Homogeneous functions of regular linear and bilinear operators

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 i,...,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 ^ : C ^ R with continuous restriction to K . The expression b(x i ,... ,x n ) can be correctly defined provided that the compatibility condition [x i ,..., x n ] C K is hold, see [5].

Denote by H V ( R N , K) and H A ( R N , K) respectively the sets of all lower semicontinuous sublinear functions ^ : R N ^ R U { + ^} and upper semicontinuous superlinear functions ^ : R N ^ R U{-^} which are finite and continuous on a fixed cone K C R N . Put H V ( R N ) : = H _V ( R N , { 0 } ) and H _A ( R N ) := H _A (R ^N , { 0 } ).

Denote by G _V ( R N , K) and G _A ( R N , K) respectively the sets of all lower semicontinuous gauges ^ : R N ^ R ₊ U { + ^} and upper semicontinuous co-gauges ^ : R N ^ R ₊ U {-to} which are finite and continuous on a fixed cone K C R N . Put G _V ( R N ) := G _V ( R N , { 0 } ) and G a ( R ⁿ ) := G _a ( R ⁿ , { 0 } ). Observe that G v ( R ⁿ ) C H ( R N ) and G _A ( R ^N ) C H _A ( R ^N ), see [4, 5].

Everywhere below E, F , and G denote vector lattices, while L ^r (E, F) and BL r (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.

• 2.    Functions of Bilinear Operators

• 2.1.    Lemma. Let E, F, and G be vector lattices, b i ,...,b N G BL r (E,F ; G), and b := (b i ,...,b N ). Let у G H _V ( R N ), ^ G H (R N ), $(b i (x ₀ ,y ₀ ),...,b N (x o ,y o )) and ^(b i (x o ,y o ),... ,b N (x o ,y o )) are well defined in G for all 0 6 x o 6 x and 0 6 y o 6 y. Denote x:= (x i ,..., x n ) G E ⁿ and y := (y i ,..., y m ) G F m , m,n G N . Then the sets

A partition of x G E + is any finite sequence (x i ,..., x n ), 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.

nm

^(b x,y):=  EE

nm

k(bx,y):=  EEib(bi(xi,yj),...,bN(xi,yj)) : n,m G N,x G Prt(x), y G Prt(y) к i=i j=i are upward directed and downward directed, respectively.

C Assume that (x i ,..., x n ) and (x i ,..., x , ₀ ) are partitions of x while (y i ,..., y m ) and (y _i ⁰ , . . . , y _m ⁰ 0 ) are partitions of y. By The Riesz Decomposition Property of vector lattices there exist finite double sequences (u i,k ) i 6 n,k 6 n 0 in E and (v j,i ) j 6 m,i 6 m 0 in F . such that

, U i,k = x i ,          U i,.. = x k (i :=1,...,n, k :=1,...,n ⁰ );

k =i                  i =i

X_l =_i ^ve = y j ,    X_j =_i v j,i = y 0 ^jj:=^i,...^,m,l ^:=^i,...^,m ⁰ ).

In particular, (ui,k)i6,, k6,0 and (vj,l)j6m, l6m0 are partition of x and y, respectively. Taking subadditivity of ϕ into consideration we obtain n,m n,m     n0,m0                    n0,m0

= Ek Eb i (U i,k ,v j,i ),...,Eb N (U i,k ,v j,l ) i,j =i      k,l =i                       k,l =i

52 i?(bi(xi,yj), . .., bN(xi, yj)) i,j=i n,m     n0,m0

= X bf X (bi(u i,j=i     k,l=i

i,k ^{, v} j,l )^{, . . . ,} b N (^u i,k ^{, v} j,l )

n,m n ⁰ ,m ⁰

⁶               $(b i (U i,k ,V j,l ), .. . ,b N (U i,k ,V j,l)) .

i,j =i k,l =i

In a similar way we get n0 ,m0                                         n0,m0 n,m

X *?(bi(xk,y'l),...,bN(xk,y°)) 6EE

• 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:

  (sup B)(x,y) = sup

  
  

  nm

  XX

  i=i j=i

  
  

  ^bk(i,j) ^(xi,yj)

  
  

  ^,

  
  

  (inf B)(x, y) = inf

  
  

  nm

  XX

  i=i j=i

  
  

  bkCijkxGyj)

  
  

  ^,

  
  

• 2.3.    Theorem. Let E, F , and G be vector lattices with G Dedekind complete, bi,..., bN E BLr(E, F; G), and b := (bi,..., bN). Assume that y E H^(RN), ^ E H^(RN), y?(b_i(xo,yo),...,bN(xo,yo)) and ^(b_i(xo,yo),...,bN(xo,yo)) are well defined in G for all 0 6 xo 6 x and 0 6 yo 6 y, y(b; x, y) is order bonded above, and ^(b; x, y) is order bounded below for all x E E+ and y E F+. Then y(bi,..., bN) and ^(bi,..., bN) are well defined in BLr (E, F; G) and for every x E E+ and y E F+ the representations

where supremum and infimum are taken over all naturals n,m,l E N, functions k : {1,..., n}x {1,...,m} ^ {1,...,l}, partitions (xi,...,xn) E Prt(x) and (yi,... , ym) E Prt(y), and arbitrary finite collections bi . . . , bi E B.

