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differential operator (Definition)

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 $ \mathbb{R}^n$, a differential operator is commonly understood to be a linear transformation of $ \mathcal{C}^\infty (\mathbb{R}^n)$ having the form

$\displaystyle f \mapsto \sum_{I} a^I f_I,\quad f\in \mathcal{C}^\infty (\mathbb{R}^n), $
where the sum is taken over a finite number of multi-indices $ I=(i^1,\ldots,i^n)\in \mathbb{N}^n$, where $ a^I\in \mathcal{C}^\infty (\mathbb{R}^n)$, and where $ f_I$ denotes a partial derivative of $ f$ taken $ i_1$ times with respect to the first variable, $ i_2$ 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 $ i_1+\ldots+i_n$ taken over all the $ I$ involved in the above summation.

On a $ \mathcal{C}^\infty $ manifold $ M$, a differential operator is commonly understood to be a linear transformation of $ \mathcal{C}^\infty (M)$ having the above form relative to some system of coordinates. Alternatively, one can equip $ \mathcal{C}^\infty (M)$ with the limit-order topology, and define a differential operator as a continuous transformation of $ \mathcal{C}^\infty (M)$.

The order of a differential operator is a more subtle notion on a manifold than on $ \mathbb{R}^n$. 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 $ T$ and $ x\in M$, we define $ \mathop{\mathrm{ord}}\nolimits _x(T)$ the order of $ T$ at $ x$, to be the smallest $ k\in\mathbb{N}$ such that

$\displaystyle T[f^{k+1}](x) = 0$
for all $ f\in\mathcal{C}^\infty (M)$ such that $ f(x)=0$. For a fixed differential operator $ T$, the function $ \mathop{\mathrm{ord}}\nolimits (T):M\rightarrow \mathbb{N}$ defined by
$\displaystyle x\mapsto \mathop{\mathrm{ord}}\nolimits _x(T)$
is lower semi-continuous, meaning that
$\displaystyle \mathop{\mathrm{ord}}\nolimits _y(T)\geq \mathop{\mathrm{ord}}\nolimits _x(T)$
for all $ y\in M$ sufficiently close to $ x$.

The global order of $ T$ is defined to be the maximum of $ \mathop{\mathrm{ord}}\nolimits _x(T)$ taken over all $ x\in M$. This maximum may not exist if $ M$ is non-compact, in which case one says that the order of $ T$ 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 $ T$ is a local operator, meaning that

$\displaystyle T[f](x) = T[g](x),\quad f,g\in\mathcal{C}^\infty (M),\; x\in M,$
if $ f\equiv g$ in some neighborhood of $ x$. A theorem, proved by Peetre states that the converse is also true, namely that every local operator is necessarily a differential operator.


References

  1. Dieudonné, J.A., Foundations of modern analysis
  2. Peetre, J. , “Une caractérisation abstraite des opérateurs différentiels”, Math. Scand., v. 7, 1959, p. 211



"differential operator" is owned by rmilson. [ full author list (2) ]
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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 MSC35-00 (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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Limit-order topology? by yzarc on 2005-06-16 04:48:54
What is the "limit-order topology"? A google search returns only one result, namely this page. Perhaps it should be defined.
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