Operator Algebra Note 1 Differential Operator and Symbol

 

Operator Algebra

Note 1

Differential Operator and Symbol 

TANAKA Akio

1

Complex variant partial differential operator     _z

Multi-index     α = （α₁, …, α_n)　∈　₀ⁿ

_z^α: = _z1^α1, …, _zn^αn

α! = α₁!...α_n!

|α| = _j

2

Topological space     X

All of X’s open sets     (X)

U ∈ (X)

Regular function f(z) : U → 

All of regular functions over U     (U)

Sheaf      (ⁿ) ∋ U  (U)

3

P (z, _z) = _α(z)_z^α, a_α(z)∈(U_cⁿ)

P’s order     Minimum of m

4

Sequence { a_α(z)}α∈N₀ⁿ⊂(U_cⁿ)

5

{^|α| ; z ∈ K } = 0

6

Differential operator of infinite order     P (z, _z) = _α(z)_z^α, a_α(z)∈(U_cⁿ)

7

 ς= (ς₁, …,ς_n) ∈ⁿ

Symbol     P (z, ς) : = _α(z)ς^α

8

All of differential operators of infinite order       Sheaf of  linear space, expressed by D _c^n ∞

[Reference]

<On infinite order, refer to the next.>

Operator Algebra / Conjecture 1 / Order of Word / February 16, 2008

Tokyo February 24, 2008

Sekinan Research Field of Language