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Trends in Mathematics
Information Center for Mathematical Sciences
Volume 2, 1999, Pages 000{000
STABILITY OF ANALYTIC OPERATOR-VALUED
FUNCTION SPACE INTEGRALS
KUN SOO CHANG
Abstract. Since the Feynman integral was introduced by Feynman in 1948,
there has been considerable progress on stability of Feynman integrals in recent
years. The rst stability theorem for the integral was introduced by Johnson
in 1984 as a bounded linear operator on 2 (R) and it was extended to the
case of p (R) with respect to measures, potentials and wave of functions. In
this paper, we will briey review the previous results and introduce the recent
results on the stability of analytic operator-valued function space integrals.
L
L
1. Introduction
In 1984, Johnson proved a bounded convergence theorem for the operator-valued
function space integral [7]. This is the rst stability theorem for the integral as a
bounded linear operator on L2 (Rn ) where n is any positive integer. In [10], Johnson
and Skoug introduced stability theorems for the integral as an L(Lp (RN ); Lp (RN ))
theory, 1 < p 2. Chang studied stability theorems for the integral as a bounded
linear operator from L1 (R) to C0 (R) [2]. In those papers mentioned above, they
treated certain functionals which involve only the Lebesgue measure on the interval
(0; t).
In [8], Johnson and Lapidus established stability theorems for the integral as an
L(L2 (RN ); L2(RN )) theory for certain functionals involving any Borel measures on
(0; t). Chang and Ryu proved theorems insuring stability with respect to potentials
and wave functions for the integral as a bounded linear operator on Lp (RN ) for
certain functionals involving some Borel measures on (0; t) [5].
Lapidus provided a nice stability theorem for the modied Feynman integral [11].
The modied Feynman integral is a variation of Nelson's approach to the Feynman
integral via the Trotter product formula [12].
Recently, Chang, Ko, and Ryu studied the stability of the operator-valued function space integral as an L(L1 (R); C0 (R)) theory for the functionals involving some
Borel measures on (0; t) with respect to potentials, wave functions and measures
[4].
0
1991 Mathematics Subject Classication. 28C20.
Key words and phrases. Feynman integral, operator-valued function space integral, stability
theorem, potential, wave function, measure, bounded linear operator.
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c 1999 Information Center for Mathematical Sciences
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KUN SOO CHANG
2. Stability Theorems involving the Lebesgue Measure
For our present purposes, t > 0 will be xed. C [0; t] will denote the space of
continuous functions on [0; t]. C0 [0; t] denotes the space of functions x in
C [0; t] such that x(0) = 0. C [0; t] will denote the product of copies of C [0; t] and
C0 [0; t], respectively. m will be referred to as Wiener measure and m will denote
the product of copies of 1-dimensional Wiener measure.
Let 1 < p 2 be given and let p0 be such that p1 + p1 = 1. Given > 0, in
Lp (R ), and in R , let
R-valued
0
(I (F ) )( ) =
Z
C0 [0;t]
F (; 12 x + ) (; 21 x(t) + ) dm (x):
This formula may dene, for each > 0, a bounded linear operator I (F ) in
L = L(Lp (R ); Lp (R )), the space of bounded linear operator from Lp (R ) to
Lp (R ). If this is so, and if the operator-valued function ! I (F ) has an analytic
continuation to C + := f 2 C : Re > 0g, we denote this analytic continuation
Ian (F ). Finally given a real parameter q, q =
6 0, the operator-valued, analytic
0
0
Feynman integral Jqan (F ) is dened by
J (F ) = Jqan (F ) := !;
limiq Ian (F )
where the limits is taken in the strong operator topology and where approaches
;iq through C + .
let
Let be a bounded complex-valued Lebesgue measurable function on R , and
(2.1)
Z t
F (x) = exp
0
(x(s)) ds :
Theorem 2.1. Let be an essentially bounded, complex-valued, Lebesgue measur-
able function on R and let F be given by (2:1). Then J (F ) exists as a bounded
linear operator on L2 (R ).
The following theorem is the rst stability theorem for the Feynman integral
introduced by Johnson in 1984 [7].
Theorem 2.2 (Stability Theorem for L2 case). Let fmg be a sequence of
complex-valued, Lebesgue measurable functions on R all of which are essentially
bounded by the number L. Suppose that m ! a.e. on R . Then is also
essentially bounded by L and, by Theorem 2.1, J (F ) and J (Fm ), m = 1; 2; ,
all exist, where F is given by (2:1) and Fm is given by replacing by m in (2:1).
Further, J (Fm ) converges to J (F ) in the strong operator topology.
