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Vladikavkaz Mathematical Journal 2019, Volume 21, Issue 4, P. 11-24
YAK 517.98
DOI 10.23671/VNC.2019.21.44607
ISOMETRIES OF REAL SUBSPACES OF SELF-ADJOINT OPERATORS IN BANACH SYMMETRIC IDEALS
B. R. Aminov1, V. I. Chilin1
1 National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan E-mail: aminovbehzod@gmail.com, vladimirchil@gmail.com, chilin@ucd.uz
Dedicated to E. I. Gordon on the occasion of his 70th birthday
Abstract. Let (CE, || • ||ce) be a Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space H. Let Ce = {x £ Ce : x = x*} be the real Banach subspace of self-adjoint operators in (Ce, || • ||ce). We show that in the case when (Ce, || • ||ce) is a separable or perfect Banach symmetric ideal (Ce = C2) any skew-Hermitian operator H : CE ^ CE has the following form H(x) = i(xa — ax) for same a* = a £ B(H) and for all x £ CE. Using this description of skew-Hermitian operators, we obtain the following general form of surjective linear isometries V : CE ^ CE. Let (Ce, || • ||ce ) be a separable or a perfect Banach symmetric ideal with not uniform norm, that is ||p||cE > 1 for any finite dimensional projection p £ CE with dimp(H) > 1, let CE = C2, and let V : Ce ^ Ce be a surjective linear isometry. Then there exists unitary or anti-unitary operator u on H such that V (x) = uxu* or V (x) = —uxu* for all x £ Ce.
Key words: symmetric ideal of compact operators, skew-Hermitian operator, isometry. Mathematical Subject Classification (2010): 46L52, 46B04.
For citation: Aminov, B. R. and Chilin, V. I. Isometries of Real Subspaces of Self-Adjoint Operators in Banach Symmetric Ideals, Vladikavkaz Math. J., 2019, vol. 21, pp. 11-24. DOI: 10.23671/VNC.2019.21. 44607.
1. Introduction
The study of linear isometries on classical Banach spaces was initiated by S. Banach. In [1, Ch. XI], he described all isometries on the space Lp[0,1] with p = 2. In [2], J. Lamperti characterized all linear isometries on the Lp-space Lp(Q, where (Q, is a measure
space with a complete ^-finite measure p,. Both S. Banach and J. Lamberti used a method for description of linear isometries on Lp-spaces that was independent of the choice of a scalar field. For studying linear isometries on the broader class of function symmetric spaces E(Q, A,p), different approaches are required that depend on a scalar field. If E(Q, A,p) is a complex symmetric space then G. Lumer's method [3] based on the theory of Hermitian operators can be effectively applied. For example, M. G. Zaidenberg [4, 5] used this method for description of all surjective linear isometries on the complex symmetric space E(Q, A, p), where p is a continuous measure. For the symmetric space E = E(0,1) of real-valued
© 2019 Aminov, B. R. and Chilin, V. I.
measurable functions on the segment [0,1] with a Lebesgue measure where E is a separable space or has the Fatou property, a description of surjective linear isometries on E was given by N. J. Kalton and B. Randrianantoanina [6]. They used methods of the theory of positive numerical operators. For real symmetric sequence spaces, a general form of surjective linear isometries was described by M. Sh. Braverman and E. M. Semenov [7, 8]. They used methods based on the theory of finite groups. For complex separable symmetric sequence spaces (symmetric sequence spaces with the Fatou property), a general form of surjective linear isometries was described in [9] (respectively, in [10]).
Naturally, the next step is to describe surjective linear isometries in the noncommutative situation, when symmetric sequence spaces are replaced by symmetric ideals of compact operators.
Assume (H, (■, ■)) is an infinite-dimensional complex separable Hilbert space. Let B(H) (respectively, K(H)) be the C*-algebra of all bounded (respectively, compact) linear operators on H. For a compact operator x € K(H), we denote by ¡(x) := {¡(n,x)}the singular value sequence of x, that is, the decreasing rearrangement of the eigenvalue sequence of \x\ = (x*x)z. We let Tr denote the standard trace on B(H). For p € [1, oo) (p = oo), we let
Cp := ix € K(H) : Tr(|x|p)
:= {x € K(H) : Tr(|x|p) < ro} (respectively, Cx = K(H))
denote the p-th Schatten ideal of B(H), with the norm
ll^llp •— Tr^icl^Jp (respectively, ||x||oo :— sup |i(n,x)|).
In 1975, J. Arazy [11], [12, Ch. 11, § 2, Theorem 11.2.5] gave the following description of all the surjective isometries of Schatten ideals Cp.
Theorem 1. Let V : Cp — Cp, 1 ^ p ^ to, p = 2, be an surjective isometry. Then there exist unitary operators u1 and u2 or anti-unitary operators v1 and v2 on H such that either Vx = u1xu2 or Vx = v1x*v2 for all x € Cp.
Recall that a mapping v : H - H is an anti-unitary operator if
v{\h + f)=Mh)+v{f) and \\v{h)\\H = \\h\\u
for every complex number A and h, f € H. If v is an anti-unitary operator then there exists an anti-unitary operator v* such that (h,v(f)) = (f, v*(h)) for all h, f € H (see, for example, [12, Ch. 11, §2]).
The Schatten ideals Cp are examples of Banach symmetric ideals (CE, || ■ \\cE) of compact operators associated with symmetric sequence spaces (E, || ■ ||E) (see Section 2.2 below). In 1981 A. Sourour [13] proved a version of Theorem 1 for separable Banach symmetric ideal (CE, || ■ ||ce) such that CE = C2. Recently [14], a variant of Theorem 1 was obtained for any perfect Banach symmetric ideals (CE, || ■ 11cE), CE = C2 (recall that (CE, || ■ 11cE) is a perfect ideals, if Ce = C^x [15] (see Section 2.2 below)).
