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Владикавказский математический журнал Январь-март, 2000, Том 2, Выпуск 1
YJ\K 517.98
CYCLICALLY COMPACT OPERATORS IN BANACH SPACES
A. G. Kusraev
The Boolean-valued interpretation of compactness gives rise to the new notions of cyclically compact sets and operators which deserves an independent study. A part of the corresponding theory is presented in this work. General form of cyclically compact operators in Kaplansky-Hilbert module as well as a variant of Fredholm Alternative for cyclically compact operators are also given.
1. Preliminaries
In this section we present briefly some basic facts about Boolean-valued representations which we need in the sequel.
1.1. Let B be a complete Boolean algebra and let A be a nonempty set. Recall (see [3]) that B{A) denotes the set of all partitions of unity in B with the fixed index set A. More precicely, assign
B(A) := | v.A^B: (Va, /3 E A) (a # /3 ^ v{a) A v{fi) = 0)
If A is an ordered set then we may order the set B(A) as well:
v < H (Va, ¡3 e A) {y(ot) A (J,(f3) # 0 a < 0) {v, // ¡E B{A)).
It is easy to show that this relation is actually a partial order in B(A). If A is directed upward (downward) then so does B(A). Let Q be the Stone space of the algebra B. Identifying an element v{a) with a clopen subset of Q, we construct the mapping P : Qv —A, Qv := : a e by letting P(q) = a whenever
q E v{a). Thus, P is a step-function that takes the value a on v(a). Moreover, v < (i(Vq e Qv n QM) (P(q) < fi(q)).
© 2000 Kusraev A. G.
1.2. Let x be a normed space. Suppose that C{X) has a complete Boolean algebra of norm one projections B which is isomorphic to B. In this event we will identify the Boolean algebras B and B, writing B c C(X). Say that x is a normed B-space if B C C(x) and for every partition of unity (fe^gs in B the two conditions hold:
(1) If bçx = 0 (£ E S) for some x E x then x = 0;
(2) If b^x = b^x^ (£ E S) for x E X and a family (a^^gs in X then ||x|| < sup{||6^|| : £ e S}.
Conditions (1) and (2) amount to the respective conditions (1') and (2'):
(1') To each x E x there corresponds the greatest projection b e 11 such that bx = 0;
(2') If x, (a^), and (bç) are the same as in (2) then ||x|| = sup{||6^x^|| : £ E S}.
From (2') it follows in particular that
for x E X and pairwise disjoint projections bi,...,bn in B.
Given a partition of unity (6^), we refer to x E X satisfying the condition (V£ E S) b^x = as a mixing of (x^) by (6^). If (1) holds then there is a unique mixing x of (x^) by (&£). In these circumstances we naturally call x the mixing of (x^) by (&£). Condition (2) maybe paraphrased as follows: The unit ball Ux of X is closed under mixing or is mix-complete.
1.3. Consider a normed /¿-space X and a net (xa)a£A in it- For every v E B{A) put xv := mix a£A(v((x)xa). If all the mixings exist then we come to a new net {xv)V£B(A) in X. Every subnet of the net {xv)v^b(a) is called a cyclical subnet of the original net (xa)a£A• If s : A —X and x : A' —B{A) then the mapping s • x : A' —X is defined by s • x(a) := xv where v = x(a). A cyclical subsequence of a sequence {xk)keE C X is a sequence of the form (a^)/^ where (^¡OfceN is a sequence in B(N) with vk <C v^+i for all feel
1.4. Let A be the bounded part of the universally complete /\-space C\,, i. e. A is the order-dense ideal in C,[ generated by the order-unity 1 := 1A E C\,. Take a Banach space X inside V<B>. Denote (see [1])
n
bkX = max || bkX
fe=l,...,n
fc=1
X\°°\= {x E X\r. g A}.