Now we introduce the stability result for Lp case. Let the numbers p; ; and r
satisfy the following restrictions :
(a) Let 1 < p 2 be given and let p0 be such that 1p + p1 = 1.
(b) = 2;p p . Note that = (p) is an increasing function taking the interval (1; 2]
onto the interval (1; +1]. We remark that is chosen so that multiplication
by a function in L (R ) is a bounded linear operator from Lp (R ) to Lp (R ).
(c) is a positive integer satisfying 1 < 2 .
(d) 22; < r +1 except that we allow 1 r +1 when p = 2 and = +1.
0
0
STABILITY OF ANALYTIC OPERATOR-VALUED FUNCTION SPACE INTEGRALS
3
The mixed norm space Lr = Lr ([0; t] R ) will play a key role throughout. A
complex-valued Lebesgue measurable function g on [0; t] R is said to be in Lr
if and only if g(s; ) is in L (R ) for almost every (a.e.) s in [0; t] and kg(s; )k is
in Lr ([0; t]). Note that Lr2 Lr1 if 1 r1 r2 +1.
Given and m , m = 1; 2; from Lr ([0; t] R ), let F and Fm , m = 1; 2; be dened on C [0; t] by
Z t
(2.2)
F (x) = exp
(2.3)
Fm (x) = exp
Z 0t
0
(t ; s; x(s)) ds
m (t ; s; x(s)) ds;
where is the potential function which is of primary interested in quantum mechanics.
Theorem 2.3 (Stability theorem for Lp case). Let h be in Lr . Let fmg be
a sequence from Lr such that m (s; u) ! (s; u) for a.e. (s; u) and such that, for
every m, jm (s; u)j h(s; u) for a.e. (s; u). Then is also in Lr and J (F ) and
J (Fm ), m = 1; 2; all exist as elements of L(Lp (R ); Lp (R )). Further, J (Fm )
converges to J (F ) in the strong operator topology.
Remark. Theorem 2.2 is the special case of Theorem 2.3 where p = p0 = 2,
= +1, and ; m 's are time-independent.
0
3. Stability Theorems involving Borel measures
Let C + and C + be the set of all complex numbers with positive real part and
all nonzero complex numbers with nonnegative real part, respectively.
Let t > 0 be given. M (0; t) will denote the space of complex Borel measures on
the interval (0; t). A measure in M (0; t) is said to be continuous if (f g) = 0
for every in (0; t) and a measure in M (0; t) is said to be discrete if there is an
at most countable subset
P fi : i = 1; 2; g of (0; t) and a summable sequence h!i i
from C such that = 1
i=1 !i i where i is the Dirac measure with total mass one
concentrated at i . Then every measure in M (0; t) has a unique decomposition,
= + into a continuous part and a discrete part .
M (0; t) will denote the subset of M (0; t) which satises the following conditions:
(a) If is the continuous part of in M (0; t), then the Radon-Nikodym derivative
djj=dm exists and is essentially bounded where m is the Lebesgue measure on
(0; t).
P
P
(b) If = 1
discrete part of in M (0; t), then 1
j!i ji; i=1 !i i is the
i
=1
converges where = N=2; 1 + 1 = 1.
For in M (0; t) , let L : ([0; t] RN ) L : be the space of C -valued Borel
measurable functionals on [0; t] RN such that
0
0
kk : nZ
(0;t)
k(s; )k djj(s)
o1=
< 1:
Let F be a functional on C [0; t]. Given > 0, in Lp (RN ), and in RN , let
[I (F ) ]( ) =
Z
C0 [0;t]
F (;1=2 x + ) (;1=2 x(t) + ) dmw (x);
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KUN SOO CHANG
where C [0; t] is the space of RN -valued continuous functions on [0; t], C0 [0; t] =
fx 2 [0; t] : x(0) = 0g and mw is the Wiener measure. If for mL-a.e. in RN ,
[I (F ) ]( ) exists in Lp (RN ) and if the map ! I (F ) gives an element of
L(Lp (RN ); Lp (RN )), we say that the operator-valued function space integral I (F )
exists for . Suppose there exists 0 (0 < 0 1) such that I (F ) exists for all
0 < < 0 and there exists an L(Lp (RN ); Lp (RN ))-valued function which is
analytic in C +;0 C + \ fz 2 C : jz j < 0 g and agrees with I (F ) on (0; 0 ),
then this L(Lp ; Lp )-valued function is called the operator-valued function space
integral of F associated with and in this case, we say that I (F ) exists for
in C +;0 . If I (F ) exists for in C +;0 and I (F ) is strongly continuous in
C
+;0 C + \ fz 2 C : jz j < 0 g, we say that I (F ) exists for in C +;0 . When is purely imaginary, I (F ) is called the analytic operator-valued Feynman integral
of F .
