Operator Algebra
Note 1
Differential Operator and Symbol
TANAKA Akio
1
Complex variant partial differential operator z
Multi-index α = (α1, …, αn) ∈ 0n
zα: = z1α1, …, znαn
α! = α1!...αn!
|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)zα, aα(z)∈(Ucn)
P’s order Minimum of m
4
Sequence { aα(z)}α∈N0n⊂(Ucn)
5
{|α| ; z ∈ K } = 0
6
Differential operator of infinite order P (z, z) = α(z)zα, aα(z)∈(Ucn)
7
ς= (ς1, …,ςn) ∈n
Symbol P (z, ς) : = α(z)ςα
8
All of differential operators of infinite order Sheaf of linear space, expressed by D cn ∞
[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
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