OPERATORS BY A CHRONOLOGICAL INTEGRAL

Memoirs on Dierential Equations and Mathematical Physics

Volume 11, 1997, 47{66

Ioseb Gogodze and Koba Gelashvili

REPRESENTATION OF

-SEMIGROUPS OF

OPERATORS BY A CHRONOLOGICAL INTEGRAL

of real variable whose domain of values is endowed with minimal algebraic and limiting structures. It is proved that an exponential function of a linear (possibly unbounded) operator can be represented by means of a chrono- logical integral which preserves a number of properties of a chronological exponent.

1991 Mathematics Subject Classication. 28B10, 47D03.

Key words and Phrases. monoid, Riemann's chronological integral, for- mula of arrow inversion, formula of integration by parts, c⁰-semigroup of operators, exponent.

reziume. namdvili cladis PunqciisaTvis, romlis mniSvnelobaTa simravleSi minimaluri algebruli da zGvariTi struqturebia Setani- li, gansazGvrulia marJvena da marcxena qronologiuri integralebi.

damtkicebulia, rom CrPivi (SesaZloa SemousazGvreli) operatoris eqs- ponenta SeiZleba Carmodgenili iqnes iseTi qronologiuri integralis saxiT, romelic inaxavs mTels rigs qronologiuri eqsponentis Tvisebe- bisa.

1. Definition of the Riemann Chronological Integral 1.1. Auxiliary Denitions and Facts. Denote by M an abstract monoid whose algebraic structure is dened by a binary associative operation^f(g¹; g²) ^{7 !} g¹g^2g : M M ^! M and by the unity e. If some g ² M is invertible, then we denote its inverse by g.

The limiting structure inM is determined by means of directedness es.

This permits us to retain simplicity and generality. As usual, a family G=^fgw^gw² is said to be a directedness inM if (;) is a directed set, and w^{7 !}gw maps intoM. Given two directednesses G=^fgw^1gw¹²¹

and F =^ffw^2gw²²² in M, G is said to be a subdirectedness of F when there exists a mappingN : ^1!2 such that:

(a)gw¹=f_N⁽_w¹⁾,⁸w¹²¹;

(b) for everyw^{2 22} there exists w^{1 21} such that from w²w¹ and ww¹it followsN(w)w².

The limiting structure inM is dened by a system ^L composed of the pairs (G;g) (g²M,Gis a directedness inM) and satisfying the following restrictions:

(i) ifG=^fgw^gw²is a directedness such that gw =g for everyw², then (G;g)^2L;

(ii) if (G;g¹)^2Land (G;g²)^2L, theng¹=g²;

(iii) ifF is a subdirectedness inG, and (G;g)^2L, then (F;g)^2L. In the sequel, using the conventional terminology, if (G;g) ^2L we will say thatGconverges tog andgis a limit of G.

The limiting and algebraic structures are compatible, i.e.,

f(g¹;g²)^{7 !}g¹g^2g:MM^!M

is a continuous mapping. Under our notation this means that if

fgw^1gw¹²¹ converges to g and ^ffw^2gw²²² converges to f, then ^fgw¹

f_w^2g(_w¹_;w²⁾²¹² converges togf. (¹²) denotes a directed product of the directed sets:

(w¹;w²)(w^e1;w^e2) is equivalent to (w¹w^e1 and w²w^e2): 1.2. Denition of the Chronological Integral and Some Examples. Let (t;s)

7 ! f(t;s) map the triangle [a;b][0;] into M, where [a;b] ^R, and is a positive number. (a;b) denotes the set of all partitions of the form =^fa=s⁰¹s¹nsn=b^g, and

si=si si ¹; ^jj= max^fsi^ji= 1;:::;n^g: (a;b) is a directed set, the relation¹² meaning^j1jj2j.

For a suciently ne partition we denote

!

X

f =f(¹;s¹)f(²;s²)f(n;sn);

X

f =f(n;sn)f(²;s²)f(¹;s¹);

the arrow showing the order of co-factors in the right-hand side, which is important in the non-commutative case.

