Operator Algebra
Note 3
Self-adjoint and Symmetry
TANAKA Akio
Hilbert space H, K
Operator from H to K A
Domain of A dom A
Graph of A G ( A ) : = { x ⊕ Ax ; x ∈ A }
Operators A, B
A ⊂ B : = G ( A ) ⊂ G ( B )
Minimum of B containing A Closure of A, described by Ā
Now closure of dom A = H
Operator from H to H Operator over H
x ∈ H <x, Ay> = <x’, y>
A*x = x’
A* that is operator over H A* is adjoint operator of A
When A ⊂ A* A is symmetric operator.
When A = A* A is self-adjoint operator.
When Ā = A** A is essentially self-adjoint.
[References]
Distance Theory Algebraically Supplemented / Distance / Tokyo October 26, 2007
Theme / Peak Symmetry and Infinity / Tokyo February 3, 2008
Tokyo April 1, 2008
Sekinan Days 