Then Xl°° is a Banach-Kantorovich space called the bounded descent of X. Since A is an order complete .1 A/-space with unity, X\°° is a Banach space with mixed norm over A. If y is another Banach space and T : X —y is a bounded linear
operator inside Vwith \ T\\ E A then the bounded descent of T is the restriction of Ti to X . Clearly, the bounded descent of T is a bounded linear operator from Xi°° to y\.°°.
1.5. A normed /¿-space X is B-cyclic if we may find in X a mixing of each norm-bounded family by any partition of unity in B.
Theorem. A Banach space X is linearly isometric to the bounded descent of some Banach space inside V^ if and only if X is B-cyclic.
According to above theorem there is no loss of generality in assuming that X is a decomposable subspace of the Banach-Kantorovich space Xwhere X is a Banach space inside V^ and every projection b E B coincides with the restriction of x{b) onto X. More precisely, we will assume that X is the bounded descent of X, i.e., X = {x E XE A}, where A is the Stone algebra S(B) identified with the bounded part of the complex algebra CIn this event a subset C C X is mix-complete if and only if C = CfJ,-
1.6. Given a sequence a : NA —Cf and x : NA —NA, the composite a I oh\ is a cyclical subsequence of the sequence a I : N —C if and only if [er o x is a subsequence of a} = 1. Given a sequence s : N —C and x : N B(N), the composite sf o xA is a subsequence of the sequence erf : NA Cf inside V^B) if and only if s • x is a cyclical subsequence of the sequence s.
2. Cyclically compact sets and operators
In this section we introduce cyclically compact sets and operators and consider some of their properties.
2.1. A subset C E X is said to be cyclically compact if C is mix-complete (see 1.5) and every sequence in C has a cyclic subsequence that converges (in norm) to some element of C. A subset in X is called relatively cyclically compact if it is contained in a cyclically compact set.
A set C C X is cyclically compact (relatively cyclically compact) if and only if Cf is compact (relatively compact) in X.
< It suffices to prove the claim about cyclical compactness. In view of [1; Theorem 5.4.2] we may assume that X = X J,- Suppose that [Cf is compact ] = 1. Take an arbitrary sequence s : N C. Then [sf : NA Cf is a sequence in Cf] = 1. By assumption Cf is compact inside so that there exist
p,x E V(li) with {p is a subsequence of sf] = [a; E Cf] = [lim(p) = = 1. Since C is mix-complete, we obtain that p\ is a cyclical subsequence of s and lim (p 1) = x E C. Conversely, suppose that C is a cyclically compact set. Take a sequence a : NA Cf in C. By assumption the sequence : N C has a cyclic subsequence p : B(N) C converging to some x E C. It remains to observe that [pf is a subsequence of the sequence cr ] = 1 and [lim(p'l') = = 1. >
2.2. Theorem. A mix-complete set C in a Banach B-space X is relatively cyclically compact if and only if for every e > 0 there exist a countable partition of unity (iTn) in the Boolean algebra 23(X) and a sequence (9n) of finite subsets 9n C C such that the set 7rn( mix (0n)) is an e-net for ttn(C) for all n G N. The last means that if
then for every x E tt n(C) there exists a partition of unity {pn,i? • • • 5 Pn,i{n)} 'm 23(X) with
l(n)
% ^^ ^ ^iil'a.i,-1'a.I,
k=1
< £.
< According to 1.5 we may assume that X := X^ for some Banach space X inside V(BK By 2.1 a set C C X is relatively cyclically compact if and only if [Cf is relatively compact] = 1. By applying the Hausdorff Criterion to Cf inside V^, we obtain that relative cyclical compactness of Cf is equivalent to [Cf is totally bounded] = 1 or, what amounts to the same, the following formula is valid inside
(VO < e g RA) (3n E NA) (3/ : n —X) (\/x E Cf) (3k E n) (\\x^f(k)\\<e).
