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A limit of products of the form
 
A limit of products of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p0750201.png" /></td> </tr></table>
+
$$
 +
\prod _  \Delta  = \
 +
e ^ {A ( s _ {n} ) ( s _ {n} - s _ {n-} 1 ) } e ^ {A
 +
( s _ {n-} 1 ) ( s _ {n-} 1 - s _ {n-} 2 ) }
 +
{} \dots e ^ {A ( s _ {1} ) ( s _ {1} - s _ {0} ) } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p0750202.png" /> is a continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p0750203.png" /> with values in the space of bounded operators on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p0750204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p0750205.png" /> is the partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p0750206.png" /> by points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p0750207.png" />. The limit is taken as the diameter of the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p0750208.png" /> and is denoted by
+
where $  A $
 +
is a continuous function on $  [ a , b ] $
 +
with values in the space of bounded operators on a Banach space $  E $
 +
and $  \Delta $
 +
is the partition of $  [ a , b ] $
 +
by points $  s _ {0} = a , s _ {1} \dots s _ {n} = b $.  
 +
The limit is taken as the diameter of the partition $  | \Delta | \rightarrow 0 $
 +
and is denoted by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p0750209.png" /></td> </tr></table>
+
$$
 +
{\int\limits _ { a } ^ { {b }  } } tilde
 +
\mathop{\rm exp}  A( s)  d s .
 +
$$
  
If the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502010.png" /> commute for different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502011.png" />, then
+
If the operators $  A ( t) $
 +
commute for different $  t $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502012.png" /></td> </tr></table>
+
$$
 +
{\int\limits _ { a } ^ { {b }  } } tilde
 +
\mathop{\rm exp}  A ( s)  d s  = \
 +
e ^ {\int\limits _ {a}  ^ {b} A ( s)  d s } .
 +
$$
  
A product integral is a convenient way of representing an [[Evolution operator|evolution operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502013.png" /> for a differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502014.png" /> (see [[#References|[1]]]). Here
+
A product integral is a convenient way of representing an [[Evolution operator|evolution operator]] $  U ( t , \tau ) $
 +
for a differential equation $  \dot{X} = A ( t) X $(
 +
see [[#References|[1]]]). Here
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
U ( t , \tau )  = \
 +
{\int\limits _  \tau  ^ { t }  } tilde  \mathop{\rm exp}  A ( s) d s .
 +
$$
  
The products whose limit is the latter integral are also the evolution operators for the equations with piecewise-constant operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502016.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502017.png" />.
+
The products whose limit is the latter integral are also the evolution operators for the equations with piecewise-constant operators $  \widetilde{A}  ( t) = A ( s _ {k} ) $
 +
for $  s _ {k-} 1 \leq  t \leq  s _ {k} $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502019.png" /> are two continuous operator-valued functions, then
+
If $  A $
 +
and $  B $
 +
are two continuous operator-valued functions, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
{\int\limits _ { a } ^ { b }  } tilde  \mathop{\rm exp}
 +
( A ( s) + B ( s) ) d s =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502021.png" /></td> </tr></table>
+
$$
 +
= \
 +
\lim\limits _ {| \Delta | \rightarrow 0 }  {\prod _ { k= } 1 ^ { {n }  } } tilde e ^ {A ( s _ {k} ) ( s _ {k} - s _ {k-} 1 ) } e ^ {B ( s _ {k} ) ( s _ {k} - s _ {k-} 1 ) } ,
 +
$$
  
where the sign <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502022.png" /> over the product means that the factors with low indices are written to the right of the factors with high indices.
+
where the sign $  \widetilde{ {}}  $
 +
over the product means that the factors with low indices are written to the right of the factors with high indices.
  
 
Formulas (1) and (2) can be generalized to certain classes of differential equations with unbounded operator functions, from which representations of the solutions of parabolic and Schrödinger-type partial differential equations in the form of integrals over the space of trajectories (path integrals, continual integrals, cf. [[Integral over trajectories|Integral over trajectories]]) are obtained (see [[#References|[2]]]).
 