Denition 1. We say thatg²M is the right (left) chronological integral (or simply, integral) of the functionf from atob, and write

g=^Zb

a f(^! ;d)

g=^Zb

a f( ;d)

; if for some⁰²(a;b) the directedness

!

X

f

⁰²⁽a;b⁾

X

f

⁰²⁽a;b⁾

: (1)

is dened correctly and converges tog. When g= ^b

Z

a f(^! ;d)

g= ^b

Z

a f( ;d)

;

is invertible, we say thatg²M is the right (left) integral of the functionf fromb toa, and write

g= ^a

Z

b f(^! ;d)

g= ^a

Z

b f( ;d)

:

The directedness (1) is dened for every⁰such that^j0j< . (a;b) is a directed set, and for every ¹, ² there exists their majorant. Therefore in Denition 1 the values of the integral do not depend on the choice of⁰. In what follows, the notation^fg02(a;b⁾means that we consider the given directedness starting from some⁰.

When the values of the right and of the left integrals coincide, we can omit the arrow and write^R_a^bf(;d). Such cases arise usually whenM is a commutative monoid, or when the values of the functionf commute, since for a suciently ne there takes place^P! f =^P f, and therefore the arrow can be omitted.

Remark 1. In case of necessity (for example, when on the support M we can determine in two ways a structure of monoid consistent with the limiting structure), in our notation the binary operation will be indicated regarding to which the integral is taken

~

!

X

f; ^~Zb

a f(;d):

Example 1. (The Riemann integral). Lett^{7 !}f(t) map [a;b] into^R. If we consider^Rwith the ordinary convergence and with the operations O, +, then Denition 1 for the function^f(t;s)^{7 !}f(t)s^g: [a;b]^R!Rprovides the Riemann's integral of the functionf.

Example 2. (the multiplicative integral). Let us consider the limit of products of the form

Y

= exp(A(sn)(sn sn ¹))exp(A(s¹)(s¹ s⁰)); (2) where =^fa=s⁰sn =b^gandA(s) is a continuous function from [a;b] to the spaceB(E) of bounded linear operators in the Banach spaceE. The limit is taken as^jj!0, is denoted as

b

x

Z

a exp(A(s)ds) and is called the multiplicative integral ([1]).

If we consider B(E) with the operation of addition, f : [a;b] ^!B(E), and apply Denition 1, then we obtain the ordinary Riemann integral (as in the foregoing example): ^R_a^bf()d =^R_a^bf()d.

ConsiderB(E) with the operation of composition and apply for (t;s)^! exp(A(t)s) Denition 1. Then from the existence of^~R_a^bexp(A)(sds) there follows that of ^R_a^xbexp(A(s)ds), and hence their equality.

Example 3. (T-exponent). LetA(t),t²[a;b], be a piecewise-continuous family in a noncommutative Banach algebra.

By the denition ([2]), theT-exponentU = Exp^R_a^tA(s)dsis the solution of the Cauchy problem for the evolution equation

@U@t ⁼A(t)U; U^jt⁼a = 1(unity of the algebra);

and the following condition is fullled:

Exp^Zt

a A(s)ds= lim

j^j!0exp(A(sn)(sn sn ¹))exp(A(s¹)(s¹ s⁰)): The Banach algebra with its limiting structure and operations of composi- tion and unity is a monoid such that one can apply Denition 1 to the map- ping (;s)^{7 !}exp(A()s). Obviously, from the existence of exp^~Rt

aA(s)ds there follows that of Exp^R_a^tA(s)ds and hence their equality (we can prove that these integrals exist simultaneously).

Example 4. LetA(t),t²[a;b] be a family of unbounded linear operators on the Banach space X. Under certain conditions (see [3]), for any a s t b and for suciently ne = ^fs = s⁰ sn = t^g the limit of products (2) is dened correctly, and there exists a strong limit

lim

j^j!0

Q

=U(s;t).

