Download 2.2 Operator Algebra

2.2. OPERATOR ALGEBRA
2.2
2.2.1
19
Operator Algebra
Algebra of Operators on a Vector Space
• A linear operator on a vector space E is a mapping L : E → E satisfying
the condition ∀u, v ∈ E, ∀a ∈ R,
L(u + v) = L(u) + L(v)
and
L(av) = aL(v).
• Identity operator I on E is defined by
I v = v,
∀v ∈ E
• Null operator 0 : E → E is defined by
0v = 0,
∀v ∈ E
• The vector u = L(v) is the image of the vector v.
• If S is a subset of E, then the set
L(S ) = {u ∈ E | u = L(v) for some v ∈ S }
is the image of the set S and the set
L−1 (S ) = {v ∈ E | L(v) ∈ S }
is the inverse image of the set A.
• The image of the whole space E of a linear operator L is the range (or the
image) of L, denoted by
Im(L) = L(E) = {u ∈ E | u = L(v) for some v ∈ E} .
• The kernel Ker(L) (or the null space) of an operator L is the set of all
vectors in E which are mapped to zero, that is
Ker (L) = L−1 ({0}) = {v ∈ E | L(v) = 0} .
• Theorem 2.2.1 For any operator L the sets Im(L) and Ker (L) are vector
subspaces.
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CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES
• The dimension of the kernel Ker (L) of an operator L
null (L) = dim Ker (L)
is called the nullity of the operator L.
• The dimension of the range Im(L) of an operator L
rank (L) = dim Ker (L)
is called the rank of the operator L.
• Theorem 2.2.2 For any operator L on an n-dimensional Euclidean space
E
rank (L) + null (L) = n
• The set L(E) of all linear operators on a vector space E is a vector space
with the addition of operators and multiplication by scalars defined by
(L1 + L2 )(x) = L1 (x) + L2 (x),
and
(aL)(x) = aL(x) .
• The product of the operators A and B is the composition of A and B.
• Since the product of operators is defined as a composition of linear mappings, it is automatically associative, which means that for any operators A,
B and C, there holds
(AB)C = A(BC) .
• The integer powers of an operator are defined as the multiple composition
of the operator with itself, i.e.
A0 = I
A1 = A,
A2 = AA, . . .
• The operator A on E is invertible if there exists an operator A−1 on E, called
the inverse of A, such that
A−1 A = AA−1 = I .
• Theorem 2.2.3 Let A and B be invertible operators. Then:
(A−1 )−1 = A ,
(AB)−1 = B−1 A−1 .
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2.2. OPERATOR ALGEBRA
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• The operators A and B are commuting if
AB = BA
and anti-commuting if
AB = −BA .
• The operators A and B are said to be orthogonal to each other if
AB = BA = 0 .
• An operator A is involutive if
A2 = I
idempotent if
A2 = A ,
and nilpotent if for some integer k
Ak = 0 .
• Two operators A and B are equal if for any u ∈ V
Au = Bu
• If Aei = Bei for all basis vectors in V then A = B.
• Operators are uniquely determined by their action on a basis.
• Theorem 2.2.4 An operator A is equal to zero if and only if for any u, v ∈ V
(u, Av) = 0
• Theorem 2.2.5 An operator A is equal to zero if and only if for any u
(u, Au) = 0
Proof: Use (w, Aw) = 0 for w = au + bv with a = 1, b = i and a = i, b = 1.
• Theorem 2.2.6
1. The inverse of an automorphism is unique.
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CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES
2. The product of two automorphisms is an automorphism.
3. A linear transformation is an automorphism if and only if it maps a
basis to another basis.
• Polynomials of Operators.
Pn (T ) = an T n + · · · a1 T + a0 I,
where I is the identity operator.
• Commutator of two operators A and B is an operator [A, B] defined by
[A, B] = AB − BA
• Theorem 2.2.7 Properties of commutators.
Anti-symmetry
linearity
[A, B] = −[B, A]
[aA, bB] = ab[A, B]
[A, B + C] = [A, B] + [A, C]
[A + C, B] = [A, B] + [C, B]
right derivation
[AB, C] = A[B, C] + [A, C]B
left derivation
[A, BC] = [A, B]C + B[A, C]
Jacobi identity
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0
• Consequences
[A, Am ] = 0
[A, A−1 ] = 0
[A, f (A)] = 0
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2.2. OPERATOR ALGEBRA
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• Functions of Operators.