Formulas (1) and (2) can be generalized to certain classes of differential equations with unbounded operator functions, from which representations of the solutions of parabolic and Schrödinger-type partial differential equations in the form of integrals over the space of trajectories (path integrals, continual integrals, cf. [[Integral over trajectories|Integral over trajectories]]) are obtained (see [[#References|[2]]]).
Line 29: Line 77:
 
Formulas of the type (2) are at the basis of certain numerical methods for solving equations.
 
Formulas of the type (2) are at the basis of certain numerical methods for solving equations.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502023.png" /> is a scalar-valued continuous function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502024.png" /> is an operator-valued function of bounded variation, then the limit
+
If $  f $
 +
is a scalar-valued continuous function and $  F $
 +
is an operator-valued function of bounded variation, then the limit
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502025.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {| \Delta | \rightarrow 0 } \
 +
{\prod _ { k= } 1 ^ { {n }  } } tilde
 +
e ^ {f ( s _ {k} ) ( F ( s _ {k} ) - F ( s _ {k-} 1 ) ) }  = \
 +
{\int\limits _ { a } ^ { {b }  } } tilde
 +
\mathop{\rm exp} ( f ( t)  d F ( t) )
 +
$$
  
exists; it is called the product Stieltjes integral. These integrals have been applied in the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502026.png" />-contracting matrices and operators (see [[#References|[3]]], [[#References|[4]]]).
+
exists; it is called the product Stieltjes integral. These integrals have been applied in the theory of $  J $-
 +
contracting matrices and operators (see [[#References|[3]]], [[#References|[4]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.L. Daletskii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.L. Daletskii,  "Functional integrals connected with operator evolution equations"  ''Russian Math. Surveys'' , '''17''' :  5  (1962)  pp. 1–107  ''Uspekhi Mat. Nauk'' , '''17''' :  5  (1962)  pp. 3–115</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.P. Potapov,  "The multiplicative structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502027.png" />-contractive matrix functions"  ''Trudy Moskov. Mat. Obshch.'' , '''4'''  (1955)  pp. 125–236  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.P. Ginzburg,  "Multiplicative representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502028.png" />-contractive operator functions I"  ''Mat. Issled. (Kishinev)'' , '''2''' :  2  (1967)  pp. 52–83  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.L. Daletskii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.L. Daletskii,  "Functional integrals connected with operator evolution equations"  ''Russian Math. Surveys'' , '''17''' :  5  (1962)  pp. 1–107  ''Uspekhi Mat. Nauk'' , '''17''' :  5  (1962)  pp. 3–115</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.P. Potapov,  "The multiplicative structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502027.png" />-contractive matrix functions"  ''Trudy Moskov. Mat. Obshch.'' , '''4'''  (1955)  pp. 125–236  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.P. Ginzburg,  "Multiplicative representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075020/p07502028.png" />-contractive operator functions I"  ''Mat. Issled. (Kishinev)'' , '''2''' :  2  (1967)  pp. 52–83  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods of modern mathematical physics" , '''2''' , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.D. Dollard,  Ch.N. Friedman,  "Product integration" , Addison-Wesley  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods of modern mathematical physics" , '''2''' , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.D. Dollard,  Ch.N. Friedman,  "Product integration" , Addison-Wesley  (1979)</TD></TR></table>