1.3. Algebraic Properties of Integrals. Directly from the denition we have Proposition 1. Let for some >0f : [a;b][0;]^!M andf(t;0) =e for everyt. Then^R_t^tf(;d) =e,⁸t²[a;b].

The following result is analogous to the formula of arrow inversion for the chronological exponent ([4]).

Proposition 2. (Formula of arrow inversion). Let f : [a;b][0;]^!M, > 0, let there exist f(t;s), ⁸(t;s) ² [a;b][0;] and let for some t¹, t²²[a;b] there exist_t^Rt2

1

f(^! ;d) and_t^Rt2

1

f( ;d). Then⁹_t^Rt1

2

f(^! ;d) and

t¹

Z

t² f(^! ;d) = ^t

2

Z

t¹ f( ;d): (3)

Proof. For the sake of simplicity, let us consider rst the case t¹ t². By our condition, there exists suciently ne ^{0 2} (t¹;t²) such that

!

P

f

⁰²⁽t¹;t²⁾and

P

f

⁰²⁽t¹;t²⁾converge respectively to

t²

R

t¹f(^! ;d) and^Rt2

t¹f( ;d). The binary operation is continuous inM. There- fore

!

X

¹ f^X

² f

(⁰;⁰⁾⁽¹;²⁾²⁽t¹;t²⁾⁽t¹;t²⁾ (4)

converges to

^Rt²

t¹f(^! ;d)

^Rt²

t¹f( ;d)

. Consequently,

!

P

f

P

f

(⁰;⁰⁾⁽;⁾²⁽t¹;t²⁾⁽t¹;t²⁾, being the subdirectedness in (4), con- verges to the same limit. Taking into account that for every ⁰

!

P

f^P f =

P

f^P! f =e, we obtain

^Rt²

t¹f(^! ;d)

^Rt²

t¹f( ;d)

=e.

Analogously we obtain that

^Rt²

t¹f( ;d)

^Rt²

t¹f(^! ;d)

=e. Thus we have determined both sides in (3), hence (3) holds.

Let nowt²< t¹. By Denition 1, from the condition of our proposition there follows the existence of the integrals

t¹

Z

t² f(^! ;d) =

^Zt²

t¹ f(^! ;d)

; ^t

1

Z

t² f( ;d) =

^Zt²

t¹ f( ;d)

: Applying the case considered above, we can see that

t²

Z

t¹ f(^! ;d) = ^t

1

Z

t² f( ;d): Consequently,

t¹

Z

t² f(^! ;d) =

^Zt²

t¹ f(^! ;d)

=

^Zt¹

t² f( ;d)

=

= ^t

2

Z

t¹ f( ;d):

Proposition 3. Let >0, f : [a;b][0;] ^! M and t¹, t², t^{3 2} [a;b].

Then:

(a) if there exist_t^Rt2

1

f(^! ;d),_t^Rt3

2

f(^! ;d) and_t^Rt3

1

f(^! ;d), then

t³

Z

t¹ f(^! ;d) =

^Zt²

t¹ f(^! ;d)

^Zt³

t² f(^! ;d)

; (5)

(b) if there exist_t^Rt2

1

f( ;d),_t^Rt3

2

f( ;d) and_t^Rt3

1

f( ;d), then

t³

Z

t¹ f( ;d) = ^t

3

Z

t² f( ;d) ^t

2

Z

t¹ f( ;d):

Proof. Let us prove the case (a) (the case (b) can be proved analogously).

Lett¹< t²< t³. Owing to the continuity of the binary operation inM, the directedness

!

X

¹ f^X!

² f

(⁰¹;⁰²⁾⁽¹;²⁾²⁽t¹;t²⁾⁽t²;t³⁾

converges to the right-hand side of (5). On the other hand, being the subdirectedness of the directedness

!

P

f

⁰²⁽t¹;t³⁾, it also converges to the left-hand side of (5). Thus (5) holds.

