CC BY
4XAK: 517.98
MSC2010: 46E30, 37A30, 47A35
ERGODIC THEOREMS FOR FLOWS IN THE IDEALS OF COMPACT
OPERATORS © A. N. Azizov, V. I. Chilin
National University of Uzbekistan, 100174, Tashkent, Uzbekistan, e-mail: azizov.07@mail.ru, vladimirchil@gmail.com
ERGODIO THEOREMS FOR FLOWS IN THE IDEALS OF COMPACT OPERATORS. Azizov, A. N., Chilin, V.I.
Abstract. Let H be an infinite-dimensional complex Hilbert space, let (B(H), || • ||X) be the C*-algebra of all bounded linear operators acting in H, and let Ce be the symmetric ideal of compact operators in H generated by the fully symmetric sequence space E C cq. If Tt : B(H) ^ B(H), t > 0, is a semigroup of positive Dunford-Schwartz operators, which is strongly continuous on Ci, then the following versions of individual and mean ergodic theorems are true: For each x £ Ce the net At(x) = 1 JQ Ts(x)ds converges to some x £ Ce with respect to the norm || • ||X, as t ^ m; moreover, if E is separable and E = 11 (as a set), then lim \\At(x) - x\\vE = 0.
t^X
Keywords: Symmetric sequence space, Banach ideal of compact operators, Dunford-Schwartz operator, individual ergodic theorem, mean ergodic theorem.
Introduction
Let (B(H), || ■ ||X) be the C*-algebra of all bounded linear operators in a complex infinite-dimensional Hilbert space H. The study of noncommutative individual ergodic theorems in the space of measurable operators affiliated with a semifinite von Neumann algebra M C B(H) equipped with a faithful normal semifinite trace t was initiated by F. Yeadon. In [16], as a corollary of a noncommutative maximal ergodic inequality in L1 = L1(M , t), it was established that for any positive L1 — M - contraction T : L1 ^ L1
n
and every x G L1 there exists x G L1 such that the averages An(T)(x) = S Tk(x)
n k=0
converge to x bilaterally almost uniformly, that is, given £ > 0, there exists a projection e G M such that t(1 — e) < £ and ||e(An(T)(x) — ¡r)e||X ^ 0 as n ^ to, where 1 is the unit of M.
The study of individual ergodic theorems beyond L1 (M, t) started much later with another fundamental paper by M. Junge and Q. Xu [11], where, among other results, individual ergodic theorem was extended to the case with a positive Dunford-Schwartz
operator acting in the space LP(M,t), 1 < p < to. In [1] ([2]) an individual ergodic theorem was proved for a positive Dunford-Schwartz operator in a noncommutative Lorentz (respectively, Orlicz) space.
Advancing Lance's extension of the pointwise ergodic theorem for actions of the group of integers on von Neumann algebras, Conze and Dang-Ngoc [4] and Watanabe [15] studied continuous extensions of Lance's results. In particular, the noncommutative individual ergodic theorems were established for actions of the semigroups R+ and R+ respectively. The corresponding ergodic theorem for actions of R+ and with respect to bilaterally almost uniform convergence was initially considered by Junge and Xu [11]. In particular, they derived that these averages converge bilaterally almost uniformly in any noncommutative L^-space for 1 < p < to and almost uniformly if 2 < p < to.
Let H be a complex infinite-dimensional Hilbert space, and let c0 be the Banach space of converging to zero sequences of complex numbers. Every symmetric sequence space (E, || ■ ||E) C c0 generates a symmetric ideal of compact operators (CE, || ■ 11ce), acting in H, by the following rule (see, for example, [12, Chapter 3, Section 3.5]):
Ce = {x e K(H) : {sn(x)} e E}, ||x||ce = ||{sn(x)}|E,
where K(H) is the two-sided ideal of compact linear operators in B(H) and {sn(x)}miX1 is the set of eigenvalues of the compact operator |x| in the decreasing order.
