Sunday, 12 July 2015

von Neumann Algebra Note 3 Compact Operator

von Neumann Algebra
Note 3
Compact Operator 

TANAKA Akio


1
Sobolev space     n
Sobolev norm  ||| |||2: = ∑|α|≤n ||Dαf||22
||| f ||||:=(∑|α|≤n|yα|2Ff(y)|2dNy
||| f ||||n 
f ∈ L∈ n
is Hilbert space by inner product corresponded with norm |||  |||.
2
Operator in Hilbert space H     A
Unit sphere of Sobolev space H     B
Compact subset of H    
Complete orthonormal system of H     {φn }n=1
Pn: = n r=1 <f, φ>φr
Pis finite class operator.
1-Pis convergent over by the next.
D is all bounded.
Arbitrary ε> 0
Finite set of D {x1, …, xs}
x  D ||– xt || < ε/ 2    1   s
N  n and x  D
||(1-Pn)x|| < ||(1-Pn)xt|| + ||(1-Pn)(xt-x)|| ≤ε/ 2 +ε/ 2 =ε
3
A is compact operator.

[References]


Tokyo April 6, 2008

Sekinan Research Field of Language

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