE(k), || ■ ||ce^ (CE(k), || ■ ||ce(k)) be a positive linear surjective isometry. Then there exists an Jordan isomorphism p: L (H) ^ L (H) such th at U (x) = ^>(x) for all x G CE(k) = L(H). In particular, U(x)U(y) = U(y)U(x) if and only if xy = yx.
The following Theorem gives a description of positive linear bijeetive isometries U : (Ce, || ■ ||ce) ^ (Cf, || ■ ||cf) when Cf is a strongly symmetric ideal.
Theorem 6 (cf. [2, 16]). Let (CE, || ■ ||ce) be a symmetric ideal and (CF, || ■ ||cf) be a strongly symmetric ideal. Let U: CE ^ CF be a positive linear surjective isometry. Then there exists an unitary or antiunitary operator u G L (H) such th at U (x) = u*xu for all x G Ce.
< By Proposition 2 we have that an inverse isometry U_1 is also a positive map. Let p, e, q,f G F(H) be an one-dimensional projections such that U(p) = q, U(e) = f (see Theorem 4). If p ■ e = 0 then by Theorem 5 we have that q ■ f = 0.
Let |Рп}П=1 С F(H) be a pairwise orthogonal one-dimensional projections and x = ЕП=1 AnPn G F (H), \n G R, n = 1,..., k. Since U (p„) ■ U (pm) = 0 n = m, n,m = 1,...,k, it follows that
/к \ к ,2) = U XnpJ = ^ ХПи(Pn) = U(x)2
Vn=1 / n=1
U (x2) = U^\2nPn) = ^ AnU (Pn) = U (x)2 and
fe fe
tr(U (x)) = ^ Antr(U (pn)) = ^ An = tr(x).
n=1 n=1
Therefore U(x2) = U(x)2 and tr(U(x)) = tr(x) for all x* = x G F(H). In addition, U is a bijection of the set P(H) of all one-dimensional projections. If p, e,q, f G P(H) and U(p) = q, U(e) = /, then
2tr(pe) = tr(pe) + tr(ep) = tr ((p + e)2 — p — e) tr (U ((p + e)2)) — 2 = tr (U ((p + e)2)) — 2 = tr ((q + /)2) — 2 = 2tr(qf).
Consequently, tr(pe) = tr(U(p)U(e)) for all p, e G P(H). Now using Theorem 3.2.8 [6, Ch. 3, § 3.2] we get that there exists an unitary or antiunitary operator u such that U(p) = u*pu for all p G P(H^us U(x) = u*xu for all x G F(H).
Let 0 ^ x G Ce and 0 ^ xn G F(H) be such a sequence that xn t x. Since U: Ce ^ Ce is an order isomorphism (see Proposition 2) it follows that u*xnu = U(xn) t U(x). Consequently, U(x) = u*xu for all x G CE. >
4. Pelchinsky problem with respect positive isometries
Consider now the following version of problem (P):
(P+): Let (Ce, || ■ IIce) and (Cf, || ■ ||cf) are symmetric ideals and let there exists a positive isometry U : (CE, || ■ ||ce) ^ (CF, || ■ ||cf). Is it true that then (E, || ■ ||E) = (F, || ■ ||F)?
Below we give a solution of the problem (P+) for the class of strongly symmetric ideals of compact operators.
Theorem 7. Let (CE, || ■ ||ce) be a symmetric ideal and let (CF, || ■ ||cf) be a strongly symmetric ideal. Let U: CE ^ CF be a positive linear surjective isometry. Then (E, || ■ ||E) =
(F, || ■ IIf)•
< By Theorem 6 there exists an unitary or an antiunitary operator u G L(H) such that U(x) = u*xu for all x G Ce- Fix an orthonormal basis |^n}^=1 in a separable Hilbert space H. Let Pn G P(H), Pn(^n) = Ф n, n G N Consider real sub space Ge — {x — S™=1 ^n'Pn : £n GR x G CE} in the space (CE, || ■ ||ce)• It is Лаг that |^n}^=1 G E and ||x||ce = ||{£n}||E f°r all x = £nPn G Ge- Consequently, the correspondence GE э x о {£n} G E identifies the
Banach spaces (Ge, || ■ ||ce) and (E, || ■ ||e)•
Since u is an unitary от antiunitary operator it follows that un = u*^nu, n G N, is an orthonormal basis in a Hilbert space H. Let qn G P(H), qn(un) = vn, n G N. Set Gf = {x = X^U nnqn : Пп G R, x G Cf}. It is clear that the corresp ondence GF Э x о {nn} G F identifies the Banach spaces (Gf, || ■ ||cf) and (F, || ■ ||f)• Since Gf = u*Geu we get that (E, || ■ ||e) = (F, || ■ IF)• >
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Received 29 November, 2017
Aminov Behzod Rasulovich National University of Uzbekistan, Phd student Vuzgorodok, Tashkent, 100174, Uzbekistan E-mail: aminovbehzod@gmail.com
Chilin Vladimir Ivanovich National University of Uzbekistan, Professor Vuzgorodok, Tashkent, 100174, Uzbekistan E-mail: vladimirchil@gmail.com, chilin@ucd.uz
ЕДИНСТВЕННОСТЬ СИММЕТРИЧНОЙ СТРУКТУРЫ В ИДЕАЛАХ КОМПАКТНЫХ ОПЕРАТОРОВ
Аминов Б. Р., Чилин В. И.
Пусть H — сепарабельное бесконечномерное комплексное гильбертово пространство, L(H) — C*-ал-гебра ограниченных линейных операторов, действующих в H, K(H) — двусторонний идеал в L(H) всех компактных операторов. Пусть (E, || • ||e) — симметричное пространство последовательностей, CE := {x £ K(H) : (sn(x)}^=! е E} — собственный двусторонний идеал в L(H), порожденный (E, || • ||e), где {sn(x)}^=! сингулярные числа компактного оператора x. Известно, что Ce — банахов симметричный идеал относительно нормы ||x||cE = ||{sn(x)}£=i||E.
Говорят, что симметричный идеал Ce имеет единственную симметричную структуру, если наличие изоморфизма из (CE, ||-||cb ) та другой симметричный идеал (CF, || • ||cf) обязательно влечет равенство CE = CF, т. е. E = F, с точностью до эквивалентных норм. На международной конференции по теории банаховых пространств и их приложений (Kent, Ohio, August 1979), А. Пельчинский поставил следующую проблему:
(Р): Каждый ли симметричный идеал имеет единственную симметричную структуру? Эта проблема получила положительное решение в работе J. Arazy и J. Lindenstrauss (1975) для идеалов Шаттена Cp, 1 < p < то. В случае произвольных симметричных идеалов проблема (Р) до сих пор не решена. Мы рассматриваем вариант проблемы (Р), заменяя наличие изоморфизма U : (Ce, || • ||ce ) ^ (Cf , || • ||cf ) на существование положительной линейной сюръективной изо-
F
каждая положительная линейная сюръективная изометрия U : (CE, || • ||ce ) ^ (CF, || • ||cf ) имеет следующий вид: U(x) = u*xu для всех x е Ce, где u е L(H) есть унитарный или антиунитарный оператор. Используя это описание положительных линейных сюръективных изометрий, доказывается, что наличие такой изометрии U : Ce ^ Cf влечет равенство (E, || • ||e) = (F, || • ||f)•
Ключевые слова: симметричный идеал компактных операторов, единственность симметричной структуры, положительная изометрия.