C See [6, Proposition 2.6]. B

b(b i ,.. ., b N )(x,y) = sup y(b; x,y),

V’Cb i ,..., b N )(x, y) = inf ^(b; x, y)

hold with supremum over upward directed set and infimum over downward directed set. If E 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.

C Denote Ьд := A_ib_i + • • • + An b N for A := (A i ,..., An ) E R ^N and observe that if the set { Ьд : A E dy } is order bounded in BL r (E, F; G), then by [5, Theorem 4.4] (b i ,..., b N ) exists in BL r (E, F; G) and the upper envelope representation y(b i ,...,b N ) = sup { b д : A E dy } holds. Take arbitrary A r := (A r ,..., A N ) E dy (r := 1,..., l), k : { 1,..., n } x { 1,..., m } ^ { 1,..., l } , x := (x i ,..., x n ) E Prt(x), and y := (y i ,..., y_m) E Prt(y). Making use of Lemma 2.2 and [5, Theorem 4.4] we deduce:

n,m                n,m N                  n,m

52 bAk(i.j) (xi,yj ) = 52 52Ak(i’j) Mxi^j ) 6 52 b(bi (x^' ),‘“,bN (xi ’У )) 6 a, i,j=i                    i,j=i s=i                      i,j=i where a is an upper bound of y(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 y(bi,...,bN) is well defined and y(bi,..., bN)(x, y) 6 y(b; x, y). Surely, in above reasoning we could take (xi,..., xn) E DPrt(x) provided that E has the principal pro jection property.

Conversely, let f (x, y) stands for the right-hand side of the first equality. Observe that if (A i ,..., A n ) E dy and u E E + , v E F + , then by [5, Theorem 4.4] we have

NN

52 Akbk(u, v) = ( 52 Akbk) (u, v) 6 «bCbi,..., bN)(u, v) k=ik=i and again y?(bi(u, v),..., bN (u, v)) 6 y(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,mn,m

52 yWx^yj),..., bN (xi,yj)) 6 52 y(bi,---,bN)(xi,yj) 6 y(bi,...,bN)(x,y) i,j=ii,j=i and thus f (x, y) 6 y(bi,..., bN)(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. B

• 2.4.    Corollary. Let E, F, G, y, ф, b i ,..., b N be the same as in 2.1, b := b(b i ,..., b N ) and b : = ^(b i ,..., b N ). Assume that, in addition, E = F has the strong Freudenthal property and b i ,... ,b N are orthosymmetric. Then for every x E E the representations

b(x,x) = sup < ^ y(b i (x i , | x | ),... ,b N (x i , | x | )) : (x i ,.. .,x n ) E DPrt( | x | ) к i=i

n

.

.,X n ) E DPrt( | x | )I,

b(x,x) = inf ^2^(bi(xi, |x|),... ,bN(xi, |x|)) : (xi,. i=i hold with supremum and infimum over upward and downward directed sets, respectively.

C It is sufficient to check the first formula. We can assume x E E+. Denote by g(x) the right-hand side of the desired equality. From Theorem 2.3 we have g(x) 6 b(bi,..., bN)(x, x). To prove the reverse inequality take two disjoint partitions of x, say x0 := (xi,..., x0) and x00 : = (x"..., xm), and let (xi,..., xn) E DPrt(x) be their common refinement. Since bi,..., bN are orthosymmetric we deduce l,m

(bi ^(x r , ^x s ),..., ^b N ^(x r , ^x s ))

r,s=i nn

= ^2i(xi,xi),... ,bN(xi,xi)) = J^ i (xi,x),... ,bN(xi,x)).

i=1                                   i=1

Passing to supremum over all x ⁰ and x ⁰⁰ we get the desired inequality. B

• 3.    Functions of Linear Operators

• 3.1.    Theorem. Let E and F be vector lattices with F Dedekind complete, T i , ..., Tn E L ^r (E,F), and T := (Ti,...,T n ). Let y E H _V ( R N ), ф E H _A (R ^N ), $(Tix ₀ ,...,T n x q ) and ф(TlX Q , ..., T n x q ) are well defined in F for all 0 6 x q 6 x. If for every x E E the sets

  ⁿ

  ^(T; x) = \ У^ bTx k ,..., T n x k ) : k =1

  
  

  (x , . .

  
  

  ⁿ

  ф(Т;x) = ^ ^^^(Tix k ,...,T n x k ): (xi,. k =1

  
  

  .

  
  

  . , x n ) E Prt(x) , .,x n ) E Prt(x) I

  
  

• 3.2.    Remark. (1) Assume that E , F , T _i ,..., T n , y, and ф are the same as in [4, Theorem

• 5.2] . Then y(T i ,..., T n )x > _ix,..., Tnx) and ^(Ti,..., Tn)x 6 ^b(T_ix,..., Tnx) for all x E E+. In particular, if RN C dom(y) П dom(^) and (p(T_ix,..., TNx) > ^(T_ix,..., TNx) for all x E E+, then y(Ti,..., Tn) > ^(Ti,..., Tn).