The remaining ve cases are reduced to that proven above. As an exam- ple, consider the caset³< t²< t¹:

t³

Z

t¹ f(^! ;d) =

^Zt¹

t³ f(^! ;d)

=

=

^Zt²

t³ f(^! ;d)

^Zt¹

t² f(^! ;d)

=

=

^Zt¹

t² f(^! ;d)

^Zt²

t³ f(^! ;d)

=

=

^Zt²

t¹ f(^! ;d)

^Zt³

t² f(^! ;d)

:

Proposition 4. Let for some > 0 fi : [a;b][0;] ^! M, i ^{2 f}1;2^g, f¹(t;s)f²(t;s) =f²(t;s)f¹(t;s),⁸(t;s)²[a;b][0;] and lett¹;t²²[a;b].

Then:

(a) if there exist_t^Rt2

1

fi(^! ;d),i^2f1;2^g, then_t^Rt2

1

(f¹(^! ;d)f²(^! ;d)) does exist and equals the product

^Rt²

t¹fi(^! ;d)

^Rt²

t¹f³ i(^! ;d)

,⁸i^2f1;2^g.

(b) if there exist_t^Rt2

1

fi( ;d),i^2f1;2^g, then_t^Rt2

1

(f¹( ;d)f²( ;d)) does exist and equals the product

^Rt²

t¹fi( ;d)

^Rt²

t¹f³ i( ;d)

,⁸i^2f1;2^g. Proof. We prove here only the case (a), because the case (b) can be proved similarly.

Consider the caset¹t². Introduce the notation =^X!

(f¹f²); ⁽¹_;²⁾=

!

X

¹ f^1X!

² f²; u() = (;); that is, u: (t¹;t²)^!(t¹;t²)(t¹;t²). By the condition, =_u⁽⁾, and we can easily see that for every (¹;²) ² (t¹;t²)(t¹;t²) ^{9b 2}

P(t¹;t²) such thatu()(¹;²),^{8 b}. Hence^fg02(t¹;t²⁾is the subdirectedness of the directedness

f⁽¹;^2)g(0;⁰⁾⁽¹;²⁾²⁽t¹;t²⁾⁽t¹;t²⁾:

Taking into consideration the continuity of operation and also the equality ⁽¹_;²⁾=

P

!²f^2P!1f¹, we get

t²

Z

t¹ (f¹(^! ;d)f²(^! ;d)) =

^Zt²

t¹ fi(^! ;d)

^Zt²

t¹ f³ i(^! ;d)

; ⁸i^2f1;2^g: Ift¹> t², then with regard for the case considered above, we arrive at

t¹

Z

t² f¹(^! ;d)f²(^! ;d)) =

^Zt¹

t² fi(^! ;d)

^Zt¹

t² f³ i(^! ;d)

; ⁸i^2f1;2^g: Equating the elements inverse to the left and right-hand sides, we com- plete the proof of the case (a).

2. One-Parameter Integral and Formulas of Partial Integration

2.1. One-parameter Integral. Let f : [0;]^!M, >0. For every interval [a;b] we may assume f to be the mapping with respect tot : (t;s)^!f(s) maps the rectangle [a;b][0;] intoM. Hence for =^fa=s⁰¹s¹

nsn=b^gsuch that^jj< ,^P! f =f(s¹)f(s²)f(sn), and ^P f =f(sn)f(s²)f(s¹) are dened correctly, and we may speak on the integrability off in terms of Denition 2.

It appears that there may exist simultaneously the left and the right integrals of such (incomplete) subintegral functions; and if they do, then they are equal (notation, of course, reects this fact). Indeed, givenab, let us dene u : (a;b) ^! (a;b): if = ^fa = s^{0 1} s¹

n sn = b^g, then u() = ^fa = u^{0 1} u¹ n un = b^g, where uk=a+(b sn k),k^2f0;1;:::;n^g,k =a+(b n k⁺¹),k^2f1;:::;n^g. Clearly, u(u()) = (i.e., u is one-to-one), ^ju()^j = ^jj, and ^P!_u⁽⁾f =

P f,^P! f =^P_u⁽⁾f for suciently ne. Thus each of the following directednesses

!