Let {Tt}t>0 be a strongly continuous on C'i semigroup of Dunford-Schwartz operators (definition of Dunford-Schwartz operator see below, Section 1) and let
t
At(x) = 1 J Ts(x)ds, x e Ch ,t> 0, 0
be the corresponding ergodic averages. We show that the operators At extend to the Dunford-Schwarz operators, in particular, At (CE) C CE for any fully symmetric sequence space E C c0. Therefore, we can talk about the convergence of the averages At(x) as t ^ to.
We prove that for each x e CE the net At(x) converges to some x e CE with respect to the uniform norm || ■ as t ^ to (noncommutative version of the individual ergodic theorem). Besides, we show that in the case when (E, || ■ ||E) is separable space and CE = C\x as sets, the averages At(x) converge to x with respect to the norm || ■ |Ice as t ^ to (noncommutative version of the mean ergodic theorem). In conclusion, we give versions of the individual and mean ergodic theorems for the Orlicz and Lorentz ideals of compact operators.
1. Preliminaries
Let l™ (respectively, c0) be the Banach lattice of bounded (respectively, converging to zero) sequences {£n}™=1 of complex numbers equipped with the uniform norm
||{£n}||™ = sup |£n|, where N is the set of natural numbers.
neN
If £ = {£n}™=1 G l™, then the non-increasing rearrangement £ * = {£n }™=1 of £ is defined by
£n = if sup |£n| : F C N, |F| <n
L n/F
The Hardy-Littlewood-Polya partial order in the space l™ is defined as follows:
m m
£ = {£n} ^ n = {nn} ^ ]T£n < ZX for all m G N.
n=1 n=1
A non-zero linear subspace E C l™ with a Banach norm || ■ ||E is called a symmetric (fully symmetric) sequence space if
n G E, £ G l™, £* < n* (respectivelly, £* ^ n*) £ G E and ||£||£ < ||nlU.
Let (H, (■, ■)) be an infinite-dimensional separable Hilbert space over the field C of complex numbers, and let (B(H), || ■ ||™) be the C*-algebra of bounded linear operators in H. Denote by K(H) the two-sided ideal of compact linear operators in B(H). Let Bh(H) = {x G B(H) : x = x*}, B+(H) = {x G Bh(H) : x > 0},
and let t : B+(H) ^ [0, to] be the canonical trace on B(H), that is,
™
t(x) = ^(x^j, ^j), x G B(H), where {^i}°=1 is an orthonormal basis in H. i=1
Let P(H) be the lattice of projections in H. If 1 is the identity of B(H) and e G P(H), we will write ef = 1 — e.
Let x G B(H), and let {e^}A>0 be the spectral family of projections for the absolute value |x| = (x*x )1/2 of x, that is, eA = {|x| < A}. If t > 0, then the t-th generalized singular number of x, or the non-increasing rearrangement of x, is defined as (see [9])
^(x) = inf{A > 0 : t(ef) < t}.
A non-zero linear subspace X C B(H) with a Banach norm || ■ ||X is called noncommutative symmetric (fully symmetric) space if the conditions
x G X, y G B(H), ^(y) < ^t(x) V t > 0
s s
(respectively, J ^t(y)dt < J ^t(x)dt V s > 0 (writing y x))
00 imply that y G X and ||y||X < ||x||X.
The spaces (B(H), || ■ ||^) and (K(H), || ■ ||^), as well as the classical Banach two-sided ideals
Cp = {x e K(H) : ||x||p = t(|x w}, 1 < p < w,
are examples of noncommutative fully symmetric spaces.
If x e K(H), then |x| = sn(x)pn (if m(x) = w, the series converges with
n=l
respect to the uniform norm || ■ ||^), where {s^x)}™^ is the set of eigenvalues of the compact operator |x| in the decreasing order, and pn is the projection onto the eigenspace corresponding to sn(x). Consequently, the non-increasing rearrangement ^t(x) of x e K(H) can be identified with the sequence {sn(x)}^=l, sn(x) ^ 0 (if m(x) < w, we set sn(x) = 0 for all n > m(x)).