Writing out Boolean truth values for the quantifiers, we see that the last claim can be stated in the following equivalent form: for every 0 < e E R there exist a countable partition of unity (bn) in B and a sequence (/n) of elements of V^ such that [ fn : nA —X ] > bn and
[ (Vz g Cf) (3k e nA)(\\x&fn(k)\\ < eA) ] > bn.
Substitute fn for mix (bnfn, b^gn), where gn is an element of V^ with fgn : nA ^ X ] = 1. Then fn meets the above properties and obeys the additional requirement Ifn ■ nA ^ X} = 1. Denote hn := So, the above implies that for every x E C holds
\J{l\\x^hn(k)\\<£Aj: k E n} > bn.
Let % : B —23(A') be the isomorphism from 1.5 and put tt^ := %(£>&). If 6n,fc := [||a;^n(fc)|| < eA ] and x' := YJlZl j{K,k)hn{k) then < eA ] = 1, or
equivalently \7rn(x | < el. Thus, putting 9n := {hn(0),..., hn(n -^1)}, we obtain the desired sequence 9n of finite subsets of C. >
2.3. Denote by Cb{X, Y) the set of all bounded £?-linear operators from X to Y. In this event W := £/>• (X.)') is a Banach space and B C W. If Y is /¿-cyclic then so is W. A projection b E B acts in W by the rule T b o T (T E W). We
call X* := CB(X,A) the B-dual of X. For every / e X* define a seminorm pf on X by pf : x i—Y ||/(a;) ||oo {x E X). Denote by aoo{X,X*) the topology in X generated by the family of seminorms {pf : f E X*}.
A mix-complete convex set C C X is cyclically aOQ(X, X*)-compact if and only if Cf is <j{x, x*)-compact inside v^bk
< The algebraic part of the claim is easy. Let the formula ij){A,u) formalize the sentence: u belongs to the weak closure of A. Then the formula can be written as
(Vn e N) (V0 e Vfin(X)) (3u e A) (Vy e 9) |(x|y)| < rr\
where uj is the set of naturals, (• | •) is the inner product in and Vnn{X) is the set of all finite subsets of X. Suppose that [V>(.4, u)] = 1. Observe that
-Pfln(Xf) = {0f: OEVnn(X)}f
Using the Maximum Principle and the above relation, we may calculate Boolean truth values and arrive at the following assertion: For any n E 00 and any finite collections 9:= {yi,..., ym} in there exists v E A-l such that
[(Vy E9A)\{u^v\y)\< l/nA] = 1.
Moreover, we may choose v so that the extra condition [||u|| < |u|] = 1 holds. Therefore,
M < |m|, |((m I yi)\ < n^1! (k:= 1,..., n; l:= 1,..., m).
There exists a fixed partition of unity (e^gs C B which depends only on u and is such that e^\u\ E A for all From here it is seen that e^u E A and e^v E A. Moreover,
\\(e^(u | yi) 1100 < rC1 (k:= l,...,n; l:= 1 ,...,m).
Repeating the above argument in the opposite direction, we come to the following conclusion: The formula ij){A,u) is true inside V^ if and only if there exist a partition of unity (e^)^gs in B and a family (u^gs such that u^ belongs to the (Too-closure of A and u = mix (e^u^).
Now, assume that A is (Too-closed and the formula tj){A, u) is true inside V^BK Then u^ is contained in A by assumption and {u^ E A} = 1. Hence e^ < \u E A] for all i.e., [u e *A] = 1. Therefore,
V(W) |= (Vu E £(X))i/j(A, u)^ueA.
Conversely, assume A to be weakly closed. If u belongs to the (Too-closure of A, then u is contained in the weak closure of A. >
2.4. Consider X**:= (X*)*:= £B{X*, A), the second B-dual of X. Given x E X and / E X*, put x** := t(x) where t(x) : f f(x). Undoubtedly, i(x) E L{X*, A). In addition,
|®##| = U(®)|=sup{|t(®)(/)|: |/|<1} = sup{|/(®)|: (V® g X)\f(x)\ < \x\} = sup{|/0r)|: / g 3(|-|)} = |4
Thus, /(./■) E X** for every x E X. It is evident that the operator i : X —X##, defined as i : x i(x), is linear and isometric. The operator i is referred to as the canonical embedding of X into the second S-dual. As in the case of Banach spaces, it is convenient to treat x and x**:= ix as the same element and consider X as a subspace of X**. A B-normed space X is said to be B-reflexive if X and X** coincide under the indicated embedding i.