X

f

⁰²⁽a;b⁾

X

f

⁰²⁽a;b⁾

is a subdirectedness of the other.

The casea > b can be easily reduced to that considered above.

Proposition 5. Let for some >0f : [0;]^!M,a;b^2R, and let there exist^R_a^bf(d). Then there exists_a^bR+t

+tf(d),⁸t^2R, and

b

Z

a f(d) = ^b

+t

Z

a⁺t f(d); ⁸t^2R: (6) Proof. Evidently, it suces to consider the caseab. Denote

1

X= (a;b); ^X2 = (a+t;b+t)

and dene u: (a;b)^!(a+t;b+t) as follows: to every =^fa=s^{0 1}s¹nsn=b^gthere corresponds

u() =^fa+t=s⁰+t¹+ts¹+tn+tsn+t=b+t^g: It is clear that u is one-to-one and ^ju()^j = ^jj, ^{8 2 P1}. More- over, for suciently ne ^{2 P2}, we have ^P_u ¹⁽⁾f =^Pf. Therefore

Pf

^2P2jj<is the subdirectedness of the directedness

Pf

^2P1jj<. Consequently, (6) holds.

Proposition 6. Let for some >0f : [0;]^!M and let for everyt0 there exist ^Rt

0

f(d). Then

^Rt

0

f(d)

t⁰ is a one-parameter semigroup in M, i.e.,

t^1+Z t²

0

f(d) =

^Zt¹

0

f(d)

^Zt²

0

f(d)

; ⁸t¹;t²0:

Proof. Combining the results of Propositions 3 and 5, we can prove the above proposition:

t^1+Z t²

0

f(d) =

^Zt¹

0

f(d)

t^Z1+t²

t¹ f(d)

=

=

^Zt¹

0

f(d)

^Zt²

0

f(d)

:

Corollary 1. If in the conditions of Proposition 6 there is alsof(0) =e, then

^Rt

0

f(d)

t⁰ is a one-parameter submonoid inM.

Denote by(^b a;b) the set of all partitions of the interval [a;b] of the form =^fa=t⁰<< tn=b^g(we imply thata < b). As usual,

ti=ti ti ¹; ^jj= max^fti^ji= 1;:::;n^g: (b a;b) is ordered as follows: ¹² if^j1jj2j.

Lemma 1. Let for some >0f : [0;]^!M, f(0) =e andt0. Then each of the following two directednesses

X

f

⁰²⁽⁰;t⁾ and

ff(¹)f(²)f(_n)^gbf0=0<<ⁿ⁼t^g2b(0;t⁾

is the subdirectedness of the other.

Proof. Denote

=^X

f; ^{0 2}(0;t); =f(¹)f(²)f(n); ⁰=^f0 =⁰<< n=t^g2(0^b ;t);

and construct the mappingsu: (0;t)^! (0^b ;t) and v :(0^b ;t)^!(0;t) as follows.

Let = ^f0 = ^{0 1 1} n n = t^{g 2} (0;t). Denote s⁰ = ⁰, s¹ = j ^{2 f0};:::;n^gjj ¹ = s⁰;j > s⁰ and so on. If the constructed in such a way set ^fs⁰;:::;sk^g does not involve ^f0;:::;n^g, thensk⁺¹=j^2f0;:::;n^gjj ¹=sk;j > sk .

Not more than innsteps we obtain^fs⁰;:::;sp^gsuch that

fs⁰;:::;sp^g=^f0;:::;n^g:

Now we can determineu:u() =^f0 =s⁰<< sp=t^g2(0^b ;t).