It is clear that for any unitary or anti-unitary operator u the linear operators V1(x) = uxu* and V2(x) = -uxu* acting in a real Banach space (Ch, || ■ 11cE) are surjective isometries, where Ch = {x €ce : x = x*}.
Our main result states that if (CE, || ■ 11cE) is a separable or a perfect Banach symmetric ideal of compact operators such that CE = C2, there are no other isometries in (Ch, || ■ ||ce):
Theorem 2. Let (CE, || ■ |ce) be a separable or a perfect Banach symmetric ideal with not uniform norm,
Ce = C2, and let V: Ch — Ch be a surjective isometry. Then there exists unitary or anti-unitary operator u on H such that V can be written in the form V(x) = uxu* (x € Ch) or in the form V(x) = -uxu* (x € Ch).
An analogous result for the space of self-adjoint traceless operators on a finite dimensionalal Hilbert space was obtained by G. Nagy [16].
2. Preliminaries
2.1. Symmetric Sequence Spaces. Let (respectively, co) be the Banach lattice of all
bounded (respectively, converging to zero) sequences of real numbers with respect
to the uniform norm = supneN |£n|, where N is the set of natural numbers. If 2N
is the ^-algebra of all subsets of N and p({n}) = 1 for each n € N, then (N, 2N, p) is a ^-finite measure space, Lœ(N, 2N, p) =
Li(N, 2n,p) = h = i {£ra}£=i C R : ||{£„}||i = Y, < ~
k n=1 )
where R is the field of real numbers. If £ = {£n}^=1 € then the non-increasing rearrangement £* : (0, to) — (0, to) of £ is defined by
£*(t) = inf{A : p({|£| > A}) < t}, t> 0,
(see, for example, [17, Ch. 2, Definition 1.5]).
Therefore the non-increasing rearrangement £* is identified with the sequence £* = {£n}, where
£n = if sup |£ra|.
FCN, n/F
card(F )<n
A non-zero linear subspace E C with a Banach norm || ■ ||E is called symmetric sequence space if conditions n € E, £ € £* ^ n* imply that £ € E and ||£||E ^ ||n||E.
If (E, || ■ ||E) is a symmetric sequence space, then l1 c E c l^, in addition, ||£||E ^ ||£|1 for all £ € I1 and ||£||^ < ||£|e for all £ € E [17, Ch. 2, §6, Theorem 6.6]. If there exists £ € (E \ co) then £* ^ a1 for some a > 0, and therefore 1 € E, where 1 = {1,1,...}. Consequently, for any symmetric sequence space E we have that E C c0 or E = l^.
2.2. Banach Symmetric Ideal of Compact Operators. Let (H, (■, )) be an infinite-dimensional complex separable Hilbert space, let B(H) (respectively, K(H), F(H)) be the *-algebra of all bounded (respectively, compact, finite rank) linear operators in H, and let P(H) = {p € B(H) : p = p* = p2}. It is known that *-algebras B(H) and K(H) are C*-algebras with respect to the uniform operator norm, which we shall denote by || ■ For a subset A C B(H), we set Ah = {x € A : x = x*}.
It is well known that F(H) c I c K(H) for any proper two-sided ideal I in B(H) (see for example, [18, Proposition 2.1]).
If (E, || ■ ||E) C c0 is a symmetric sequence space, then the set
Ce := {x € K(H) : {p(n,x)}~=1 € E}
is a proper two-sided ideal in B(H) (see [18, Theorem 2.5]). In addition, (CE, ||-||cE) is a Banach space with respect to the norm ||x||cE = ||{p(n,x)}||E [19] (see also [20, Ch. 3, §3.5]), and the norm || ■ ||Ce has the following properties:
1) ||xzy||ce ^ ||x|U||y|U||z||ce for all x,y € B(H) and z € Ce;
2) ||x||CE = ||x||^ if x € F(H) is of rank 1.
In this case we say that (CE, || ■ ||ce) is a Banach symmetric ideal (cf. [18, Ch. 1, §1.7], [21, Ch. III]). It is known that Ci cCE C K(H) and ||x||ce ^ ||x|i, ||y|U ^ ||y||cE for all x € Ci, y €ce .
If (E, || ■ ||e) is a symmetric sequence space (respectively, (CE, || ■ ||ce) is a Banach symmetric ideal), then the Kothe dual Ex (respectively, CX) is defined as
Ex = {£ = {£„}~i € l^ : £n = {£nnn}~=i € li for all n = {nn}~=i € e}, ^respectively, CX = |x € B(H) : xy € Ci for all y € CE jj,
||£||e* = sup{ ^ |e„nn| : n = {nn}~=i € E, ||n||E < A, £ € Ex,
and
^respectively, ||x||Cx = sup{Tr(|xy|) : y € Ce, ||y||cB ^ l}, x € C^.
It is known that (Ex, || ■ ||Ex) is a symmetric sequence space [22, Ch. II, §4, Theorems 4.3, 4.9] and IX = 1те. In addition, if E = li then Ex с co. Therefore, if E = lb the space (CX, || ■ ||Cx) is a symmetric ideal of compact operators.
A Banach symmetric ideal (CE, || ■ ||ce) is said to be perfect if CE = CXX (see, for example, [15]). It is clear that CE is perfect if and only if E = Exx.
