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differential operator
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(Definition)
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Roughly speaking, a differential operator is a mapping, typically understood to be linear, that transforms a function into another function by means of partial derivatives and multiplication by other functions.
On
 , a differential operator is commonly understood to be a linear transformation of
 having the form
where the sum is taken over a finite number of multi-indices
 , where
 , and where  denotes a partial derivative of  taken  times with respect to the first variable,  times with respect to the second variable, etc. The order of the operator is the maximum number of derivatives taken in the above formula, i.e. the maximum of
 taken over all the  involved in the above summation.
On a
 manifold  , a differential operator is commonly understood to be a linear transformation of
 having the above form relative to some system of coordinates. Alternatively, one can equip
 with the limit-order topology, and define a differential operator as a continuous transformation of
 .
The order of a differential operator is a more subtle notion on a manifold than on
 . There are two complications. First, one would like a definition that is independent of any particular system of coordinates. Furthermore, the order of an operator is at best a local concept: it can change from point to point, and indeed be unbounded if the manifold is non-compact. To address these issues, for a differential operator  and  , we define
 the order of  at  , to be the smallest
 such that
for all
 such that  . For a fixed differential operator  , the function
 defined by
is lower semi-continuous, meaning that
for all  sufficiently close to  .
The global order of  is defined to be the maximum of
 taken over all  . This maximum may not exist if  is non-compact, in which case one says that the order of  is infinite.
Let us conclude by making two remarks. The notion of a differential operator can be generalized even further by allowing the operator to act on sections of a bundle.
A differential operator  is a local operator, meaning that
if  in some neighborhood of  . A theorem, proved by Peetre states that the converse is also true, namely that every local operator is necessarily a differential operator.
References
- Dieudonné, J.A., Foundations of modern analysis
- Peetre, J. , “Une caractérisation abstraite des opérateurs différentiels”, Math. Scand., v. 7, 1959, p. 211
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"differential operator" is owned by rmilson. [ full author list (2) ]
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(view preamble)
Cross-references: converse, neighborhood, sections, act on, even, infinite, lower semi-continuous, fixed, unbounded, point, independent, transformation, continuous, topology, coordinates, manifold, derivatives, operator, order, variable, multi-indices, finite, sum, linear transformation, multiplication, partial derivatives, function, Transforms, mapping
There are 18 references to this entry.
This is version 7 of differential operator, born on 2002-02-15, modified 2008-02-10.
Object id is 1984, canonical name is DifferentialOperator.
Accessed 11553 times total.
Classification:
| AMS MSC: | 35-00 (Partial differential equations :: General reference works ) | | | 53-00 (Differential geometry :: General reference works ) | | | 47E05 (Operator theory :: Ordinary differential operators) | | | 47F05 (Operator theory :: Partial differential operators) |
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Pending Errata and Addenda
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 = 0$](http://images.planetmath.org:8080/cache/objects/1984/l2h/img24.png "$\displaystyle T[f^{k+1}](x) = 0$"){kind=small}
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