First, we consider stability with respect to the potentials.
P
Let be in M (0; t) and let H belong to L : . Let = + 1
i=1 !i i be
the decomposition of into its continuous part and discrete part and let hn i be a
sequence in L : . For nonnegative integer n, we let
0
0
0
0
(3.1)
Fn (y) =
and
(3.2)
Fn(m) (y) =
Z
n
(0;t)
Z
(0;t)
(s; y(s)) d(s)
m (s; y(s)) d(s)
for y in C [0; t]
n
for y in C [0; t]:
Theorem 3.1. Suppose m (s; u) ! (s; u) as m ! 1 for mL-a.e. (s; u) and
jm (s; u)j H (s; u) for all positive integer m and for mL-a.e. (s; u). Then is
also in L : and for all positive integers m; n, I (Fn ) and I (Fn(m) ) exist for all in C + and
I (Fn(m) ) ! I (Fn ) strongly as m ! 1:
P
n
Let f (z ) = 1
n=0 an z be an analytic function in C + \ fz 2 C : jz j < 0 g for
some positive real number 0 . For positive integer m and y in C [0; t], let
Z
(3.3)
F (y) = f
(3.4)
F (m) (y) = f
(0;tZ)
(s; y(s)) d(s)
(0;t)
and
m (s; y(s)) d(s) :
P1
(m)
Theorem
3.2.
Suppose
for
all
in
C
and
for
all
positive
integer
m
,
+
;
n=1 jan jkI (Fn )k
0
P1
and n=1 kI (Fn )k are convergent. Then I (F ) and I (F (m) ); m = 1; 2; , exists
for all in C +;0 and
I (F (m) ) ! I (F ) strongly as m ! 1 in C +;0 :
Now, we consider stability with respect to the wave functions.
STABILITY OF ANALYTIC OPERATOR-VALUED FUNCTION SPACE INTEGRALS
5
Theorem 3.3. Suppose
(m) ! in Lp (RN ) as m ! 1. Then for positive integer
m, I (F ) and I (F (m) ) (m) exist in Lp (RN ) for in C +;0 and I (F (m) ) (m) !
I (F ) in Lp (RN ) as m ! 1.
0
0
Lastly, we treat the stability theorem with respect to the measures. Let and
m ; m = 1; 2; , be in M (0; t) such that m converges to in the total variation
norm and let be bounded which is in L : \ (\1
m=1 L :m ). Let
Z
(3.5)
Fn (y) =
(3.6)
Fn(m) (y) =
(0;t)
Z
(s; y(s)) d(s)
(0;t)
n
and
(s; y(s)) dm (s)
n
P
n
Let f (z ) = 1
n=0 an z and let
F m (y) = f
(3.7)
Z
(0;t)
(s; y(s)) dm (s)
for y in C [0; t]:
for y in C [0; t]:
Theorem 3.4. Suppose that there are positive real number 0 and K such that for
all in C +;0 and all positive real number m,
1
X
n=0
jan jkI (Fnm )k < K and
1
X
n=0
jan jkI (Fn )k < K:
Then for in C +;0 and for positive integer m, I (F ) and I (F m ) exist and
I (F m ) ! I (F ) uniformly in on all compact subsets of C +;0 in the operator
norm topology. Moreover for all 0 < < 0 ,
kI (F m ) ; I (F )k 2t0 Tm for m = 1; 2; where Tm = supfjf (z1) ; f (z2 )j : jz1 ; z2 j Mmg with Mm = k ; m kkk1. Here,
kk1 denotes the supremum norm of on (0; t) RN .
4. Stability Theorems: L(L1 (R); C0 (R)) Theory
Let C0 (R) will denote the space of C -valued continuous functions on R which
vanish at 1 with the supremum norm. L1(R) is the space of Borel measurable,
C -valued functions on R such that j jRis integrable with respect to the Lebesgue
measure m on R with the norm k k1 = j j dm. L(L1 (R); C0 (R)) will denote the
space of bounded linear operators from L1 (R) to C0 (R). Let M~ (0; t) denote the
space of complex Borel measures on the interval (0; t) which satisfy the following
conditions;
jj exists
(1) If is the continuous part of , the Radon-Nikodym derivative ddm
and is essentially bounded, where m is the Lebesgue measure on (0; t).
(2) =
k
X
wj j + , where j is the Dirac measure at j
j =1
< k < t and wj 2 C for j = 1; 2; ; k.