A symmetric sequence space (E, || ■ ||E) (a Banach symmetric ideal (CE, || ■ ||ce)) is said to possess Fatou property if the conditions
0 ^ £k ^ £k+b £k € E (respectively, 0 ^ xk ^ xk+1, xk € CE) for all k € N
and supfc^1 ||£k ||e < oo (respectively, supfc^1 ||xk ||ce < oo) imply that there exists an element £ € E (respectively, x € CE) such that £k t £ and ||£||E = supk^1 ||£k||E (respectively, xk t x and ||x|ce = supk^1 ||xk||cb).
It is known that (E, || ■ ||E) (respectively, (CE, || ■ ||ce)) has the Fatou property if and only if E = Exx [23, Vol.II, Ch. 1, Section a] (respectively, ce = CXX [24, Theorem 5.14]). Therefore (CE, || ■ ||ce) is a perfect Banach symmetric ideal if and only if (CE, || ■ ||ce) has the Fatou property.
If y € Cx, then a linear functional fy(x) = Tt(x■ y), x € CE, is continuous on (CE, || ■ ||ce), in addition, ||fy = ||y |Cx, where (C|,, || ■ ||ce) is the dual of the Banach space (CE, || ■ ||ce)
(see, for example, [15]). Identifying an element y € Cx and the linear functional fy, we may assume that Cx is a closed linear subspace in C^. Since F(H) с Cx, it follows that Cx is a total subspace in C|,, that is, the conditions x € CE, f (x) = 0 for all f € Cx imply x = 0. Thus, the weak topology ct(ce, Cx) is a Hausdorff topology, in addition F(H) (respectively, F(H)h) is ct(ce,Cx)-dense in CE (respectively, C^).
3. Skew-Hermitian Operators in Banach Symmetric Ideals
Let X be a linear space over the field K of real or complex numbers. A semi-inner product on a space X is a K-valued form [■, ■]: X x X — K which satisfies
(i) [ax + y, z] = a ■ [x, z] + [y, z] for all a € K and x, y, z € X;
(ii) [x, ay] =a-[x, y] for all a € K and x, y € X;
(iii) [x, x] ^ 0 for all x € X and [x, x] = 0 implies that x = 0;
(iv) | [x, y] |2 ^ [x,x] ■ [y,y] for all x,y € X
(see, for example, [25, Ch. 2, §1]).
The function ||x|| = y/[x, x] is the norm on a linear space X. Conversely, if (X, || • ||x) is a normed linear space, then there exists semi-inner product [•, •] on X compatible with the norm || • ||x, that is, = y/[x, x] [25, Ch. 2, § 1]. In particular, the semi-inner product (compatible with the norm || ■ ||X) can be defined using the equation [x,y] = ^y(x), where <£y € X*, ||^>y||x* = ||y||x and ^>y(y) = ||y||X (such functional is called a support functional at y € X) [25, Ch. 2, §1, Theorem 10].
Let (X, || ■ ||x) be Banach space over field K, and let [■, ■] be a semi-inner product on X which is compatible with the norm || ■ ||X. A linear bounded operator H: X — X is said to be skew-Hermitian, if Re([H(x),x]) = 0 for all x € X, where Re(a) is the real part of number a € K [12, Ch. 9, §4]. In particular, if K = R then (H(x)) = [H(x),x] = 0 for every x € X.
The following Proposition is well known [12, Ch. 9, §4, Proposition 9.4.2].
Proposition 1. Let (X, || ■ ||X) be a real Banach space and let H be a skew-Hermitian operator on X .If V: X — X is a surjective isometry then an operator V ■ H ■ V-1 is a skew-Hermitian.
It is clear that in the case (X, || ■ ||X) = (Cg, || ■ ||ce) every linear operator H : Cg — Cg defined by H(x) = i(xa — ax),x € Cg, where a € B(H)h, i2 = —1 is a skew-Hermitian operator.
The following Theorem gives a description of skew-Hermitian operators acting on Cg when CE is a separable or perfect Banach symmetric ideal other than C2.
Theorem 3. Let (CE, || ■ ||ce) be a separable or perfect Banach symmetric ideal, and let CE = C2. Then for any skew-Hermitian operator H : Cg — Cg there exists a € B(H)e such that H(x) = i(xa — ax) for all x € Cg.
< We slightly modify the original proof of Sourour [13]. For vectors £,n € H, denote by £®n the rank one operator on H given (£®n)(h) = (h, n)£, h € H. It is easily seen (x, £®n) := Tr((n ® £) ■ x) = (x(n), £) for any x € B(H)e and £, n € H. If y = £ ® £, ||£||h = 1, then y is an one dimensional projection on H and ||y||cE = ||y||^ = 1. Thus for a linear functional fy(x) := (x,y) = Tr(y*x),x € Cg, we have that fy(y) = Tr(y2) = Tr(y) = (£,£) = 1 = ||y||£E. In addition, if x € Cg and ||x||ce ^ 1 then |fy(x)| = |Tr(yx)| = |(x(£),£)| ^ ||x|^ ^ ||x|ce ^ 1. Consequently, ||fy|(ce)* = 1 = ||y|cE. This means that fy is a support functional at y € Cg,
and [x,y] = fy(x) is a semi-inner product on Cg compatible with the norm || ■ [25, Ch. 2, §1, Theorem 10].
Step 1. If £, n € H, (n, £) = 0, then (H(n ® n), £ ® £) = 0.