Fix an orthonormal basis {^n}^=l in H. Let pn be the one-dimensional projection on the subspace C ■ <pn C H. If (X, || ■ ||X) C K(H) is a symmetric space then the set
E(X) = je = {U~=! e co: xc = ^ CnPn e x|
(the series converges uniformly) is a symmetric sequence space with respect to the norm ||£||e(x) = ||x^||x. Consequently, each symmetric space (X, || ■ ||X) C K(H) generates a symmetric sequence space (E(X), || ■ ||e(x)) C c0. The converse is also true: every symmetric sequence space (E, || ■ ||E) C c0 generates a symmetric space (CE, || ■ IIce) C K(H) by the following rule (see, for example, [14, Chapter 3, Section 3.5]):
CE = {x e K(H) : K(x)} e E}, ||x||cb = ||K(x)}||e•
The
pair (CE, || ■ IIce) is called a Banach ideal of compact operators (cf. [10, Chapter III]). It is known that (Cp, || ■ ||p) = (C/p, || ■ ||cip) for all 1 < p < w and
(K(H), II ■ U = (CC0, II ■ \\Veo).
Hardy-Littlewood-Polya partial order in the Banach ideal K(H) is defined by
x ^ y, x, y e K(H) ^^ {sn(x)} ^ {sn(y)}.
Using Lemma 2.5 (ii) [9] and Theorem 4.4 (iii) [9] we get
x ^ y, z « y, x, y, z e K(H) x + z 2y. (1)
We say that a Banach ideal (CE, || ■ He.) is fully symmetric, if conditions y e CE, x e K(H), x y entail that x e CE and ||x|c.b < HyllcE. It is clear that (CE, || ■ He.) is a fully symmetric ideal if and only if (E, || ■ ||E) is a fully symmetric sequence space.
Note that, along with any Schatten ideals Cp, 1 < p < to, of compact operators, the family of such fully symmetric ideals CE contains many noncommutative counterparts of classical symmetric sequence spaces, examples of which are given in the last section of this note.
A linear contraction T : B(H) ^ B(H) is called a Dunford-Schwartz operator (writing T G DS), if T(C) C C and ||T(x)||i < ||x||i for all x G C. We will write T G DS+ if T is a positive Dunford-Schwartz operator, that is, T G DS and T(B+(H)) C B+(H).
Any fully symmetric ideal CE is an exact interpolation space in the Banach pair (C, B(H)) (see [5, Theorem 2.4]). It then follows that T(CE) C and ||T||ce< 1 for all T G DS. In particular, T(K(H)) C K(H) and the restriction of T on K(H) is a linear contraction (also denoted by T). In addition, T(x) x for all T G DS and x G K(H) (see [6, Theorem 4.7]).
We will utilize the next fundamental fact, which can be found, for example, in [13, Corollary 2.9].
Theorem 1. Let A and B be C*-algebras, and let 1 be the unit of A. If T : A ^ B is a positive linear map, then ||T|| = ||T(1)||.
The following theorem establishes an extension of any positive linear contraction T : C ^ C with the property ||T(x)||™ < ||x||™ for all x G C up to the DunfordSchwarz operator T : B(H) ^ B(H).
Theorem 2. Let T : C ^ C1 be a positive linear contraction such that ||T(x)|™ < ||x|™ for all x G C1. Then there exists a unique operator T G DS such that T(x) = T(x) for all x G C1, and T is ct(B(H),C1) -continuous.
Proof. Since (C1)* = B(H), the adjoint operator T* acts in B(H) and is o*(B(H), C1) -continuous. Moreover, since
t(T*(x)y) = t(xT(y)) V x G B(H), y G C1,
it follows that the linear operator T* is positive. Choose pn G P(H), n = 1,2,..., satisfying
pn < pn+1, t (pn) < to for all n and sup pn = 1.
n> 1
Since ||T(pn)|™ < ||pn||™ < 1 and T(pn) > 0 it follows that T(pn) < 1 for each n G N. Consequently, for any x G C1 R B+ (H) we have that
||T*(x)||1 = t(T*(x)) = lim t(T*(x)pn) = lim t(xT(p*)) < t(x) = ||x||1.
