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Владикавказский математический журнал 2009, Том 11, выпуск 3, С. 38-43
UDC 517.98
HOMOGENEOUS FUNCTIONS OF REGULAR LINEAR AND BILINEAR OPERATORS1
To Yuri G. Reshetnjak on the occasion of his 80th birthday
A. G. Kusraev
Using envelope representations explicit formulae for computing (p(Ti,... ,TN) for any finite sequence of
regular linear or bilinear operators Ti,..., TN on vector lattices are derived.
Mathematics Subject Classification (2000): 46A40, 47A50, 47A60, 47A63, 47B65.
Key words: regular linear operator, regular bilinear operator, homogeneous functional calculus, envelope
representation.
1. Introduction
This paper is a continuation of [5]. We apply the upper envelope representation method (or the quasilinearization method) in vector lattices developed in [4, 5] to the homogeneous functional calculus of linear and bilinear operators. Explicit formulae for computing (p(Ti,..., Tn) for any finite sequence of regular linear or bilinear operators Ti,..., Tn are derived.
For the theory of vector lattices and positive operators we refer to the books [1] and [3]. All vector lattices in this paper are real and Archimedean.
Consider conic sets C and K with K C C and K closed. Let H(C; K) denotes the vector lattice of all positively homogeneous functions p : C ^ R with continuous restriction to K. The expression (p(xi,..., Xn) can be correctly defined provided that the compatibility condition [xi,..., Xn] C K is hold, see [5].
Denote by HV(RN,K) and HA(RN,K) respectively the sets of all lower semicontinuous sublinear functions p : RN ^ R U and upper semicontinuous superlinear functions
^ : Rn ^ RU which are finite and continuous on a fixed cone K C RN. Put HV(RN) : =
Hv(Rn, {0}) and HA(Rn) := HA(RN, {0}).
Denote by GV(RN, K) and GA (RN ,K) respectively the sets of all lower semicontinuous gauges p : RN ^ R+ U {+^} and upper semicontinuous co-gauges ^ : RN ^ R+ U which are finite and continuous on a fixed cone K C RN. Put GV(RN) := GV(RN, {0}) and Ga(Rn) := Ga(Rn, {0}). Observe that Gv(Rn) C HV(RN) and GA(RN) C HA(RN), see [4, 5].
Everywhere below E, F, and G denote vector lattices, while Lr (E,F) and BLr (E,F; G) stand for the spaces of regular linear operators from E to F and regular bilinear operator from E x F to G, respectively.
© 2009 Kusraev A. G.
iSupported by a grant from Russian Foundation for Basic Research, project № 09-01-00442.
2. Functions of Bilinear Operators
A 'partition of x G E+ is any finite sequence (x\,..., xn), n G N, of elements of E+ whose sum equals x. Denote by Prt(x) and DPrt(x) the sets of all partitions of x and all partitions with pairwise disjoint terms, respectively.
2.1. Lemma. Let E, F, and G be vector lattices, b1,...,bN G BLr (E,F; G), and b := (bi,...,bN). Let p G HV(RN), ^ G HA(RN), £(bi(xo,Vo),...,bN(xo,yo)) and tp(b1 (x0,y0),... ,bN (x0,y0)) are well defined in G for all 0 ^ x0 ^ x and 0 ^ y0 ^ y. Denote x : = (x1,..., xn) G En and y : = (y1}..., ym) G Fm, m,n G N. Then the sets
f n m n
p(b; x,y):= I J2fi(bi(xi,yj ),...,bN (xi,yj)) : n,m G N, x G Prt(x), y G Prt(y) L ^ i=1j=1 '
f n m
^(b; x,y):= i yE^(b1(xi,yj),... ,bN(xi,yj)) : n,m G N, x G Prt(x), y G Prt(y) ^ i=1 j = 1
are upward directed and downward directed, respectively.
< Assume that (x1,..., xn) and (x1,..., x'n,) are partitions of x while (y1,..., ym) and (y1,..., y'm,) are partitions of y. By The Riesz Decomposition Property of vector lattices there exist finite double sequences (ui,k)i^n,k^n' in E+ and (vj,i)j^m,i^m> in F+ such that
En' ST^-n I ! I \
k=1 Ui,k = xi, 2^i=i Ui,k = xk (i := 1,... ,n, k :=1,...,n );
Em r—y/fi . ,
l=1 vj,i = yj, V = vji = yi j:=1,...,m, l:=1,...,m).