We can assume that ||n||w = ||£|h = 1. Since p = n ® n is one dimensional projections and H is a skew-Hermitian operator, it follows that
0 = [H (p),p] = fp(H (p)) = (H (p),p). (1)
By Lemma 9.2.7 ([12, Ch. 9, §9.2], see also the proof of Lemma 11.3.2 [12, Ch. 9, §11.3]), there exists a vector £ = {£1,£2} € (R2, || ■ ||E), £1 > 0,£2 > 0, ||£||e = 1, such that the functional f ({n1,n2}) = n1£1 + n2£2, {n1,n2}€ R2, is a support functional at £ for space (R2, || ■ |E).
Let us show that the linear functional
^(y) = (y, x), y € CE, x = £1p + £2q, is a support functional at x for (Cg, || ■ ||ce ).
Since f is support functional at £ for (R2, || ■ ||E) and ||£||e = 1, it follows that £2 + £2 = f({£i,£2}) = f(£) = ||£NE = 1. Furthermore, by ||f || = ||£||e = 1, we have that |f({ni,n2})| = |£ini + £2n2| ^ 1 for every {ni,%} € R2 with ||{ni,n2}||e ^ 1. Further, by [21, Ch. II, §4, Lemma 4.1], we have
l(y(n),n)l < ^(1,y), l(y(£),£)l < M1,y), l(y(n),n)l + |(y(£),£)| < M1,y) + M2,y),
that is, {(y(n),n), (y(£), £)} {^(1,y),^(2,y)}. Since (E, || ■ ||E) is a fully symmetric sequence space, it follows that
||{(y(n), n), (y(£),£)}||E < ||{^(1,y),^(2,y)}|E < ||y•
Consequently, if y € CE and 11y |^ 1, then
№(y)l = Ky,x)| = |£iTr(py) + £2Tr(qy)| = |f ({(y(n),n), (y(£),£)})| < 1, that is, |, IHI-e)* ^ 1. Since ||x||cE = ||£||E = 1 and
<p(x) = (x, x) = (£ip + £25, £ip + £2?) = Tr(£ip + £2q)(£ip + £2?) = £2 + £2 = 1,
it follows that |M|(Ce )* = 1 = ||x|and <^(x) = ||x|. This means that ^ is a support functional at x for space (Ch, || ■ ||ce ). Hence,
0 = [H (x), x] = <^(H (x)) = <H (x), x> = <£iH (p) + £2H (q),£i p + £25).
Since (H(p),p) = (H(q),q) = 0 (see (1)), it follows that
(H (p),q) = -<H (q),p>. (2)
We extend ni = n, n2 = £ up to an orthonormal basis {ni}|=i, and let p» = n ® n». Now we replace our operator H with another skew-Hermitian operator Ho. Let u be a unitary operator such that u(ni) = n2, u(n2) = ni and u(nk) = nfc if k = 1,2. It is clear that u* = u-i = u, upiu = p2, up2u = pi, up»u = p», i = 1,2, and V(x) = uxu* = uxu is an surjective isometry on Ch, in addition, V-i = V.
By Proposition 1, a linear operator Hi = VHV-i is a skew-Hermitian operator, in particular, (Hi(pk),pk) = 0 for all k € N (see (1)). If i, j = 1,2, then
(Hi(p»),p^ = (uH(p»)u,p^ = Tr(pjuH(p»)u) = (uH(pi)u(nj), nj)
= (H (pj )u(nj ),u* (nj)) = (H (pi)(nj ),nj) = Tr(pj H (p»)) = (H (pj),pj >.
If i = 1, j = 1, 2, then
(Hi(pi),pj) = (uH (p2)u,pj) = Tr(pj uH (p2)u) = (uH (p2)u(nj ),nj)
= (H(p2)u(nj),u*(nj)) = (H(p2)(nj),nj) = Tr(pjH(p2)) = (H(p2), pj)•
Similarly, we get the following equalities
(i) (Hi(p2),pj) = (H(pi),pj) if i = 2, j = 1, 2;
(ii) (Hi(p»),pi) = (H(pj),p2) if j = 1, i = 1, 2;
(iii) (Hi(pi),p2> = <H(p2),Pi) if i = 1, j = 2;
(iv) <Hi(p2),Pi) = <H(pi),p2) if i = 2, j = 1.
It is clear that Ho = \{H — H\) is a skew-Hermitian operator, and if i,j / 1,2, then (Ho(pi),Pj) = \((H{pi),pj) - (Hi(pi),pj)) = 0. Similarly, if i = 1, j / 1,2 (respectively, i = 2, j = 1,2) we get
(Ho{Vl),Vj) = \((H{p 1),p3) - (H{p2),pj))
(respectively, (H0(p2),pj) = ^({H(p2),pj) - (H{pi),pj))),
that is, <Ho(pi),Pj) + <H0(p2),pj) = 0 in the case j = 1,2. Similarly, <H0(pj),pi) + <H0(pj),p2) =0 if j = 1, 2. Since
(Ho(Pl),P2) = \((H(Pl),p2) - (H(P2),Pl)), (H(Pl),P2) = -(H(p2),pi)
(see (2)), it follows that <H0(pi),p2) = <H(pi),p2). Similarly, we get that <H0(p2),pi) = — <H(pi),p2). Finally, since H0 is a skew-Hermitian operator, we have <H0(pk),pk) = 0 for all
k € N (see (1)).
Let n be the smallest natural number such that the norm || ■ ||E is not Euclidian on Rn. Then there exist (see, [10, Lemma 5.4]) linear independent vectors £ = ... ,£n), n =
(ni, n2,..., nn) € ||£||e = 1, such that
ii£iie = If IE = fn (£) = 1, (3)
where fn(Z) = Y^i=i Z^, Z = (Zb Z2, • • •, Zn) € Rn. By rearranging the coordinates we may assume that £in2 = £2ni.