Therefore, T* is || ■ ||l-continuous on Cl R B+(H), hence on Cl.
Next, replacing in the above argument T by T*, we obtain that the operator (T*)* : C1 ^ C1 is positive || ■ ||1 -continuous linear operator. Since
t(x(T*)*(y)) = t(T*(x)y) = t(xT(y)) V x,y G Ci, it follows that T(x) = (T*)*(x) for all x G Cl. Consequently, (T*)* coincides with T on
Ci.
Furthermore, as T = (T*)* is o-(B(H),C1) -continuous and C1 is o-(B(H),C1) -dense in B(H), T uniquely extends to a linear ct(B(H), Cl) - continuous operator T on B(H) for which T(x) = T(x) for all x G C1.
Let us now show that ||T||,b(H)^,b(H) < 1. Indeed, given x G C1 R B+(H), we have
t(xT(1)) = t(T*(x)1) < t(x),
and we conclude that T(1) < 1, hence ||T(1)||^ < 1. Therefore, in view of Theorem 1 with A = B = B(H), we have
hT||B(HHB(H) = l|T(1)IU < 1.
This completes the proof of the Theorem 2, since the operator T : B(H) ^ B(H) defined by above is Dunford-Schwartz operator. □
2. Individual and Mean Ergüdic theorems for flows in Banach
ideals of compact operators
Let R be the set of real numbers and let R+ = {t G R, t > 0}. In what follows, {Tt}ígR+ C DS+ is a semigroup such that T0(x) = x for all x G B(H). A semigroup {Tt}igR+ is said to be strongly continuous on fully symmetric ideal Cl, if lim ||Tt(x) — Ts(x)|c'1 = 0 for each x G Cl.
If {Tt}gR+ C DS+ is a strongly continuous semigroup on Cl then for any x G Cl and y G B(H) the function (t) = t(Tt(x)y) is continuous on R+. Therefore, for the Lebesgue measure ^ on R+ we have that the map : R+ ^ Cl defined as (t) = Tt(x) is weakly ^-measurable, that is, the complex function t(Ux(t)y) is a measurable function on (R+,^) for all y G B(H) (recall that (Cl)* = B(H) and every f G (Cl)* has the following form f (x) = t(xy) for some y G B(H)). Since, in addition, (R+) is a separable subset in Cl, Pettis theorem [18, Chapter V, §4] entails that the real function ||Ux(t)|1 = ||Tt(x)|1 is ^-measurable on R+. Using the inequality ||Tt(x)|1 < ||x|1, we obtain that ||Ts(x)|1 is an integrable function on [0,t] for any t > 0. By [18, Chapter V, §5, Theorem 1], the function Ts(x) is Bochner ^-integrable on [0,t] for every t > 0. Consequently, for any x G C1 and t > 0 there exists the Bochner integral
At(x) = 1 Jg Ts(x)ds 6 Ci. It is clear that ||At(x)|i < and < ||x||^ for
all x 6 Ci. Consequently, by Theorem 2, there exists a unique operator 6 DS + such that At(x) = At(x) for all x 6 Ci, and is ct(B(H), CJ -continuous. Below, the operator is denoted by At.
Theorem 3. Let (CE, ||-||ce ) be a fully symmetric Banach ideal, and let {Tt}ig C DS+ be a strongly continuous semigroup on C1. Then given x 6 CE, the averages At(x) converge to some x 6 CE with respect to the uniform norm || ■ as t ^ to.
