Operators

[drufae]

While using several operators(e.g.grad,div,curl) we often separate them and treat them as vectors in their own right,performing most algebraic and vector operation on them.How is this possible? are not operators and their operands inexorably linked together.In order to separate them we would need to define a whole set of specialized rules just to use them.

This is not an isolated example several times while solving differential equations we replace the differential operator with a variable ,say 's'(and hence this technique in some areas gets its name as the s-operator method) (ref:http://en.wikibooks.org/wiki/Circuit_Theory/Second-Order_Solution)and proceed to manipulate it as a variable.

If there is indeed a technique or theory which allows us to dichotomize operands and operators then it is that which i hope to discuss in this post.If not then how do these techniques hold up?

 

p.s.I'm sorry if I have posted this in the wrong place but the topic seems to belong to general mathematics rather than a specialized field.

[ajb]

While using several operators(e.g.grad,div,curl) we often separate them and treat them as vectors in their own right,performing most algebraic and vector operation on them.

 

You mean that the set of endomorphisms of a vector space over a field form a ring over that field.

 

You can add linear operators as well as multiply by scalars and still get a linear operator.

 

Moreover they form an algebra under composition.

 

How is this possible? are not operators and their operands inexorably linked together.In order to separate them we would need to define a whole set of specialized rules just to use them.

 

Rings/algebras are quite wide objects in a sense.

 

What is often more important is that the set of endomorphisms is a Lie algebra where the Lie bracket is given by the commutator of linear operators.

 

Anyway, the point is that you can consider algebras and Lie algebras of linear operators quite independent of the space they act on.