2 (0; t), 0 < 1 <
6
KUN SOO CHANG
Let r 2 (2; 1] and 2 M~ (0; t). Let L1r: ([0; t] R) L1r: be the space of all
Borel measurable C -valued functions on [0; t] R such that
k k1r: Z
1=r
(0;t)
k (s; ) kr1 djj(s)
is nite. If is in L1r: and = + is the Lebesgue decomposition, it is not
dicult to show that 2 L1r: \ L1r: : Let 2 M~ (0; t). A Borel measurable
C -valued function on [0; t] R is said to belong to L11: if
kk11: =
Z
(0;t)
k(s; )k1 djj(s)
is nite.
Let F be a functional from C [0; t] to C . Given > 0; 2 L1 (R) and 2 R, let
(I (F ) )( ) =
Z
C0 [0;t]
F (;1=2 x + ) (;1=2 x(t) + ) dm(x):
If I (F ) is in C0 (R) as a function of and if the correspondence ! I (F ) gives
an element of L L(L1 (R); C0 (R)), we say that the operator-valued function space
integral I (F ) exists. Next suppose that there exists 0 (0 < 0 < 1) such that
I (F ) exists for all in (0; 0 ) and further suppose that there exists an L-valued
function which is analytic in C +;0 and agree with I (F ) on (0; 0 ). Then this
L-valued function is denoted by Ian (F ) and is called the operator-valued analytic
Wiener integral of F associated with . Finally, let q be in R with 0 < jqj < 0 .
Suppose there exists an operator Jqan (F ) in L such that for every in L1 (R),
Jqan (F ) is the weak limit of Ian (F ) as ! ;iq through C +;0 . Then Jqan (F ) is
called the operator-valued function space integral of F associated with q.
Firstly, we establish the stability for the operator-valued Feynman integral of
functionals involving some Borel measures on (0; t) with respect to potentials.
k
X
Theorem 4.1. Let be in M~ (0; t) with = + wpp . Let H 2 L1r: and
p=1
H (p ; ) be essentially bounded for each p = 1; 2; ; k. Let (N ) ; N = 1; 2; , be
Borel measurable functions on [0; t] R such that for m-a.e.
(N ) ;! as N ! 1
and
j(N ) j jH j for N = 1; 2; :
Then and (N ) belong to L1r: . For nonnegative integer n, let
Fn (x) =
and
Fn(N ) (x) =
Z
n
(0;t)
Z
(0;t)
(s; x(s)) d(s)
(N ) (s; x(s)) d(s)
n
for x in C [0; t]
for x in C [0; t]:
STABILITY OF ANALYTIC OPERATOR-VALUED FUNCTION SPACE INTEGRALS
Then for all real q > 0, Jqan (Fn ) and Jqan (Fn(N ) ) exist for each N 2
N ! 1,
Jqan (Fn(N ) ) ! Jqan (Fn ) in the operator norm.
Let
Z
F (y) = f
(s; y(s)) d(s) for y in C [0; t]
N
7
and as
(0;t)
and
F (N )(y) = f
Z
(0;t)
(
N
)
(s; y(s)) d(s)
for y in C [0; t]:
Now, we consider the stability for the operator-valued Feynman integral of functionals with respect to wave functions.
Theorem 4.2. Let f (N )g be a sequence in L1(R) and k (N ) ; k1! 0 as
N ! 1. Then for N 2 N ; Jqan (F ) and Jqan (F (N ) ) (N ) exist in C0 (R) for real q
with 0 < jqj < 0 . Moreover,
kJqan (F (N ) )
(N ) ; J an (F )
q
k1 ! 0 as N ! 1:
Finally, we treat the stability theorem for the operator-valued Feynman integral
with respect to measures.
Theorem 4.3. Let be a continuous function bounded by c and let and N ; N =
1; 2; be in M~ (0; t). Assume that
N ! weakly:
Let
FN (y) = f
Then
Z
(0;t)
(s; y(s)) dN (s)
for y in C [0; t]:
Ian (FN ) ! Ian (F ) in the operator norm,
uniformly in on all compact subset of C +;0 :
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
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J.S.Chang and G.W.Johnson, The Feynman integral and Feynman's operational Calculus;
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L ;C
L
L
;C
L
;C
;C
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KUN SOO CHANG
[9] G.W.Johnson and D.L.Skoug, The Cameron-Storvick function space integral: an L( p p )
theory, Nagoya Math. J. 60 (1976), 93-137.
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L ;L 0
Department of Mathematics
Yonsei University
Seoul 120-749, Korea
E-mail : [email protected]