In particular, (ui,k)i^n,k^n' and (vj,i)j^m>i^m' are partition of x and y, respectively. Taking subadditivity of p into consideration we obtain
n,m , n',m' n',m' ^
((b1(xi,yj ),...,bN (xi, yj)) = Yj Y b1 (ui,k ,vj,i),...^Y bN (ui,k ,vj,i)J
i,j=1 i,j=1 k,i=1 k,i=1
n,m , n',m' x n,m n',m'
= Y Yj (b1(ui,k ,vj,i ),...,bN (ui,k ,vjti)^J ^^ Yl PP(b1(ui,k ,vj,i),...,bN (uitk ,vj,i)).
i,j=1 k,l=1 i,j=1 k,l=1
In a similar way we get
n ,m n,m
Yj ( (b1 (xk ,y'l),...,bN (x'k ,y'l)) ^Y Y ((b1 (ui,k ,vj,i),...,bN (ui,k ,vj,i)), k, =1 i,j=1 k, =1
so that the first set is upward directed. Similarly, the second set is downward directed. >
2.2. Lemma. Let Let E, F, and G be vector lattices with G Dedekind complete and B be an order bounded set of regular bilinear operators from E x F to G. Then for every x G E+ and y G F+ we have:
{n m ^
YYbk(ij (xi,yj) r> i=1 j=1 J
{n m
YYbk(i>j)(xi,yj) r> i=1 j=1
n ,m
where supremum and infimum are taken over all naturals n,m,l G N, functions k : {1,... ,n}x {1,...,m} ^ {1,...,l}, partitions (x\,... ,xn) G Prt(x) and (yi,...,ym) G Prt(y), and arbitrary finite collections b\ ... ,bi G B.
< See [6, Proposition 2.6]. >
2.3. Theorem. Let E, F, and G be vector lattices with G Dedekind complete, bi,...,bN G BLr (E,F; G), and b := (bi,...,bN). Assume that <p G HV(RN), ^ G Ha(rn ), (p(b i(xo ,y0),... ,bN (xo,yo)) and tp(b i(x0,y0),...,bN (x0,y0)) are well defined in G for all 0 ^ xo ^ x and 0 ^ yo ^ y, <fi(b; x, y) is order bonded above, and ^(b; x, y) is order bounded below for all x G E+ and y G F+. Then (p(bi,..., bN) and ip(bi,..., bN) are well defined in BLr (E, F; G) and for every x G E+ and y G F+ the representations
(p(b i,...,bN )(x,y) = sup v(b; x,y), tp(bi,.. .,bN)(x,y) = inf ^(b; x,y)
hold with supremum over upward directed set and infimum over downward directed set. IfE and F have the strong Freudenthal property (or principal projection property) then Prt(x) and Prt(y) may be replaced by DPrt(x) and DPrt(y), respectively.
< Denote b\ := Aibi + ■ ■ ■ + \nbN for A := (Ai,..., \n) G RN and observe that if the set {b\ : A G d<^} is order bounded in BLr (E, F; G), then by [5, Theorem 4.4] p(bi,..., bN) exists in BLr (E,F; G) and the upper envelope representation (p(bi,... ,br) = sup{bA : A G d<^} holds. Take arbitrary Ar := (Ai,..., ArN) G d<£ (r := 1,... ,l), k : {1,... ,n} x {1,...,m} ^ {1,..., l}, x := (xi,..., xn) G Prt(x), and y := (yi,..., ym) G Prt(y). Making use of Lemma 2.2 and [5, Theorem 4.4] we deduce:
n,m n,m N n,m
Y bxk(i'j)(x^yj) = Y YAk(i'J)bs(xi,yj) ^ Y p(bi^^yj),bN(xi,yj)) ^а,
i,j=i i,j=i s=i i,j=i
where a is an upper bound of b; x,y). Passing to supremum over all (Ai....,Al), k, x, and y and taking [5, Theorem 4.4] into account we get that (p(bi,...,br) is well defined and (p(bi,..., br)(x, y) ^ <fi(b; x, y). Surely, in above reasoning we could take (xi,..., xn) G DPrt(x) provided that E has the principal projection property.