Theorem. A normed B-space is B-reflexive if and only if its unit ball is cyclically (joo {X, X*)-compact.
< The Kakutani Criterion claims that a normed space is reflexive if and only if its unit ball is weakly compact. Hence, the result follows from 2.3. >
2.5. Let X and Y be normed /¿-spaces. An operator T E Cb{X,Y), is called cyclically compact (in symbols, T E K,b{X,Y)) if the image T(C) of any bounded subset C C X is relatively cyclically compact in Y. It is easy to see that )Cb{X, Y) is a decomposable subspace of the Banach-Kantorovich space Cb{X, Y).
Let X and y be Boolean-valued representations of X and Y. Recall that the immersion mapping T of the operators is a linear isometric embedding of the
lattice-normed spaces Cb{X,Y) into CB(X,y)l, see [1; Theorem 5.5.9]. Assume that Y is a /¿-cyclic space.
(1) A bounded operator T from X into Y is cyclically compact if and only if [T~ is a compact operator from X into = 1.
< Observe that C is bounded in X if and only if [C~ is bounded in .1'] = 1. Moreover, according to [1: 3.4.14],
V{B) | =T(CT =T~(C~).
It remains to apply 2.1. >
(2) )Cb{X, Y) is a bo-closed decomposable subspace in £b(X, Y). < Let X and y E V^ be the same as above and let tC^ (X, y) be the space of compact operators from X into y inside V^BK As was shown in [1; Theorem 5.9.9 the mapping T is an isometric embedding of Cb{X,Y) into C^B\X
It follows from (1) that this embedding maps the subspace Kb(X,Y) onto the bounded part of tC^ (X, Taking into consideration the ZFC-theorem claiming the closure of the subspace of compact operators, we have \K^B\X ,y) is a closed subspace in
£<*>(*,3>)] = 1.
From this we deduce that K^B\X, y) I is 6o-closed and decomposable in &B\X,y)\,. Thus, the bounded part of K,^ (X, y) I is also 6o-closed and decomposable. >
(3) Let T G jC-b{X, Y) and S G Cb(Y, Z). If either T or S is cyclically compact then S oT is also cyclically compact.
< We need only to immerse the composite S oT inside V^ and, taking into account (1) and [1; 3.4.14], apply therein the ZFC-theorem about compactness of the composite of a bounded operator and a compact operator. The subsequent descent leads immediately to the desired result. >
(4) A bounded operator T is cyclically compact if and only if its adjoint T* is cyclically compact.
< Apply the above procedure, immersion into a Boolean-valued model and the subsequent descent. Observe that the operator (T*)~ is the adjoint of inside V^ and use the corresponding ZFC-theorem on compactness of the adjoint of a compact operator. >
3. Cyclically compact operators in Kaplansky^Hilbert modules
Now we consider general form of cyclically compact operators in Kaplansky-Hilbert modules.
3.1. Let A be a Stone algebra and consider a unitary A-module X. The mapping (■ | ■} : X x X A is a A-valued inner product, if for all x,y,z G X and a G A the following are satisfied:
(1) (x | x) > 0; (x | x) = 0 x = 0;
(2) (x\y) = (y\x)*]
(3) (ax | y) = a(x | y);
(4) (x + y\ z) = (x\z) + (y\z).
Using a A-valued inner product, we may introduce the norm in X by the formula
(5) lll^m:= v^ll(a;la:;)ll (x e
and the vector norm
(6) |ar|: = yj(x\x) (x E X).