Proof. Using Theorem 6.4 (i) [11], we get that for each x 6 C2 there exists x 6 C2 such that the averages At(x) converge to x 6 CE almost uniformly, that is, given e > 0, there exists a projection e 6 P(H) such that t(1 — e) < e and ||(At(x) — x)e||^ ^ 0 as t ^ to, where t is the canonical trace on B(H). In particular, for e = 2 we obtain that 1 — e = 0, thus ||At(x) — ^ 0 as t ^ to. Consequently,
||At(x) — Afl(x)||M ^ 0 as t,s ^to V x 6 C2. (2)
Since C2 contains the linear subspace of all finite-dimensional operators, it follows that C2 is everywhere dense in K(H). Therefore, for each x 6 CE and e > 0, there exists x£ 6 C2 such that ||At(x) — At(x£)||^ < ||x — x£||^ < e for all t > 0 . Using (2) and following inequalities
||At(x) — As(x)|U < ||At(x) — A^xJH«, + ||At(xj — A^xJH«, + P^) — As(x)|U =
= ||At(x — xJIU + ||At(x£) — A^xJH«, + ||Aa(x£ — x)|U < 2e + P^) — A^xJH«, we obtain that
||At(x) — Afl(x)||M ^ 0 as t,s ^ to.
Since (K(H), || ■ ||^) is the Banach space it follows that there exists x 6 K(H) such that ||At(x) — ^ 0 as t ^ to.
By [10, Chapter II, §2, Corollary 2.3] we have that |sn(At(x)) — sn(x)| < ||At(x) — for all n 6 N. Consequently, sn(At(x)) ^ sn(x) as t ^ to for all n 6 N. Since 6 DS+ it follows that At(x) x (see [6, Theorem 4.7]), that is
m m
^sn(At(x)) sn(x) for all m 6 N and t> 0.
Thus
sn(x) < sn (x) for all m 6 N,
3nV t\ /) < / Jn\ n=1 n=1
mm
sn(x) < ^^ sn (x) lor all m
n=1 n=1
that is x x 6 CE. Finally, using that (CE, || ■ \\ce ) is a fully symmetric Banach ideal we obtain that x 6 CE. □
n—1
Remark 1. An analogue of Theorem 3 for the averages An(T) = n ^ Tk was obtained in
n k=0
[3], where it was proved that for any T G DS and fully symmetric Banach ideal (CE, || ■ 11ce) for each x G CE the averages An (T)(x) converge to some x G CE (as n ^ to) with respect to the uniform norm || ■ ||^.
Using the reflexivity of CP-spaces, 1 < p < to, and the well-known mean ergodic theorem for linear contractions of reflexive spaces (see, for example, [8, Chapter VII, §5, Corollary 4]), we have the following version of the mean ergodic theorem for the DunfordSchwarz operators:
n— 1
If T G DS and 1 < p < to, then the averages An(T) = n ^ Tk converge strongly in
n k=0
CP, that is, given x G C, there exists x G C such that ||An(T)(x) — x||p ^ 0 as n ^ to. If p =1, this is not true in general. As a consequence, mean ergodic theorem may not hold in some fully symmetric Banach ideals (CE, || ■ ||ce ).
It is known that every separable symmetric sequence space (E, || ■ ||E) is a fully symmetric sequence space. In this case a symmetric Banach ideal (CE, || ■ 11ce ) is a fully symmetric ideal. Using Yeadon's paper [17], we have the following version of the mean ergodic theorem for the Dunford-Schwarz operators.
Theorem 4. Let (E, || ■ ||E) be a separable symmetric sequence space, let T G DS + , and let T(e ^ 0 for any increasing sequence of projections (en}^=1 C C1, 0 < t(en) < to, with t(en) ^ to. Then for any x G CE there exists x G CE such that ||An(T)(x) — xice ^ 0 as n ^ to.
It is clear that Theorem 4 cannot be used for the ideal (C1, || ■ ||1). The following theorem essentially extends the class of fully symmetric ideals for which a version of the non-commutative mean ergodic theorem for flows (Tt}igR+ C DS+ is true.
Theorem 5. Let (E, || ■ ||E) be a separable symmetric sequence space and E = l1 as sets. Then for any strongly continuous semigroup (Tt}igR+ C DS+ on C the averages converge strongly in CE.
To prove the Theorem 5, we need the following property of separable symmetric sequence spaces [7, Proposition 2.2].
Proposition 1. Let (E, || ■ ||E) be a separable symmetric sequence space and E = l1 as sets. If yn G CE, yn x G CE for every n G N and ^ 0 as n ^ to, then
^ 0 as n ^ to.