Conversely, let f (x, y) stands for the right-hand side of the first equality. Observe that if (Ai,..., An) G and u G E+, v G F+, then by [5, Theorem 4.4] we have
N N
Y Ak bk (u, v) = (Y Ak bkj (u, v) ^ (p(b i,..., br )(u, v) k=i k=i
and again (p(bi(u,v),... ,bN(u,v)) ^ (p(bi,... ,bN)(u,v) by [5, Theorem 4.4]. Now, given (xi,... ,xn) in Prt(x) or DPrt(x) and (yi,... ,yn) in Prt(y) or DPrt(y), we can estimate
n,m n,m
Y <p(bi(xi,yj),...,br (xi, yj)) ^Y P(bi,...,bN )(xi,yj) < p(bi ,...,br )(x,y)
i'j=i i'j=i
and thus f (x, y) ^ p(bi,..., br)(x, y). Thus the first equality is hold true. By Lemma 2.1 the supremum on the right-hand side of the required formula is taken over upward directed set. The second representation is proved in a similar way. >
2.4. Corollary. Let E, F, G, p, 61,..., 6n be the same as in 2.1, 6 := p?(bi,..., 6n) and 6:= ^(bi,..., 6n). Assume that, in addition, E = F has the strong Freudenthal property and bi,..., 6n are orthosymmetric. Then for every x G E the representations
b(x,x) = sup ^Tp(bi(xi, |x|),... ,6n(xi, |x|)) : (xi,... ,xn) G DPrt(|x|) >
^ i=i > [ n
6(x, x) = inf < ^ ^(bi(xi, |x|),..., bN(xi; |x|)) : (xi;..., xn) G DPrt(|x|) ^ i=i
hold with supremum and infimum over upward and downward directed sets, respectively.
< It is sufficient to check the first formula. We can assume x G E+. Denote by g(x) the right-hand side of the desired equality. From Theorem 2.3 we have g(x) ^ ..., b^)(x, x). To prove the reverse inequality take two disjoint partitions of x, say p' := (xi,..., xj) and p'' : = (xi',..., x^,), and let (xi,..., xn) G DPrt(x) be their common refinement. Since bi,..., 6n are orthosymmetric we deduce
l,m
^ (p(bi (xj ,xS'),...,bN (xj ,x"))
r,s=1
n n
= ^ <^(bi(xi,xi),... ,bN (x»,xi)) = ^ <^(bi(xi,x),...,bN (xi, x)). i=1 i=1
Passing to supremum over all p' and p'' we get the desired inequality. >
3. Functions of Linear Operators
The above machinery is applicable to the calculus of order bounded operators.
3.1. Theorem. Let E and F be vector lattices with F Dedekind complete, Ti,...,Tn G Lr (E, F), and T := (Ti,...,Tn ). Let p G HV(RN), ^ G HA(RN), £(Tixo,... ,Tn xo) and i/>(Tixo,..., Tnxo) are well defined in F for all 0 ^ xo ^ x. If for every x G E+ the sets
<(T; x) = i <?(Tixfc,... ,Tn xfc) : (xi,... ,xn) G Prt(x) L ^ k=i >
^(T; x) = i Y^ i/"(Tixfc ,...,Tn xfc) : (xi,...,xn) G Prt(x) \ ^ fe=i ^
are order bounded from above and from below respectively, then <>(Ti,...,Tn) and i/>(Ti,..., Tn) exist in Lr (E, F), and the representations
p(TL,... ,TN)x = sup <(T; x), ^(Tl,...,Tn )x = inf ^(T; y)
hold with supremum over upward directed set and infimum over downward directed set. If E has the principal projection property then Prt(x) may be replaced by DPrt(x). < Follows immediately from 2.3. >
3.2. Remark. (1) Assume that E, F, Ti,..., Tn, and ^ are the same as in [4, Theorem
5.2]. Then y(Ti,.. .,TN )x ^ (p(Txx,.. .,TN x) and $(Ti,.. .,TN )x ^ tp(T1x,.^. .,TN x) for all x £ E+. In particular, if R+ C dom(y) n dom(i) and (p(T\x,... ,TNx) ^ ip(Tix,... ,TNx) for all x £ E+, then y(Ti,... ,Tn) ^ i(Ti,... ,Tn).
(2) Assume that y £ H(C; [x]) and y(0, t2,... ,tN) = 0 for all (t1,..., tN) £ dom(y). Then evidently (p(xi,... ,xn) £ provided that [x] C dom(y). This simple observation
together with [4, Theorem 5.2] enables one to attack the nonlinear majorization problem for wider variety of majorants y(Ti,..., Tn), cp. [2].
3.3. Let E and F be vector lattices with E relatively uniformly complete and F Dedekind complete. Then for Ti,..., Tn £ L+(E, F), xi,... ,xn £ E+, and ai,..., aN £ R+ with ai + ■ ■ ■ + aN = 1 we have
(T^1 . ..T^N )(xa . ..xNN) < (Tixi)ai ... (Tn Xn )an.
The reverse inequality holds provided that ai +---+aN = 1, (-1)k(1 -ai-----ak)ai-.. .-ak ^
0 (k := 1,... ,N - 1), and xi » 0, f (x,) > 0 for all i with ai < 0.