Let x = ¿n=i , y = Yjj=i njPj, and let ^y(z) = <z y) = E"=i nj ■ Tr(pjz) z € cl. Let us show that ^>y is a support functional at x for , || ■ ||E). Since ||fn||E* = 1 (see (3)), it follows that |fn(Z)| = | E"=i niZil < 1 for every Z = {Zi}^ € Rn with ||Ziie < 1. Note that
iixice = ii£iie = 1.
We should show that ||^>y || = ||x||cE = 1 and ^>y(x) = ||x||CE = 1. Indeed,
In n \ n
^y(x) = <x,y) = ('52 Pj^Pj } =Y nj = fn(£) = 1 = ||x||CE. \j=i j=i / j=i
If z € Ch, ||z||cE ^ 1 then |^y(z)| = |£n=i nj(z(nj),nj)| ^ 1. The last inequality follows from
{fcto nl), (z(n2), n2),..., (z(nn), n^} ^ M1, M2, z),... , z) }
(see [21, Ch. II, §4, Lemma 4.1]). Therefore ||^y|| = ||x|cE = 1 and ^>y(x) = ||x||CE = 1. This means that ^>y is a support functional at x for (C^,, || ■ ||E). Consequently,
0 = (H0(x),y) = (£iH0(pi) + ... + £nH0(pn),niPi + ... + nnPn) = (£in2 — 6ni)( H0(pi),p^ + (£ins — 6ns)( H0(pi),pa>
+ ... + (£inn — 6nn)( H0(pi),Pn) + (6ni — &%)( H0(pa),pi> + ... + (£nni — £nnaK H0(p n), pi .
Let now x = £ipi + £2p2 - £3p3 - • • • - £rap«. and y = nipi + n2p2 - n3p3 - • • • - n«p«. As above, we have that = (-,y) is a support functional at x. Consequently,
0 = (Ho(x),y> = (£in2 - £2ni)(Ho(pi),p^ + (-£in3 + £2n3)(Ho(pi),p3)
+ • • • + (-£inn + £2n«K Ho(pi),p^ + (-£3ni + £3naX Ho(p3),p0 (5)
+ • • • + (-£nni + £nna)( Ho(p ra^ pi / •
Summing (4) and (5) we obtain 2(£in2 - £2ni)(Ho(pi),p2) = 0, that is, (H(pi),p2) = (Ho(pi),p2) = 0.
Step 2. Let n € H, ||n||w = 1, p = n ® n, x € K(H)h, and let Tr(xq) = 0 for any one dimensional projection q with qp = 0. Then there exists f € H such that x = n ® f + f ® n -
(n ® n)(f ® n), ||f ||h < ||x|^.
Indeed, if q is an one dimensional projection with qp = 0 then qxq = aq for some a € R, and 0 = Tr(xq) = Tr(qxq) = Tr(aq) = a, that is, a = 0 and qxq = 0. Let e € P(H), dim e(H) = 1, ep = 0, eq = 0, y = (q + e)x(q + e). If y = 0 then there exists r € P(H), dim r(H) = 1 such that r ^ q + e and rxr = ryr = ^r for some 0 = ft € R. Since rp = 0, it follows that 0 = Tr(xr) = Tr(rxr) = ft = 0. Thus y = 0. Continuing this process, we construct a sequence of finite-dimensional projections t (I - p) such that gnxgn = 0 for all n € N, where I(h) = h, h € H. Consequently, (I - p)x(1 - p) = 0. If f = x(n) then xp = f ® n and px = n ® f .In addition,
(I - p)xp(h) = (I - p)x((h, n)n)) = (h, n)(1 - p)f, h € H,
that is, (I - p)xp = (I - p)f ® n. Therefore,
x = px + (I - p)xp = n ® f + (I - p)f ® n and ||f ||h ^ ||x|^•
Step 3. Let n € H, ||n||w = 1, p = n ® n. Then there exists f € H such that
h(n<g> n) = n® f + f ® n, ||f ||h < ||h|
Indeed, if x = H(n ® n), £ € H, (n, £) =0, q = £ ® £, then by Step 1 we obtain that (x(£),£) = (x, £ ® £) = Tr(x ■ £ ® £) = 0. Using Step 2, we have that there exists f € H such that H(n ® n) = x = n ® f + f ® n - (n ® n)(f ® n). Since H is a skew-Hermitian operator, it follows that
0 = (H(n ® n), n ® n) = (n ® f + f ® n - (n ® n)(f ® n), n ® n) = Tr((n ® nKn ® f + f ® n - (n ® n)(f ® n))) = ir((n ® n)(n ® f )) = ((n ® f )(n), n) = (n, f )•
Thus (n, f) = 0 and x = n ® f + f ® n - (n ® n)(f ® n) = n ® f + f ® n. In addition, |h < ||x|^ < ||x||cB = ||H(n ® n)|cE < ||H|| ■ ||n ® n|cE = ||H|| ■ ||n ® n|U = ||H|
i
Step 4. There exists a € B(H) such that H(x) = ax + xa* for every x € Ch.
Let {pj}°=i = {n ® n»}ro=i be a basis in real linear space F(H)h, where {nj}i= is an orthonormal basis of H. For every n € H there exists f € H such that H(n ® n») = n»® f + f ®n», and ||fj|H ^ ||H|| for all i € N (see Step 3). Define a linear operator a: H — H setting a(n») = f». Since 11f»|H ^ ||H|| for all i € N, it follows that a € B(H), in addition,
H(pj) = n» ® a(n») + a(n») ® n». Since n» ® a(n») = (n ® n»)a* and a(n») ® n» = a(n» ® n»), it follows that H(x) = ax + xa* for all x € F(H)h.