The above machinery is applicable to the calculus of order bounded operators.

are order bounded from above and from below respectively, then

n ) and ф(Т,..., T n ) exist in L r (E, F ), and the representations

b(Ti,..., T n )x = sup ^(T; x), ф(T₁,...,T N )x = inf ф(Т; 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) .

C Follows immediately from 2.3. B

• (2)    Assume that y E H(C; [x]) and y(0, t ₂ ,..., t N ) = 0 for all (t ₁ ,..., t N ) E dom(y). Then evidently y(x i ,..., xn ) E { x i }^ 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 b(T i ,..., Tn ), cp. [2].

• 3.3.    Let E and F be vector lattices with E relatively uniformly complete and F Dedekind complete. Then for T i ,...,Tn E L + (E,F), x i ,...,x N E E + , and a i ,...,a N E R + with a i + • • • + a N = 1 we have

• 3.4.    Theorem. Let E and F be vector lattices with F Dedekind complete and T,..., Tn E L + (E, F). Suppose that y E G _V ( R ^N , r + ) and ^ E G _A (R ^N , r + ) are increasing and [T i ,..., T N ] C dom(y) П dom(^). Then for every x E E + the representations hold

(T i ^a 1 .. .T N N Xx ? ¹ . ..x N N ) 6 (T i x i ) ^a 1 ... ( Tnxn ) ^a N .

The reverse inequality holds provided that a _i + • • Bon = 1, ( — 1) ^k (1 — a _i — • —a k )ap.. . • a k > 0 (k := 1,..., N — 1), and x i ^ 0, f (x i ) » 0 for all i with a i < 0.

C Apply [4, Corollary 6.7] with K = R + , C = 1, y o (t) = ^ i (t) = y ₂ (t) = t “ 1 ... t a N . B

r N

. ,xn ) 6 x},

. ,xn )> x},

№,.

. ,T N )x = sup ^T k x k : x i ,... ,xn E E ₊ , C(x i ,. k=i

^(Ti,... ,TN)x = inf ^Tkxk : xi,...,xN E E+, ^o(xi,. k=i with supremum over upward directed set and infimum over downward directed set.

C Suppose that y(T i , ...,T N ) exists and x E E + . If x i ,...,x N E E + and y ^o (x _i ,..., xn ) 6 x, then making use of the Bipolar Theorem, positivity of y(Ty..., Tn ), and [4, Corollary 6.8] we deduce

N

X T k x k 6 y (T i ,... ,Tn )(y ^o (x i ,... ,xn )) 6 y(T i ,... ,Tn )x. k=i

To prove the reverse inequality take (x i ,...,x _n ) E Prt(x), A ^k = (A k ,..., A N ) E dy = { y ^o 6 1 } (k := 1,..., n), and put u i := 52n=_i A ^k x _k . If a := (a i ,..., a N ) E dy ^o = { y 6 1 } , then h a,A ^k i 6 y(a)y ^o (A ^k ) 6 1 and thus

N   Nn    n

£ a i U i = £ ai £ A ^k x k = J> A ^k i x k 6 x.

i=i         i=i   k=i         k=i

It follows from [5, Theorem 5.4] that y ^o (u _i ,..., Un ) 6 x.

Denote S (A) := Ai Ti + • • • + An TN with A := (Ai,..., An )• Let f (x) is the right-hand side of the first equality. Then nN

^ S (A ^k )(x k ) = 5^T i U i 6 f (x).

k=i               i=i

It remains to observe that y(T i ,..., Tn ) = sup { S(A) : A E dy } by [5, Theorem 4.4]. B

• 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 ^ E H (C, K ). Assume that S i ,..., S n E V(E, F ) and [S i ,..., S n ] C K. Then [S i о R,..., S n о R] c K and

• 3.6.    Proposition. Let E and F be vector lattices with F Dedekind complete. Assume that ^ E H(C, K ), S i ,..., S n E V(E, F ), and [S i ,..., S n ] C K. If S * denotes the restriction of the order dual S 0 to F n , the order continuous dual of F, then [S * ,..., S N ] C K and

b(S 1 , .. .,Sn ) о R — b(S i о R,...,Sn о R).

If, in addition, G is Dedekind complete, then [T о S i ,..., T о S n ] c K and

T о b(S i , ...,S n ) = b(T о S i ,...,T о S n ).

C Under the indicated hypotheses the operators S ^ S о R from L ^r (G,F ) to L^T(E,F ) and S ^ T о S from L ^r (E, G) to L ^r (E,F ) are lattice homomorphisms, see [1, Theorem 7.4 and 7.5]. Therefore, it is sufficient to apply [5, Proposition 2.6]. B

∗  ∗∗

WSl^.^S N ) — ^⁽S i ^,...^,S N ).

C By Krengel-Synnatschke Theorem [1, Theorem 5.11] the map S ^ S * is a lattice homomorphism from L ^r (E, F) into L ^r (F n ,E ⁿ ), see [1, Theorem 7.6]. Thus, we need only to apply [5, Proposition 2.6]. B

• 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.