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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: [email protected], [email protected], [email protected]
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. >
References
1. Banach, S. Theorie des Operations Lineaires, Warsaw, 1932.
2. Lamperti, J. On the Isometries of Some Function Spaces, Pacific Journal of Mathematics, 1958, vol. 8, no. 3, pp. 459-466. DOI: 10.2140/pjm.1958.8.459.
3. Lumer, G. On the Isometries of Reflexive Orlicz Spaces, Annales de l'Institut Fourier, 1963, vol. 13, no. 1, p. 99-109. DOI: 10.5802/aif.132.
4. Zaidenberg, M. G. On Isometric Classification of Symmetric Spaces, Doklady Akademii Nauk SSSR, 1977, vol. 234, pp. 283-286 (in Russian).
5. Zaidenberg, M. G. A Representation of Isometries of Functional Spaces, Zhurnal Matematicheskoi Fiziki, Analiza, Geometrii [Journal of Mathematical Physics, Analysis, Geometry], 1997, vol. 4, no. 3, pp. 339347.
6. Kalton, N. J. and Randrianantoanina, B. Surjective Isometries on Rearrangment Invariant Spaces, The Quarterly Journal of Mathematics, 1994, vol. 45, no. 3, pp. 301-327. DOI: 10.1093/qmath/45.3.301.
7. Braverman, M. Sh. and Semenov, E. M. Isometries on Symmetric Spaces, Doklady Akademii Nauk SSSR, 1974, vol. 217, pp. 257-259 (in Russian).
8. Braverman, M. Sh. and Semenov, E. M. Isometries on Symmetric Spaces, Trudy NII Matem. Voronezh. Gos. Univ., 1975, vol. 17, pp. 7-18 (in Russian).
9. Arazy, J. Isometries on Complex Symmetric Sequence Spaces, Mathematische Zeitschrift, 1985, vol. 188, no. 3, pp. 427-431. DOI: 10.1007/BF01159187.
10. Aminov, B. R. and Chilin, V. I. Isometries and Hermitian Operators on Complex Symmetric Sequence Spaces, Siberian Advances in Mathematics, 2017, vol. 27, no. 4, pp. 239-252. DOI: 10.3103/S105513 4417040022.
11. Arazy, J. The Isometries of Cp, Israel Journal of Mathematics, 1975, vol. 22, no. 3-4, pp. 247-256. DOI: 10.1007/BF02761592.
12. Fleming, R. J. and Jamison, J. E. Isometries on Banach Spaces: Vector-Valued Function Spaces, Chapman-Hall/CRC, 2008.
13. Sourour, A. Isometries of Norm Ideals of Compact Operators, Journal of Functional Analysis, 1981, vol. 43, no. 1, pp. 69-77. DOI: 10.1016/0022-1236(81)90038-0.
14. Aminov, B. R. and Chilin, V. I. Isometries of Perfect Norm Ideals of Compact Operators, Studia Math., 2018, vol. 241(1), pp. 87-99. DOI: 10.4064/sm170306-19-4.
15. Garling, D. J. H. On Ideals of Operators in Hilbert Space, Proceedings of the London Mathematical Society, 1967, vol. 17, no. 1, 115-138. DOI: 10.1112/plms/s3-17.1.115.
16. Nagy, G. Isometries of the Spaces of Self-Adjoint Traceless Operators, Linear Algebra and its Applications, 2015, vol. 484, pp. 1-12. DOI: 10.1016/j.laa.2015.06.026.
17. Bennett, C. and Sharpley, R. Interpolation of Operators, Academic Press Inc., 1988.
18. Simon, B. Trace Ideals and Their Applications, Mathematical Surveys and Monographs, vol. 120, 2nd edition, Providence, R. I., Amer. Math. Soc., 2005.
19. Kalton, N. J. and Sukochev, F. A. Symmetric Norms and Spaces of Operators, Journal für die Reine und Angewandte Mathematik, 2008, vol. 621, pp. 81-121. DOI: 10.1515/CRELLE.2008.059.
20. Lord, S., Sukochev, F. and Zanin, D. Singular Traces. Theory and Applications, Berlin/Boston, Walter de Gruyter GmbH, 2013.
21. Gohberg, I. C. and Krein, M. G. Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, vol. 18, Providence, R. I., Amer. Math. Soc., 1969.
22. Krein, M. G., Petunin, Ju. I. and Semenov, E. M. Interpolation of Linear Operators, Translations of Mathematical Monographs, vol. 54, Providence, R. I., Amer. Math. Soc., 1982.
23. Lindenstrauss, J. and Tzafriri, L. Classical Banach Spaces, Berlin and N. Y., Springer-Verlag, 1996.
24. Dodds, P. G., Dodds, T. K. and Pagter, B. Noncommutative Köthe Duality, Transactions of the American Mathematical Society, 1993, vol. 339, no. 2, pp. 717-750. DOI: 10.2307/2154295.
25. Dragomir, S. S. Semi-Inner Products and Applications, N. Y., Hauppauge, Nova Science Publishers Inc., 2004.
26. Ayupov, Sh. and Kudaybergenov, K. 2-Local Derivations and Automorphisms on B(H), Journal of Mathematical Analysis and Applications, 2012, vol. 395, no. 1, pp. 15-18. DOI: 10.1016/j.jmaa. 2012.04.064.
27. Dolinar, G., Guterman, A., Kuzma, B. and Oblak, P. Extremal Matrix Centralizers, Linear Algebra and its Applications, 2013, vol. 438, no. 7, pp. 2904-2910. DOI: 10.1016/j.laa.2012.12.010.
28. Schatten, R. Norm Ideals of Completely Continuous Operators, Berlin and N. Y., Springer-Verlag, 1960.
29. Baksalary, J. K. and Baksalary, O. M. Idempotency of Linear Combinations of Two Idempotent Matrices, Linear Algebra and its Applications, 2000, vol. 321, no. 1-3, pp. 3-7. DOI: 10.1016/S0024-3795(00)00225-1.
30. Bratteli, O. and Robinson, D. W. Operator Algebras and Quantum Statistical Mechaniks, N. Y.-Heidelber-Berlin, Springer-Verlag, 1979.
Received 13 June, 2019
Behzod R. Aminov National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan, Teacher
E-mail: [email protected]
Vladimir I. Chilin National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan Professor
E-mail: [email protected], [email protected]
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, [email protected], [email protected]
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.