The mappingv can be dened in a more simple manner: to every =

f0 =⁰<< n=t^gthere corresponds

v() =^f0 =⁰¹=¹n=n=t^g2(0;t):

Taking into account the properties of u and v, the following identities complete the proof:

_v⁽⁾=; ⁰ =^f0 =⁰<< n =t^g2(0^b ;t); _u⁽⁾=; ⁰²(0;t):

Corollary 2. Under the conditions of the lemma,g²M is the integral of the function f on[0;t] (from 0 to t) if and only if the directedness

ff(¹)f(²)f(n)^g0f0=⁰<<ⁿ⁼t^g2b(0;t⁾

converges to g.

Proposition 7. Let for some > 0f : [0;] ^!M, f(0) = eand t 0.

Then the existence of each of the following two integrals

t

Z

0

f(d) and

1

Z

0

f(td) implies the existence of the other one and their equality.

Proof. Consider the nontrivial caset >0 and introduce the notation:

=f(s¹)f(s²)f(sm); ⁰ =^f0 =s⁰<< s_m=t^g2(0^b ;t); =f(t¹)f(t²)f(tn); ⁰ =^ff0 =⁰<< n= 1^g2(0^b ;1)^g; where⁰and⁰are xed suciently ne partitions.

Determineu:(0^b ;1)^!(0^b ;t) as follows: to every=^f0 =⁰<<

n = 1^g2 (0^b ;1) there corresponds u() = ^f0 = t⁰ < < tn = t^g2 (0b ;t). Obviously,uis one-to-one and

ju()^j=t^jj; ⁸²(0^b ;1); ^ju ¹()^j=t ^1jj; ⁸²(0^b ;t): The identities

_u⁽⁾=; ⁸²(0^b ;1); ⁰; _u ¹⁽⁾=; ⁸²(0^b ;t); ⁰; with regard for the properties of the mappinguprove that

f^g02b(0;t⁾ and ^fg02b(0;¹⁾

are subdirectednesses of each other, which by virtue of Lemma 1 proves out proposition.

2.2. Formulas of \Partial Integration". For every invertibleg ²M, let us determine an automorphism of the monoidM:

Adgf =gf g; ⁸f ²M:

It is easily seen thatAdge=e,Adg(f¹f²) =Adgf¹Adgf²,Adg(f) =Adgf whenf is invertible inMand if^fp(t)^gt⁰ is a one-parameter semi-group in M, then^fAdgp(t)^gt⁰ is also a one-parameter semi-group.

Proposition 8. Let g : [0;] ^!M, > 0,^fp(t)^gt⁰ be a one-parameter subgroup inM, and for somea;b^2Rlet there exist the integral^Rb

aAd_p(

!⁾g(d)

^Rb

a Ad_p(⁾g(d)

. Then⁹_a^bR+t

+tAd_p(

!⁾g(d)

b^R+t

a⁺tAd_p(⁾g(d)

, and⁸t^2R we have.

b⁺t

Z

a⁺t Ad_p(

!⁾g(d) =Adp⁽t⁾

^Zb a Ad_p(

!⁾g(d)

(7)

^Zb⁺t

a⁺t Ad_p(⁾g(d) =Ad_p⁽_t^)Zb

a Ad_p(⁾g(d)

: Proof. Letab and let there exist ^Rb

a Ad_p(

!⁾g(d). Take somet ^2R and determineu: (a+t;b+t)^!(a;b) as follows: to every =^fa+t=s^{0 1}s¹n sn =b+t^gthere corresponds u() = ^fa=s⁰ t ¹ ts¹ tn tsn t=b^g. Clearly, ^ju()^j=^jj anduis one-to-one. For the sake of simplicity we also denote f(;s) =Adp⁽⁾g(s).

The evident equality^P! f =Adp⁽t^)P!u⁽⁾f, with regard for the properties ofu, proves that

!

X

f

²⁽a⁺t;b⁺t⁾;^jj< (8) is a subdirectedness of the directedness

Ad_p⁽_t^)P! f

²⁽a;b⁾;^jj<

which converges to the right-hand side of (7). Consequently, (8) also con- verges to the right-hand side of (7). By the denition of the integral, this means that (8) converges to the left-hand side (7) as well. Thus, by virtue of the uniqueness of the limit, (7) holds.