Proof of Theorem 5. By Theorem 3, for every x £ there exists x £ such that ||At(x) — ^ 0 as t ^ to. Moreover, (—x) x x and At(x) x for every t > 0 (see proof of Theorem 3). Using (1) we obtain that At(x) + (—x) 2x £ . Consequently, by Proposition 1, for every sequence 0 < tn ^ to we have that
l|Atn(x) — x||ce ^ 0 as n ^ to. It means that ||At(x) — x||cE ^ 0 as t ^ to.
□
3. Applications to Orlicz and Lorentz Banach ideals
In this section we give applications of Theorems 3 and 5 to Orlicz and Lorentz ideals of compact operators.
1. Let $ be an Orlicz function, that is, $ : [0, to) ^ [0, to) is left-continuous, convex, increasing and such that $(0) = 0 and $(u) > 0 for some u = 0. Let
Z$(N) = j f = {fn}~=i £ : ^ $ ^ ^^ < to for some a > üj
be the corresponding Orlicz sequence space, and let ||f = inf a > ü : $ ^— 1
be the Luxemburg norm in Z$(N). It is well-known that (Z$(N), || ■ ||$) is a fully symmetric sequence space.
If $(u) > ü for all u = ü, then J2 $ Q) = to for each a > ü. Hence
n=1
1 = {1,1,...} £ 1$(N) and Z$(N) C c0. If $(u) = ü for all ü — u < u0, then 1 £ Z$ and Z$(N) =
It is said that an Orlicz function $ satisfies (A2)-condition at ü if there exist u0 £ (ü, to) and k > ü such that $(2u) < k ■ $(u) for all ü < u < u0 .It is well known that an Orlicz function $ satisfies (A2)-condition at ü if and only if (Z$(N), || ■ ||$) is a separable space.
We also note that /$(N) = l1 as sets if and only if lim sup > ü [14, Chapter 16, §16.2].
Set = and ||x||$ = HxH^(N), x £ Theorems 3 and 5 are yield the
following.
Theorem 6. Let $ be an Orlicz function, let {Tt}igR+ C DS + be a strongly continuous semigroup on C1. Then
(i). If $(u) > ü for all u > ü and x £ then the averages At(x) converge to some x £ with respect to the uniform norm as t ^ to;
(ii). If lim = ü and the Orlicz function $ satisfy (A2)-condition at ü, then ||At(x) — x||$ ^ ü as t ^ to.
2. Let - be a concave function on [0, to) with -(0) = 0 and -(t) > 0 for all t > 0, and let
A^(N) = j£ = {&}£=! 6 : lieik = £ en(t)(-(n) — -(n — 1)) < to|
be the corresponding Lorentz sequence space. It is well-known that (A^(N), || ■ ||a^) is a fully symmetric sequence space (see, for example, [14, Part III, Ch.9, § 9.1]); moreover, if -(to) = to, then 1 6 A^(N) and A^(N) C c0. If -(to) < to, then 1 6 A^(N) and A^ (N) = In addition, the space (A^(N), || ■ ||a^) is separable if and only if -(+0) = 0 and -(to) = to [14, Ch.9, §9.3, Theorem 9.3.1]. It is clear that lim ^^ > 0 if and only if the
norms || ■ ||a^ and || ■ ¡1 are equivalent on A^(N), i.e. the equality A^(N) = l1 (as sets) is true.
Set Ca^ = Ca^(n) and ||x||a^ = ||x||cA (N), x 6 Ca^ . Theorems 3 and 5 are imply the following.
Theorem 7. Let - be a concave function on [0, to) with -(0) = 0, -(t) > 0 for all t > 0, and let {Tt}tg r+ C DS + be a strongly continuous semigroup on C1. Then
(i). If -(to) = to, then the averages At(x) converge to some x 6 Ca^ with respect to the uniform norm;
(ii). If -(+0) = 0, -(to) = to and lim ^ = 0, then ||At(x) — x||A, ^ 0 as t ^ to.
t^oo v
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