If (CE, || ■ ||ce) is a separable space then F(H)h is dense in (Ch, || ■ ||ce). Consequently, H(x) = ax + xa* for all x € Ch.
Let now (CE, || ■ ||ce) be a perfect Banach symmetric ideal. Repeating the proof of Theorem 4.4 [14] that establishes the ct(ce, C^)-continuity of the Hermitian operators acting in (CE, || ■ ||ce), we obtain that the skew-Hermitian operator H also , (CE)h)-continuous. Since the space F(H)h is a(Ch, )-dense in Ch, it follows that H(x) = ax + xa* for all x €Ch.
Step 5. a = ib for some b € B(H)h. Indeed, if a = ai + ia2, ai, a2 € B(H)h, then
H(x) = ax + xa* = aix + xai + i(a2x - xa2) = Si(xi) + S2(x),
where Si(x) = aix + xai, S2(x) = i(a2x - xa2), x € Ch. Since H and S2 are skew-Hermitian, it follows that Si = H - S2 is also skew-Hermitian.
If p € P(H), dim p(H) = 1, then the lineal functional <^(y) = (y,p) = Tr(yp), y € Ch, is support functional at p. Thus Tr(paip + pai) = Tr(Si(p)p) = 0, that is, -Tr(pai) = Tr(paip) = Tr(pai)^ This means that Tr(pai) = 0 for all p € P(H) with dimp(H) = 1. Consequently, Tr(xai) = 0 for all x € F(H), and by [26, Lemma 2.1] we have ai = 0. Therefore, a = ia2. >
4. The Proof of Theorem 2
Let (CE, || ■ ||ce) be a Banach symmetric ideal. We say that a bounded linear operator T: Ch — Ch has the property (P) if for any a € B(H)h there are operators b € B(H)h and c € B(H)h such that T(i(bx - xb)) = i(aT(x) - T(x)a) and T(i(ax - xa)) = i(cT(x) - T(x)c) for all x € CEh.
It is clear that a bounded linear bijection T: Ch — Ch has the property (P) if and only if T-i has the property (P).
Lemma 1. Let (CE, || ■ ||ce) be a separable or a perfect Banach symmetric ideal other than C2, and let V : Ch — Ch be a surjective isometry. Then an isometry V has the property (P).
< If a € B(H)h then the linear operator H : Ch — Ch defined by H(x) = i(xa - ax), x € Ch, is a skew-Hermitian operator. By the Proposition 1 the operator V-i ■ H ■ V is also skew-Hermitian. Using the Theorem 3 we obtain that there exists b € B(H)h such that V-i ■ H ■ V(x) = i(bx - xb), that is, i(aV(x) - V(x)a) = V(i(bx - xb)) for all x € Ch•
Similarly, V ■ H ■ V-i is a skew-Hermitian operator. Hence, there exists an operator c € B(H)h such that V H -V-i(y) = i(cy - yc) for all y € Ch. If V-i(y) = x, then V (i(ax - xa)) = i(cV(x) - V(x)c) for all x € Ch. >
Let (CE, || ■ ||ce) be a Banach symmetric ideal, 0 = x € Ch, and let Z(x) = {x}' nB(H)h = {y € B(H)h : xy = yx}- A non-zero operator x € Ch is said to be a Ch-maximal if Z(x) = Z(y) for any 0 = y € Ch with Z(x) C Z(y) (cf. [27, Definition 1.4]).
Lemma 2. The following conditions are equivalent:
(i) x € Ch is a Ch-maximal operator;
(ii) x = ap, where 0 = p € P(H) n F(H), 0 = a € R.
< (i) (ii). Since x € C^, it follows that x = ^t=i Aipi, t € N or t = to (the series converges with respect to the norm || ■ ||^), where 0 = pi € P(H) n F(H), pipj = 0, i = j, 0 = Ai € R, for all i,j = 1,...,t. If y € Z(x) then ypi = piy [28, Ch. 1, §4, p. 17], that is, Z(x) C Z(pi) for all i = 1,...,t. Since, x is a C|—maximal operator, it follows that Z(x) = Z(pi), thus Z(pi) = Z(pfc) for all i, k = 1,..., t.
Suppose that t ^ 2. As Z(pi) = Z(p2), we have
{Pi}'' = W = {a ■ P2 + £ ■ (I — P2) : a,£ € C},
that is, pi = a0 ■ p2 + £0 ■ (/ — p2) for some a0, £0 € C. Consequently, 0 = pip2 = a0 ■ p2, and a0 = 0. Therefore pi = £0 ■ (/ — p2), which contradicts the inclusion pi € F(H). Thus t = 1 and x = Aipi.
(ii) (i). Let x = ap, where 0 = p € P(H) n F(H), 0 = a € R. If 0 = y € C| and Z(x) C Z(y) then Z(p) = Z(x) C Z(y), and y € {y}'' C {p}'' = {a■ p + £ ■ (I — p) : a,£ € C}, that is, y = a0 ■ p + £0 ■ (/ — p) for some a0, £0 € C. Since y is a compact operator, it follows that £0 = 0, that is, y = a0 ■ p and Z(x) = Z(y). >
Lemma 3. Let T: C| — C| be a bounded linear bijective operator with the property (P). Then T(x) is a C^-maximal operator for any C^-maximal operator x € C^.
< Suppose that x € C^ is a C^-maximal operator, but T(x) is not C^-maximal, that is, there exists z € C| such that Z(T(x)) C Z(z) and Z(T(x)) = Z(z). Since T is a bijection, z = T(y) for some y € C|. Hence, Z(T(x)) C Z(T(y)) and Z(T(x)) = Z(T(y)).