Let nowa > b, and let there exist^Rb

a Ad_p(

!⁾g(d). Hence there also exists its inverse^R_a^bAd_p(

!⁾g(d). Sinceb < a, for everytwe have

a⁺t

Z

b⁺t Ad_p(

!⁾g(d) =Ad_p⁽_t^)Za

b Ad_p(

!⁾g(d)

;

and both sides are invertible. Equating their inverse elements, we obtain (7).The version of Proposition 8 given in square brackets is proved in a similar

way.

Proposition 9. Let^fp(t)^gt⁰ and ^fq(t)^gt⁰ be one-parameter subgroups inM, and for somea;b^2Rlet there exist the integral^R_a^bAd_q(

!⁾p(d). Then there exists^Rb

a Ad_p(

!⁾q(d), and the equality

b

Z

a Ad_p(

!⁾q(d) =p(a)q( a)

^Zb a Ad_q(

!⁾p(d)

q(b)p( b) (9) takes place.

Proof. Letab and there exist^Rb

a Ad_q(

!⁾p(d). We constructu:(a;b)^! (a;b) as follows: to every=^fa=s⁰¹s¹nsn=b^gthere corresponds u() =^fa=⁰ =s^{0 1}s¹n sn =n⁺¹ =b^g. By the construction,^ju()^j2^jj.

!

X

Adp⁽⁾q(d)=

=p(¹)q( s⁰)q(s¹)p( ¹)p(n)q( sn ¹)q(sn)p( n)=

=p(⁰)q( s⁰)[q(s⁰)p( ⁰)p(¹)q( s⁰)]

[q(s¹)p( ¹)p(²)q( s¹)]

[q(sn ¹)p( n ¹)p(n)q( sn ¹)]

[q(sn)p( n)p(n⁺¹)q( sn)]q(sn)p( n⁺¹) =

=p(a)q( a)

!

X

u⁽⁾(Adq⁽⁾p(d))

q(b)p( b):

Consequently,

!

X

(Ad_p⁽⁾q(d))

²⁽a;b⁾ (10)

is a subdirectedness of the directedness

p(a)q( a)

!

X

(Ad_q⁽⁾p(d))

q(b)p( b)

²⁽a;b⁾;

which, by the conditions of the proposition and due to the continuity of the binary operation in M, converges to the right-hand side of (9). Hence, (10) converges to the right-hand side of (9) which, by the denition of the integral, proves (9).

The case wherea > beasily follows from the above proven.

Proposition 10. Let^fp(t)^gt⁰and^fq(t)^gt⁰be one-parameter subgroups in M and for some a;b ^{2 R} let there exist the integral ^R_a^bAd_q( ⁾p( d).

Then there exists^Rb

a Ad_p(⁾q(d), and

b

Z

a Ad_p(⁾q(d) =p(b)q(b)

^Zb

a Ad_q( ⁾p( d)

q( a)p( a): Proof. Just as in the case of Proposition 9, the proof is actually a simple checking.

The results proved in Propositions 9 and 10 are naturally associated with the formulas of partial integration. As is seen, the formula for the left integral is of more familiar form.

3. Integral representation ofc⁰-subgroups of operators LetA: D(A)^!X be a linear operator acting in the Banach spaceX. One of the basic results of the theory of subgroups of operators, the Hille- Iosida-Phillips theorem ([5], Ch. VIII. p. 1), states that the linear (possibly unbounded) operatorAin the Banach spaceXgenerates a strongly contin- uous semi-group of operators ^fU(t)^gt⁰ (i.e., ac⁰-semi-group) if and only ifAis densely dened, closed and has a resolvent satisfying

j( A) ^njB⁽x⁾( ) ⁿ; n= 1;2;:::; > ; (11) for some constants1 and0.

In this case,U(t) is a strong limit of operators of the typeI _tnA ⁿ as n ^{! 1} ([6], Ch. IX), where I is the identical mapping of the space X onto itself. This fact is contained in the denition of the exponent of