3.2. Let X be a A-module with an inner product (• | •) : X x X —A. If X is complete with respect to the mixed norm |||-|||, it is called a C*-module over A. A Kaplansky-Hilbert module or an AW* -module over A is a unitary C*-module over A that enjoys the following two properties:
(1) let x be an arbitrary element in X, and let be a partition of unity in ^3(A) with e^x = 0 for all £ G S; then x = 0;
(2) let (^)^gs be a norm-bounded family in X, and let (e^gs be a partition of unity in ^3(A); then there exists an element x E X such that e^x = e^x^ for all
e g s.
The element of (2) is the 6o-sum of the family (e^x^^s- According to the Cauchy-Bunyakovskii-Schwarz inequality (x\y) < |/| |y| the inner product is bo-continuous in each variable. In particular,
If X is a C*-module than the pair (X, |||-|||) is a /¿-cyclic Banach space if and only if (X, |-|) is a Banach-Kantorovich space over A := S(B), see [1; Theorem 6.2.7].
3.4. Theorem. The bounded descent of an arbitrary Hilbert space in \(B) is a Kaplansky-Hilbert module over the Stone algebra S(B). Conversely if X is a Kaplansky-Hilbert module over S(B), then there is a Hilbert space X in \(B) whose bounded descent is unitarily equivalent with X. This space is unique to within unitary equivalence inside V^BK
< The proof can be found in [1; Theorem 6.2.8] >
3.5. Theorem. Let T in 1Cb{X,Y) be a cyclically compact operator from a Kaplansky-Hilbert module X to a Kaplansky-Hilbert module Y. There are orthonormal families {e^ken in X, (/fc)fceN in Y, and a family (/ifc)fceN in A such that the following hold:
(1) fik+i < fJ-k (k E N) and o -linifc^oo
IJ'k = 0;
(2) there exists a projection 1r^ in A such that itoo^k is a weak order-unity in -¡Too A for all k E N;
(3) there exists a partition (7Tfc)^L0 of the projection 1r^ such that 7r0/ii = 0, TTfc < [fik], and 7rfc/ifc+i = 0, k E N;
(4) the representation is valid
oo oo n
T = TToobo-^2, ¡'k'i- ® fk + ^ /lk<i ® fk-
k=l n= 1 k=l
< By virtue of 3.4 we may assume that X and Y coincide with the bounded descents of Hilbert spaces X and y, respectively. The operator T:= Tf: X y is compact and we may apply inside V^ the ZFC-theorem on the general form of a compact operator in Hilbert space. Working inside V^ we may choose orthonormal sequences [ek)kem in X, (fk)k£N in y, and a decreasing numeric sequence {pk)ken in Tl+ \ 0 such that limpk = 0 and the presentation holds:
oo
T = Ik-
k=l
Moreover, either (Vfc e N) pk > 0 or (3k E N) pk = 0. Since {pi < ||T|| ] = 1 we have pi < \T\ E A, whence (pk) C A. Let tt^ := [T be an infinite-rank compact operator from a Hilbert space A" to a Hilbert space = 1. If p'k := itoo^k then H/4>0] = [/4> Pk+ii = [lim/ijfc = 0] = iToo, so that p'k is a weak order-unity in ffooA, p'k > pk+1, and o-limpk = 0. From the above-indicated presentation for T we deduce
oo
TTooT = bo-^2 l''k('k ® ./'/■'•
k=l
Consider the fragment tt¿T. From the definition of tt^ it follows that 7r^ = [ T is a finite-rank operator] = 1. The operator T has finite rank if and only if pn = 0 for some n E N. Thus,
oo
Put pn := {pn = 0], ttq := Pi, 7Tn := pn+1 (n E N). Since -rrn = {pn+i =
Ok, pn 0], we have construct a countable partition (Trn)^L0 of the projection 7r^ with irnpn+i = 0. Therefore, ttnT = ^n^k^t ® fk n e ^ remains
to observe that T = tt^T + -khT. >
4. Fredholm £?-aIternative
A variant of the Fredholm Alternative holds for cyclically compact operators. We will call it the Fredholm B-Alternative.