Revision as of 08:07, 6 June 2020


A limit of products of the form

$$ \prod _ \Delta = \ e ^ {A ( s _ {n} ) ( s _ {n} - s _ {n-} 1 ) } e ^ {A ( s _ {n-} 1 ) ( s _ {n-} 1 - s _ {n-} 2 ) } {} \dots e ^ {A ( s _ {1} ) ( s _ {1} - s _ {0} ) } , $$

where $ A $ is a continuous function on $ [ a , b ] $ with values in the space of bounded operators on a Banach space $ E $ and $ \Delta $ is the partition of $ [ a , b ] $ by points $ s _ {0} = a , s _ {1} \dots s _ {n} = b $. The limit is taken as the diameter of the partition $ | \Delta | \rightarrow 0 $ and is denoted by

$$ {\int\limits _ { a } ^ { {b } } } tilde \mathop{\rm exp} A( s) d s . $$

If the operators $ A ( t) $ commute for different $ t $, then

$$ {\int\limits _ { a } ^ { {b } } } tilde \mathop{\rm exp} A ( s) d s = \ e ^ {\int\limits _ {a} ^ {b} A ( s) d s } . $$

A product integral is a convenient way of representing an evolution operator $ U ( t , \tau ) $ for a differential equation $ \dot{X} = A ( t) X $( see [1]). Here

$$ \tag{1 } U ( t , \tau ) = \ {\int\limits _ \tau ^ { t } } tilde \mathop{\rm exp} A ( s) d s . $$

The products whose limit is the latter integral are also the evolution operators for the equations with piecewise-constant operators $ \widetilde{A} ( t) = A ( s _ {k} ) $ for $ s _ {k-} 1 \leq t \leq s _ {k} $.

If $ A $ and $ B $ are two continuous operator-valued functions, then

$$ \tag{2 } {\int\limits _ { a } ^ { b } } tilde \mathop{\rm exp} ( A ( s) + B ( s) ) d s = $$

$$ = \ \lim\limits _ {| \Delta | \rightarrow 0 } {\prod _ { k= } 1 ^ { {n } } } tilde e ^ {A ( s _ {k} ) ( s _ {k} - s _ {k-} 1 ) } e ^ {B ( s _ {k} ) ( s _ {k} - s _ {k-} 1 ) } , $$

where the sign $ \widetilde{ {}} $ over the product means that the factors with low indices are written to the right of the factors with high indices.

Formulas (1) and (2) can be generalized to certain classes of differential equations with unbounded operator functions, from which representations of the solutions of parabolic and Schrödinger-type partial differential equations in the form of integrals over the space of trajectories (path integrals, continual integrals, cf. Integral over trajectories) are obtained (see [2]).

Formulas of the type (2) are at the basis of certain numerical methods for solving equations.

If $ f $ is a scalar-valued continuous function and $ F $ is an operator-valued function of bounded variation, then the limit

$$ \lim\limits _ {| \Delta | \rightarrow 0 } \ {\prod _ { k= } 1 ^ { {n } } } tilde e ^ {f ( s _ {k} ) ( F ( s _ {k} ) - F ( s _ {k-} 1 ) ) } = \ {\int\limits _ { a } ^ { {b } } } tilde \mathop{\rm exp} ( f ( t) d F ( t) ) $$

exists; it is called the product Stieltjes integral. These integrals have been applied in the theory of $ J $- contracting matrices and operators (see [3], [4]).

References

[1] Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)
[2] Yu.L. Daletskii, "Functional integrals connected with operator evolution equations" Russian Math. Surveys , 17 : 5 (1962) pp. 1–107 Uspekhi Mat. Nauk , 17 : 5 (1962) pp. 3–115
[3] V.P. Potapov, "The multiplicative structure of -contractive matrix functions" Trudy Moskov. Mat. Obshch. , 4 (1955) pp. 125–236 (In Russian)
[4] Yu.P. Ginzburg, "Multiplicative representations of -contractive operator functions I" Mat. Issled. (Kishinev) , 2 : 2 (1967) pp. 52–83 (In Russian)

Comments

References

[a1] M. Reed, B. Simon, "Methods of modern mathematical physics" , 2 , Acad. Press (1972)
[a2] J.D. Dollard, Ch.N. Friedman, "Product integration" , Addison-Wesley (1979)
How to Cite This Entry:
Product integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Product_integral&oldid=48305
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article