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Abstract spectral theory 2
Hu xiyu
September 8, 2016

1

Adjoint operator

definition:for a Hilbert space H,for a linear operator A : H → H,we have
the adjoint operator of A,called A∗ ,we have map:
B(H) → B(H) : (A, D(A)) −→ (A∗ , D(A∗ ))
where D(A∗ ) is definite as:
D(A∗ ) = {y ∈ H|∃η ∈ H, ∀x ∈ D(A), < Ax, y >H =< x, η >H }
A∗ : D(A∗ ) → H, y → A∗ y = η

2

A theorem of Von Neumann

Theorem 1 For every unbounded operator A with domain D(A) dense in
H,we have: ker(A∗ ) = (Im(A))⊥
...

remark:If A is bounded
...
in
fact:
D(A∗ ) = {y ∈ H|∃η ∈ H, ∀x ∈ H, < Ax, y >=< x, η >}
1

x →< Ax, η > is a bounded linear functional
...

beside we also have:|||A∗ ||| = |||A|||,but i do not konw how to prove it
now
...

every self-adjoint operator is symmetric
...

definition:unbounded symmetric operator is positive iff:
< Ax, x >H ≥ 0, ∀x ∈ D(A)

4

Two theorem concerning |||T |||

Theorem 2 |||T ||| = sup | < T x, x >H |
||x||H ≤1

proof:

|||T ||| =

sup

||T x||H

||X||H ≤1

=

sup ||T x||H ||X||H
||x||H ≤1

≥

sup | < T x, x >H |
...
(2)
||x||H ≤1

(1),(2) ⇒ Q
...
D
Theorem 3 ||T x||2 ≤ |||T ||| · | < T x, x >H |
H
proof:
definite a(x, y) =< T x, y >H
a(x, y) bilinear form , positive-definite
...
e:||T x||2 ≤ | < T x, x >H | · |||T |||
...
E
...


5

A crucial prop

prop:
(A, D(A)) symmetric operator,∀x ∈ D(A), ∀λ = µ + iγ ∈ C, γ = 0,
||(λI − A)x||2 ≥ γ 2 ||x||2
H
H
3

proof:
||(λI − A)x||2 = < (µI − A)x + iγx, (µI − A)x + iγx >
= ||(µI − A)x||2 + i < γx, (µI − A)x > −i < γx, (µI − A)x > −||γx||2
= ||(µI − A)x||2 − ||γx||2

⇒ ||γx||2 ≤ ||(λI − A)x||2
and we have ||γx||2 = γ 2 ||x||2
...
E
...

proof:
Ker(λI − A) = 0
(λI − A)x||2 ≥ γ 2 ||x||2 ⇒ ker(λI − A) = 0
...
e to prove: if (λI − A)xn = yn , ∀n ∈ N ∗
...
E
...
the following prop
are equivalent:
(1)A is self-adjoint
...

(3) Im(A ± iI)=H
...

(1) ⇒ (2):
(x, Ax),||x||H + ||Ax||H
=< x, Ax >=< Ax, x >⇒< Ax, x >∈ R
(2) ⇒ (3):
< Ax, y >=< x, A∗ y >⇔< (A ± iI)x, y >=< x, (A∗
¯
A = A ± iI,we have,by theorem 2:
∗
¯
(ImA )⊥ = (kerA)
...
e ∀x ∈ D(A) = D(A∗ )
< Ax, y >=< x, Ay >
∀y ∈ H,∃x1 , x2 ∈ H:
(A + iI)x1 = y, (A − iI)x2 = y
< Ax, y >=< Ax, (A + iI)z >=
Q
...
D

5



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Title: abstract spectral theory 2
Description: a brief introduction to symmetric operator

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