4.1. Let X be a Banach space with the dual X*. Take a bounded operator T : X —X and consider the equation of the first kind
Tx = y (x, y E X)
and the conjugate equation
y = x (x ,y EX).
The corresponding homogeneous equations are defined as Tx = 0 and T*y* = 0. Let <po(T), (pi(n,T), (p2(n,T), and (p3(n,T) be set-theoretic formulas formalizing the following statements.
(po(T): The homogeneous equation Tx = 0 has a sole solution, zero. The homogeneous conjugate equation T*y* = 0 has a sole solution, zero. The equation Tx = y is solvable and has a unique solution given an arbitrary right side. The conjugate equation T*y* = x* is solvable and has a unique solution given an arbitrary right side.
ipi(n,T): The homogeneous equation Tx = 0 has n linearly independent solutions ..., xn. The homogeneous conjugate equation T*y* = 0 has n linearly independent solutions ..., y*.
T): The equation Tx = y is solvable if and only if y*(y) = • • • = y* (y) = 0. The conjugate equation T*y* = x* is solvable if and only if x*(xi) = ••• = x*{xn) = 0.
</?3(n, T): The general solution x of the equation Tx = y is the sum of a particular solution x0 and the general solution of the homogeneous equation; i.e., it has the form
n
X = x0 + ^ Xkxk (Afc g C). k=l
The general solution y* of the conjugate equation T*y* = x* is the sum of a particular solution yl and the general solution of the homogeneous equation; i.e., it has the form
n
y* =Vo + Yl Vkvl feeC).
fc=i
Using this notation, the Fredholm Alternative can be written as follows (see
W):
(T) V (3n G N) (n, T) & (n, T) & (n, T).
Thus, the conventional Fredholm Alternative distinguishes the cases n = 0 and n^O. (If n = 0 then the formula
Vi{n,T)kv2{n,T)kvz{n,T)
is equivalent to (fio(T).)
4.2. Consider now a /¿-cyclic Banach space X and a bounded £?-linear operator T in X. In this case X and X* are modules over the Stone algebra A:= S(B) and
T is A-linear (= module homomorphism). A subset £ C X is said to be locally linearly independent if whenever e±,..., en G £, Ai,..., Xn G C, and tt G B with
7r(Ai c.i + • • • + A„c„) = 0 we have TrA/,.c/,. = 0 for all k := 1.....//. We say that
the Fredholrn B-Alternative is valid for an operator T if there exists a countable partition of unity (bn) in B such that the following conditions are fulfilled:
(1) The homogeneous equation 60 ° Tx = 0 has a sole solution, zero. The homogeneous conjugate equation 60 ° T*y* = 0 has a sole solution, zero. The equation 60 ° Tx = b0y is solvable and has a unique solution given an arbitrary y G X. The conjugate equation 60 ° T*y* = box* is solvable and has a unique solution given an arbitrary x* G X*.
(2) For every n G N the homogeneous equation bnoTx = 0 has n locally linearly independent solutions ■^l,m • • • 5 and the homogeneous conjugate equation bn o
_ q ^^ n iocaHy linearly independent solutions yfn,..., y* n (hence have nonzero solutions).
(3) The equation Tx = y is solvable if and only if bn o y*n(y) = 0 (n G N, k < n). The conjugate equation T*y* = x* is solvable if and only if bn o x*(xk,n) = 0 (n G N, k < n).
(4) The general solution x of the equation Tx = y has the form
OO / n
x = bo-Y, Klxn + Y 'V.,,.•'•/,•.».
n=1 ^ fc=1
where is a particular solution of the equation bn o Tx = bny and (Ak,n)n£N,k<n are arbitrary elements in A.