We show that Z(x) C Z(y). Since an operator T has property (P), it follows that for a € Z(x) there exists b € B(H)h such that
T (i(ac — ca)) = i(bT (c) — T (c)b) (6)
for all c € C|. Using equations (6) and T(i(ax — xa)) = T(0) = 0, and the injectivity of the mapping T, we obtain that bT(x) = T(x)b, that is, b € Z(T(x)) C Z(T(y)). Consequently, T(i(ay — ya)) = 0 and ay — ya = 0 (see (6)), i.e. a € Z(y). Therefore Z(x) C Z(y), and by the C^-maximality of the operator x we obtain that Z(x) = Z(y).
Since Z(T(x)) = Z(T(y)), there exists an operator a € Z(T(y)) such that a / Z(T(x)). By the property (P) we can choose b € B(H)h such that
T (i(bc — cb)) = i(aT (c) — T (c)a) (7)
for all c € C|. Thus T(i(by — yb)) = 0, and by — yb = 0, that is, b € Z(y). Besides, aT(x) — T(x)a = 0 implies that bx — xb = 0 (see (7)), that is, b / Z(x), which contradicts the equality Z(x) = Z (y). >
Lemma 4. Let V : C| — C| be a surjective linear isometry with the property (P). Then for every p € P(H) n F(H)) there exists qp € P(H) n F(H) such that V(p) = qp or V (p) = — qp.
< Let 0 = pi € P(H) n F(H), i = 1,2, pip2 = 0. Since pi is a C|-maximal operator (Lemma 2), it follows that V(pi) is a C^-maximal operator too, i = 1,2 (Lemma 3). Consequently, there exist 0 = qi € P(H) n F(H), and 0 = ai € R such that V(pi) = aiqi, i = 1,2 (Lemma 2). Since pip2 = 0, it follows that (pi + p2) € P(H) n F(H) and V(pi + p2) = a3q3 for some non-zero projection q3 € P(H)nF(H) and 0 = a3 € R (Lemma 2). Therefore + = <?3- By [29] there are four possibilities:
« fi = it = 1 if QIQ2 = 0;
(11) fi = 1, || = -1 if qlQ2 = q2]
(m) St = _1' Sf = 1 and 9192 = qi]
(iv) it + St = 1 and (91 - <&)2 = 0 if qiq2 + q2qi.
The case (iv) is impossible because ||(qi - q2)||ro = 11(qi - q2)2||ro = 0, which contradicts the bijectivity of V. In other cases we have V(p2) = aq2 or V(p2) = -aq2, where a = ai. Consequently, V(p) = aqp or V(p) = -aqp for an arbitrary 0 = p € P(H) n F(H), pip = 0^
Let now 0 = e € P(H) n F(H) and pie = 0. Then there exists a non-zero finite dimensional projection f, such that pif = 0 and ef = 0. According to above, we have aiqi = V(pi) = a/qpi or V(pi) = -a/qpi and V(e) = a/qe or V(e) = -a/qe for some non-zero finite dimensional projections q/, qe and for non-zero real number a/. Consequently, qi = qp1 and ai = ±a/. In particular, V(e) = aiqe or V(f) = -aiqe.
If e € P(H) and dime(H) = 1, then 1 = ||e||cB = ||V(e)|cB = |a|||qe||ce ^ |a||qe|ro = |a|, that is, |a| ^ 1.
Replacing the isometry V with V-i, we get that V-i(p) = ^rp or V-i(p) = -^rp for arbitrary p € P(H)nF(H), where rp € P(H)nF(H) and ft does not depend on the projection p. In particular, if e € P(H) n F(H) and dime(H) = 1, then 1 = ||e||CE = ||V-i(e)||cE =
i^iNreNce ^ ^N^U = |£|, i.e. |£| < 1.
Therefore, for p € P(H) n F(H) we obtain that V(p) = ±aqp, and p = V-i(±aq) = ±(a^)rq• Hence |a^| = 1 and |a| = 1. >
We say that the norm || ■ ||ce is a not uniform if ||p||cE > 1 for any p € P(H) nF(H) with dim p(H) > 1.
Lemma 5. Let (Ce, || ■ ||cE) be a Banach symmetric ideal with not uniform norm, and let V: Ch — Ch be a surjective isometry with the property (P). Then V(p) or (-V)(p) is one dimensional projection for any one dimensional projection p.
< Let p € P(H) n F(H), dimp(H) = 1. By Lemma 4 we have that there exists qp € P(H) n F(H) such that V(p) = qp or V(p) = -qp. If dim qp(H) > 1 then 1 = ||p||cE = ||V(p)Nce = ||qp||cE > 1, what is wrong. >
Lemma 6. Let (CE, || ■ ||ce) and an isometry V be the same as in the conditions of the Lemma 5. Then
V(P(H) n F(H)) C P(H) n F(H)
or
(-V)(P(H) n F(H)) C P(H) n F(H)
< Let Pi(H) = {p € P(H) : dimp(H) = 1}, and let p,e € Pi(H). By Lemma 5, there exists q, r € P1(H) such that V(p) = q or V(p) = -q and V(e) = r or V(e) = -r. If V(p) = q, V(e) = -r then q - r = V(p + q) = ±f for some 0 = f € P(H) (see Lemma 4), which is not possible because q,r € Pi(H). Similarly, the case V(p) = -q, V(e) = r is also impossible. Consequently, V(Pi(H)) C Pi(H) or (-V)(P1(h)) C Pi(H). Since each projector p € P(H) n F(H) is the final sum of one-dimensional projectors, it follows that V(P(H) n F(H)) C P(H) n F(H) or (-V)(P(H) n F(H)) C P(H) n F(H). >
Corollary 1. Let (CE, || ■ ||cE) and V be the same as in the conditions of the Lemma 5. Then
(i) V(p) V(e) = 0 for any p, e € P(H) n F(H) with pe = 0;
(ii) V is a bijection from Pi(H) onto Pi(H).