< Apply [4, Corollary 6.7] with K = R+, C = 1, yo(t) = yi(t) = ^(t) = t^1 .. .t%N. >
3.4. Theorem. Let E and F be vector lattices with F Dedekind complete and Ti,...,TN £ L+(E,F). Suppose that y £ GV(RN,RN) and i £ GA(RN,RN) are increasing and [Ti,..., Tn] C dom(y) n dom(i). Then for every x £ E+ the representations hold
y(Ti,..., Tn )x = sup] £ Tkxk : xi,...,xn £ E+, y°(xi,.. .,xn ) ^ x\, ^ k=i >
ip(Ti,.. .,Tn )x = inf Tk xk : xi,...,xn £ E+, i°(xi,.. .,xn ) ^ x\,
k=1
with supremum over upward directed set and infimum over downward directed set.
< Suppose that y(Ti,...,Tn) exists and x £ E+. If xi,...,xn £ E+ and y°(xi,... ,xn) ^ x, then making use of the Bipolar Theorem, positivity of y(Ti,... ,Tn), and [4, Corollary 6.8] we deduce
N
Y^Tkxk < y(Ti,...,TN)(y°(xi,...,xN)) < y(Ti,...,TN)x. k=1
To prove the reverse inequality take (xi,...,xn) £ Prt(x), \k = (Ak,...,\N) £ dy = {y° ^ 1} (k := 1,..., n), and put u, := ^n^ Ak xk. If a := (ai,..., aN) £ dy° = {y ^ 1}, then {a, \k) ^ y(a)y°(\k) ^ 1 and thus
N N n n
^ aiUi = aiY \k xk = Y {a, \k )xk ^ x. i=1 i=1 k=1 k=1
It follows from [5, Theorem 5.4] that y°(ui,..., Un) ^ x.
Denote S (A) := Xi Ti + ■ ■ ■ + Xn Tn with A := (Ai,..., Xn ). Let f (x) is the right-hand side of the first equality. Then
nN
(Ak)(xkTiUi < f (x).
k=1
It remains to observe that y(Ti,... ,Tn) = sup{S(A) : A £ dyy} by [5, Theorem 4.4]. >
3.5. Proposition. Let E, F, and G be vector lattices with F Dedekind complete, R : E ^ G an order interval preserving operator, T : G ^ F an order continuous lattice homomorphism, and p £ H(C, K). Assume that S1,..., Sn £ Lr(E, F) and [Si,..., Sn] C K. Then [Si o R,..., Sn o R] C K and
((Si,..., Sn) o R = ((Si o R,..., Sn o R).
If,in addition, G is Dedekind complete, then [T o Si,..., T o Sn] C K and
T o £(Si,..., Sn) = C(T o Si,..., T o Sn).
< Under the indicated hypotheses the operators S ^ S o R from Lr (G, F) to Lr (E, F) and S ^ T o S from Lr (E, G) to Lr (E, F) are lattice homomorphisms, see [1, Theorem 7.4 and 7.5]. Therefore, it is sufficient to apply [5, Proposition 2.6]. >
3.6. Proposition. Let E and F be vector lattices with F Dedekind complete. Assume that p £ H(C, K), Si,..., SN £ Lr(E, F), and [Si,..., SN] C K. If S* denotes the restriction of the order dual S' to F~, the order continuous dual of F, then [S*,..., S*] C K and
((Si ,...,Sn )* = ((S*,...,SN).
< By Krengel-Synnatschke Theorem [1, Theorem 5.11] the map S ^ S* is a lattice homomorphism from Lr(E, F) into Lr(F~,E~), see [1, Theorem 7.6]. Thus, we need only to apply [5, Proposition 2.6]. >
3.7. Proposition. The second formula in Theorem 3.4 and Proposition 3.6 were obtained by A. V. Bukhvalov [2] under some additional restrictions.
References
1. Aliprantis C. D., Burkinshaw O. Positive Operators.—N. Y.: Acad. Press, 1985.—367 p.
2. Bukhvalov A. V. Nonlinear majorization of linear operators // Dokl. Acad Nauk SSSR.—1988.—Vol. 298, № 1.—P. 14-17.
3. Kusraev A. G. Dominated Operators.—Dordrecht: Kluwer, 2000.—446 p.
4. Kusraev A. G. Homogeneous Functional Calculus on Vector Lattices.—Vladikavkaz, 2008.—34 p.— (Preprint / IAMI VSC RAS; № 1).
5. Kusraev A. G. Functional calculus and Minkowski duality on vector lattices // Vladikavkaz Math. J.—2009.—Vol. 11, № 2.—P. 31-42.
6. Kusraev A. G., Tabuev S. N. On Disjointness Preserving Bilinear Operators // Vladikavkaz Math. J.—2004.—Vol. 6, № 1.—P. 58-70.
Received August 4, 2009.
Kusraev Anatoly Georgievich South Mathematical Institute Vladikavkaz Science Center of the RAS, Director 22 Markus Street, Vladikavkaz, 362027, Russia E-mail: [email protected]