The general solution y* of the conjugate equation T*y* = x* has the form
OO / n
y* = bo-j2 bnUt + Yl ^-"Hk.n
n=1 ^ k=1
where y* is a particular solution of the equation bn o T*y* = bnx*, and Afcin are arbitrary elements A for n G N and k < n.
4.3. Theorem. If S is a cyclically compact operator in a B-cyclic space X then the Fredholrn B-Alternative is valid for the operator T:= Ix S.
< Again we assume, without loss of generality, that X is the bounded part of the descent of a Banach space X G V{B) and T is the restriction onto X of the descent of a bounded linear operator T G V^BK Moreover, [T = Ix ^¿"J = 1 and [5 is a compact operator in .1'] = 1. We may assume that also X = X*\°° and T = T*4°°, see [1; 5.5.10]. The Fredholrn Alternative 4.1 is fulfilled for T inside
V(B) by virtue of the Transfer Principle. In other words, the following relations hold:
= [MT)l v\/W<r)]a {V2{n\ T) 1 A i^{n\T) ].
nGN
Denote b0 := [MT)J and6n:= [tpi(nA, T) ] A [ <£>2(nA, T) ] A [ 993(nA, T) ]. Since the formulas <£>o(T) and <pi(n, T) k T) k <£>3(n, T)) for different n are inconsistent, the sequence (bn)%L0 is a partition of unity in B. We will now prove that 4.2 (1-4) are valid.
(1): The claim 4.2 (1) is equivalent to the identities ker(T) = {0} and im(T) = X that are ensured by the following easy relations:
|= ker(T)f= ker(T) = {0}, |= im(T)f= im(T) = X.
(2): The part of the assertion </?i(nA, T) concerning the solution of the equation Tx = 0 is formalized
as
(3a;) ((x : {1,..., n}A ^ X) k (Vfc e {1,..., n}A) (Tx(k) = 0) &the set a;({l,..., n}A) is linearly independent)).
Moreover, there is no loss of generality in assuming that ||a;(fc)|| < 1, k E {1,..., n}A. Using the Maximum Principle and the properties of the modified descent we may find a mapping x from {1,..., n} to X such that the image of the mapping 6nx : k i—Y bnx(k) is a locally linearly independent set in X and [Tx(fc) = 0] > bn for each k E {1,..., n}. Put xk,n '•= b„,x-(k). Further,
{TxKn = 0] = [Tx(fc) = 0] A [x(fc) = xk,n] > bn,
so that bnTxk^n = 0. The conjugate homogeneous equation is handled in the same fashion.
(3): Necessity of the stated conditions can be easily checked; prove sufficiency. We confine exposition to the equation Tx = y, since the conjugate equation is considered along similar lines. Suppose that ykn{y) = 0 for k, n E N and k < n. Then
bn < hin(y) = o] = [y*J (y) = 0] (kE{ 1,..., n}).
At the same time, in view of (2), f{ykn : k = l,...,n}f is a maximal linearly independent set of solutions of the equation T*y* = 0] = 1. All this implies that [the equation Tx = y is solvable] > bn, whence the equation bn o Tx = bny has at
least one solution xn. It is then easy to check that x:= J^^Li bn
X yi IS ct solution of
the equation Tx = //.
(4): If x is a solution of the equation Tx = y then [ Tx = y ] = 1. Taking into account the inequality [</?3(nA, T) ] > bn- we arrive at
nA
bn < [(3A) (A : {1 ,...,n}A ^Ukx = x* + J2 A(fcMfc))],
k=l
where u is the ascent of the mapping k Xk,n (k = 1, ...,n). The Maximum Principle guarantees the existence of a mapping ln from {1,..., n} to A such that
nA
Ix = x + J2^nt (k)ll(k)] = 1.
fc=1
Putting Ak,n'-= bn£n(k), we obtain
n
bnX bnXn ^ ^ k=1
whence the desired representation follows. The general form of the solution of the conjugate equation is established by similar arguments. >
5. Concluding remarks
5.1. The bounded descent of 1.4 appeared in the research by G. Takeuti into von Neumann algebras and C*-algebras within Boolean-valued models [5, 6] and in the research by M. Ozawa into Boolean-valued interpretation of the theory of Hilbert spaces [7]. Theorem 3.4 on Boolean-valued representation of Kaplansky-Hilbert modules was proved by M. Ozawa [7].
