Operator Algebra Note 3 Self-adjoint and Symmetry / April 1, 2008

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 Research Field of Language