< (i). By Lemma 5, V(p) = qp € P(H) n F(H) for all p € P(H) n F(H) or V(p) = -qp € P(H) n F(H) for all p € P(H) n F(H). In the first case for p, e € P(H) n F(H) with pe = 0, we have that V(p) = qp, V(e) = qp, qr + qe = V(r + e) = qr+e, that is, V(r)V(e) = qrqe = 0.
The case V(p) = -qp € P(H) n F(H) for all p € P(H) n F(H) is proved similarly. Item (ii) directly follows from Lemma 5. >
< Proof of Theorem 2. We suppose that V(P(H) nF(H)) c P(H) nF(H) (the case (-V)(P(H) n F(H)) c P(H) n F(H) is proved by replacing V with (-V)). Let
k
x = A™pn € F(H)h, p„ € Pi(H), p„pm = 0,
n=1
n = m, 0 = An € R, n,m = 1,...,k.
Since V(pn) ■ V(pm) =0, n = m (Corollary 1 (i)), it follows that
V(x2) = v( £ A^n) = £ A^V(pn) = V(x)2
\n=1 / n=1
and
k k Tr(V(x)) = £ A„Tr(V(p„)) = £ А„ = Tr(x).
n=1 n=1
If p, e, q, f € P1(H), V(p) = q, V(e) = f, then
2 Tr(pe) = Tr(pe) + Tr(ep) = Tr((p + e)2 - p - e)
= Tr(V((p + e)2)) - 2 = Tr(V(p + e))2 - 2 = Tr((q + f)2) - 2 = 2Tr(qf).
Consequently, Tr(pe) = Tr(V(p)V(e)) for all p, e € P1(H). By [30, Ch. 3, § 3.2, Theorem 3.2.8] we obtain that there exists an unitary or anti-unitary operator u such that V(p) = upu* for all p € P1(H) . Thus V(x) = u*xu for all x € F(H)h.
If (CE, || ■ ||ce) is a separable space then F(H)h is dense in (C^, || ■ ||ce). Consequently, V(x) = u*xu (respectively, V(x) = -uxu*) for all x € Cg.
If (CE, || ■ ||ce) is a perfect Banach symmetric ideal, then V is ct(ce, Cg)-continuous (see proof of Step 4 in Theorem 4). Since F(H)h is ct(ce, Cg)-dense in (Cg, || ■ ||ce), it follows that V(x) = u*xu (respectively, V(x) = -uxu*) for all x €Cg.
In the case (-V)(P(H) П F(H)) С P(H) П F(H) we get that V(x) = -uxu* for all x € Cg. >
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Received 13 June, 2019
Behzod R. Aminov National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan, Teacher
E-mail: aminovbehzod@gmail.com
Vladimir I. Chilin National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan Professor
E-mail: vladimirchil@gmail.com, chilin@ucd.uz
Владикавказский математический журнал 2019, Том 21, Выпуск 4, С. 11-24
ИЗОМЕТРИИ ДЕЙСТВИТЕЛЬНЫХ ПОДПРОСТРАНСТВ САМОСОПРЯЖЕННЫХ ОПЕРАТОРОВ В БАНАХОВЫХ СИММЕТРИЧНЫХ ИДЕАЛАХ
Аминов Б. Р.1, Чилин В. И.1
1 Национальный университет Узбекистана, Узбекистан, 100174, Ташкент, Вузгородок
E-mail: aminovbehzod@gmail. com, vladimirchil@gmail.com, chilin@ucd.uz
Аннотация. Пусть (Ce, || • ||ce ) банахов симметричный идеал компактных операторов, действующих в комплексном сепарабельном бесконечномерном гильбертовом H. Пусть = {x G CE : x = x*} действительное банахово подпространство самосопряженных операторов в (Ce, || • ||ce). Доказывается, что в случае, когда (Ce, || • ||ce ) есть сепарабельный или совершенный банахов симметричный идеал (Ce = C2) каждый косоэрмитовый оператор H : ^ имеет следующий вид H (x) = i(xa — ax) для некоторого a* = a G B(H) и для всех x G C%. Используя это описание косоэрмитовых операторов мы получаем следующий общий вид сюръективных линейных изометрий V : C % ^ C % : Пусть (Ce , || • ||ce ) сепарабельный или совершенный банахов симметричный идеал с неравномерной нормой, т. е. ||p||cE > 1 для всех конечномерных проекторов p G C e с dimp(H) > 1, пусть C e = C2, и пусть V : C % ^ C % сюръективная линейная изометрия. Тогда существует такой унитарный или антиунитарный оператор u на H, что V(x) = uxu* или V(x) = —uxu* для всех x G CE.
Ключевые слова: симметричный идеал компактных операторов, косоэрмитовый оператор, изо-метрия.
Mathematical Subject Classification (2010): 46L52, 46B04.
Образец цитирования: Aminov B. R., Chilin V. I. Isometries of Real Subspaces of Self-Adjoint Operators in Banach Symmetric Ideals // Владикавк. мат. журн.—2019.—Т. 21, № 4.—C. 11-24 (in English). DOI: 10.23671/VNC.2019.21.44607.