5.2. Cyclically compact sets and operators in lattice-normed spaces were introduced in [8] and [3], respectively. Diffrernt aspects of cyclical compactness see in [9-12]. A standard proof of Theorem 2.4 can be extracted from [3] wherein more general approach is developed for the case of lattice normed space. Certain variants of Theorems 3.5 and 4.3 for operators in Banach-Kantorovich spaces can be also found in [3].
5.3. The famous result by P. G. Dodds and D. H. Fremlin [13] asserts that if a positive operator acting from a Banach lattice whose dual has order continuous norm to a Banach lattice with order continuous norm is dominated by a compact operator then the initial operator is also compact, see [14] for proof and related results. As regards cyclical compactness, we observe the conjecture of [15] that if a dominated operator T between spaces with mixed norm is cyclically compact and \T\ < S with S compact then T is also compact on assuming some conditions on the norm lattices like in the Dodds-Fremlin Theorem. This problem remains open.
References
1. Kusraev A. G. and Kutateladze S. S. Boolean Valued Analysis.—Dordrecht: Kluwer Academic Publishers, 1999.
2. Bell J. L. Boolean-Valued Models and Independence Proofs in Set Theory.— New York etc.: Clarendon Press, 1985.
3. Kusraev A. G. Vector Duality and Its Applications [in Russian],—Novosibirsk: Nauka, 1985.
4. Kutateladze S. S. Fundamentals of Functional Analysis.—Dordrecht: Kluwer Academic Publishers, 1998.
5. Takeuti G. Von Neumann algebras and Boolean valued analysis // J. Math. Soc. Japan.—1983.—"V. 35, No. 1, P. 1-21.
6. Takeuti G. C*-algebras and Boolean valued analysis // Japan. J. Math. (N.S.).—1983,—V. 9, No. 2. P. 207-246.
7. Ozawa M. Boolean valued interpretation of Hilbert space theory // J. Math. Soc. Japan.—1983.—"V. 35, No. 4. P. 609-627.
8. Kusraev A. G. Boolean valued analysis of duality between universally complete modules // Dokl. Akad. Nauk SSSR.—1982,—V. 267, No. 5. P. 1049-1052.
9. Kusraev A. G. and Kutateladze S. S. Nonstandard methods for Kantorovich spaces // Siberian Adv. Math.—1992.—"V. 2, No. 2. P. 114-152.
10. Kutateladze S. S. Cyclic monads and their applications.—Sib. Mat. Zh.— 1986.—"V. 27, No. 1. P. 100-110.
11. Kutateladze S. S. Monads of ultrafilters and extensional filters // Sib. Mat. Zh.—1989.—"V. 30, No. 1. P. 129-133.
12. Gutman A. E., Emel'yanov E. Yn., Kusraev A. G. and Kutateladze S. S. Nonstandard Analysis and Vector Lattices.—Novosibirsk: Sobolev Institute Press, 1999.
13. Dodds P. G. and Fremlin D. H. Compact operators in Banach lattices // Israel J. Math.—1979.—"V. 34. P. 287-320.
14. Aliprantis C.D. and Burkinshaw O. Positive Operators.—New York: Academic Press, 1985.
15. Kusraev A. G. Linear operators in lattice-normed spaces // in: Studies on Geometry in the Large and Mathematical Analysis.—Novosibirsk, 1987.—V. 9— P. 84-123.
г. Владикавказ
Статья поступила 26 марта 2000 г.
