ADJOINT

Question 1

PROPERTIES SELFADJOINT

Answer 1

• real eigenvalues
• eigenvalues corresponding to different eigenvalues are orthogonal
• are diagonalisable

Question 2

EXISTENCE OF ADJOINT THEOREM

Answer 2

let X,Y be Hilbert spaces and T∈B(X,Y)
there exists a unique adjoint operator T*∈B(X,Y) such that
- <Tx,y> = <x,T*y>, ∀x∈X, ∀y∈Y
- ||T*||<=||T||

Question 3

PROPERTIES OF ADJOINTS

Answer 3

1. (T*)* =T
2. ||T*|| = ||T||
3. ||T*T|| <= ||T||^2
4. T,S ∈B(X,Y)
   ⟹ (λT + μS)* = λ(-)T*+ μ(-)S*
5. T ∈B(X,Y) and S∈B(Y,Z)
   ⟹ (ST)* = T*S*
6. T∈K(X,Y) ⟹ T*∈K(Y,X)

where X,Y,Z hilbert spaces

Question 4

INVERSE OF ADJOINT

Answer 4

(T*)^(-1) = (T^(-1))*

if T∈B(X) invertible, then so is T*

Question 5

SPECTRUM OF ADJOINT

Answer 5

ρ(T*) = (ρ(T))(-)
and
δ(T*) = (δ(T))(-)

for T∈B(X)

Question 6

ORTHOGONALITY OF ADJOINT
T∈B(X)

Answer 6

(ran(T-λ))^(⊥) = ker(T*-λ(-))
(ran(T*-λ(-)))^(⊥) = ker(T-λ)

Question 7

ORTHOGONALITY OF ADJOINT
T∈B(X,Y)

Answer 7

(ran(T))^(⊥) = ker(T*)
(ran(T*))^(⊥) = ker(T)

Question 8

NORMAL OPERATOR

Answer 8

if TT*=T*T

Question 9

SELFADJOINT OPERATOR

Answer 9

if T=T*

Question 10

UNITARY OPERATOR

Answer 10

if T*T = I_X and TT* = I_Y

Question 11

PROPERTIES OF NORMAL OPERATORS

Answer 11

1. ||Tx|| = ||T*x||, ∀x∈X
2. ker(T-λ) = ker(T*-λ(-)), ∀λ∈K
3. ρ(T) = {λ∈K : ∃c>0: ||(T-λ)x||>=c||x||, ∀x∈X}
4. δ(T) = {λ∈K : ∃(xn): ||xn||=1 and (T-λ)xn->0}
5. Tx= λx and Ty= μy with λ ≠ μ
   ⟹ <x,y>=0

Question 12

ORTHOGONAL PROJECTION

Answer 12

P∈B(X) if
1. P^2=P
2. kerP ⊥ ranP

*X hilbert

Question 13

PROJECTION P∈B(X) IS ORTHOGONAL PROJECTION ⟺

Answer 13

⟺ P*=P

*X hilbert

Question 14

NONNEGATIVE OPERATOR

Answer 14

T∈B(X) if <Tx,x> >= 0, ∀x∈X

* in particular T is self-adjoint since K=C

*X hilbert

Question 15

T>=0 ⟹

Answer 15

1. ||Tx||^2 <= ||T||<Tx,x>, ∀x∈X
2. ||T||= sup_(||x||=1) <Tx,x>

Question 16

THE NUMBERS a AND b

Answer 16

T∈B(X) selfadjoint
a:= inf_(||x||=1) <Tx,x>
b:= sup_(||x||=1) <Tx,x>

* do not have ||T||=b since not assuming T>=0

Question 17

PROPERTIES OF THE SPECTRUM

Answer 17

1. Tx= λx and Ty= μy with λ ≠ μ
   ⟹ <x,y>=0
2. δ(T) only contains approximate eigenvalues
3. δ(T) ⊆ [a,b]
4. a,b ∈ δ(T)

Question 18

THE OPERATOR NORM THEOREM

Answer 18

if T selfadjoint, then
||T|| = sup_(||x||=1) |<Tx,x>| = max{|a|, |b|}

Question 19

EXISTENCE OF AN EIGENVALUE

Answer 19

T compact and selfadjoint then
1. -||T|| or ||T|| is an eigenvalue
2. there is y≠0 with Ty=ay

Question 20

DIAGONALISATION THEOREM

Answer 20

if X is hilbert space and T=T*∈K(X), then there exist:
1. countably many real eigenvalues λi
(if infinitely many then λi->0)
2. countably many orthonormal eigenvectors ei such that Tx = Σλi<x,ei>ei

Question 21

x ∈ Ker T ⟺

Answer 21

Tx = 0

Question 22

y ∈ RanT ⟺

Answer 22

<Tx,y>=0

Question 23

x∈ (RanT)^⊥ ⟺

Answer 23

<x,Ty>=0