Negative powers
T m = ������
T ···T
m
T
−m
= (T −1 )m
T m T n = T m+n
(T m )n = T mn
Let f be an analytic function given by
∞
�
f (k) (x0 )
f (x) =
(x − x0 )k
k!
k=0
Then for an operator T
∞
�
f (k) (x0 )
f (T ) =
(T − x0 I)k
k!
k=0
• Exponential
∞
�
1 k
T
exp(T ) =
k!
k=0
• Example.
2.2.2
Derivatives of Functions of Operators
• A time-dependent operator is a map
H : R → End (V)
Note that
• Example.
[H(t), H(t� )] � 0
A : R2 → R2
A(x, y) = (−y, x)
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CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES
Note that
so
A2 = −I
A2n = (−1)n I,
A2n+1 = (−1)n A
Therefore
exp(tA) = cos tI + sin tA
which is a rotation by the angle t
exp(tA)(x, y) = (cos t x − sin t y, cos t y + sin t x)
So A is a generator of rotation.
• Derivative of a time-dependent operator is an operator defined by
dH
H(t + h) − H(t)
= lim
h→0
dt
h
• Rules of the differentiation
dA
dB
d
(AB) =
B+A
dt
dt
dt
• Example.
d
exp(tA) = A exp(tA)
dt
• Exponential of the adjoint. Let
X(t) = etA Be−tA
It satisfies the equation
d
X = [A, B]
dt
with initial condition
X(0) = I
Let AdA be defined by
Then
AdA B = [A, B]
∞
�
tk
[A, · · · [A, B]
X(t) = exp(tAdA )B =
k! ����������������������
k=0
k
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2.2. OPERATOR ALGEBRA
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• Duhamel’s Formula.
d
exp[H(t)] = exp[H(t)]
dt
�
0
1
exp[−sH(t)]
Proof. Let
Y(s) = e−sH
dH(t)
exp[sH(t)]ds
dt
d sH
e
dt
Then
Y(0) = 0
and
We compute
So,
d
exp[H] = exp[H]Y(1)
dt
dY
= −[H, Y] + H �
ds
(∂ s + AdH )Y = H �
Therefore
Y(1) = exp(−AdH )
�
1
0
exp(sAdH )H � ds
which can be written in the form
� 1
e−(1−s)H H � e(1−s)H ds
Y(1) =
0
By changing the variable s → (1 − s) we get the desired result.
• Particular case. If H commutes with H � then
∂t eH(t) = eH ∂t H
• Campbell-Hausdorff Formula.
exp A exp B = exp[C(A, B)]
Consider
U(s) = eA e sB = eC(s)
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CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES
Of course
C(0) = A
We have
d
U = UB
ds
Also,
∂sU = U
Let
�
1
0
F(z) =
Then
exp[−τC]∂ sC exp[τC] dτ
�
1
e−τz dτ =
0
1 − e−z
z
∂ s U = UF(AdC )∂ sC
Therefore
F(AdC )∂ sC = B
Now, let
Ψ(z) =
Note that
Then
z log z
1
=
F(log z)
z−1
eAdC = AdeC = AdeA · Ade sB = eAdA e sAdB
Ψ(eAdA e sAdB )F(AdC ) = I
Therefore, we get a differential equation
∂ sC = Ψ(eAdA e sAdB )B
with initial condition
C(0) = A
Therefore,
C(1) = A +
�
0
1
Ψ(eAdA e sAdB )Bds
This gives a power series in AdA , AdB .
• Particular case. If [A, B] commutes with both A and B then
�
�
1
A B
A+B
e e = e exp [A, B]
2
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2.2. OPERATOR ALGEBRA
2.2.3
27
Self-Adjoint and Unitary Operators
• The adjoint A∗ of an operator A is defined by
(Au, v) = (u, A∗ v),
∀u, v ∈ E.
• Theorem 2.2.8 For any two operators A and B
(A∗ )∗ = A ,
(AB)∗ = B∗ A∗ .
(A + B)∗ = A∗ + B∗
(aA)∗ = āA∗
• An operator A is self-adjoint (or Hermitian) if
A∗ = A
and anti-selfadjoint if
A∗ = −A
• Every operator A can be decomposed as the sum
A = AS + AA
of its selfadjoint part AS and its anti-selfadjoint part AA
1
AS = (A + A∗ ) ,
2
1
AA = (A − A∗ ) .
2
• Theorem 2.2.9 An operator H is Hermitian if and only if (u, Hu) is real
for any u.
• An operator A on E is called positive, denoted by A ≥ 0, if it is selfdadjoint
and ∀v ∈ E
(Av, v) ≥ 0.
• An operator H is positive definite (H > 0) if it is positive and
(u, Hu) = 0
only for u = 0.
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CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES
• Example.
H = A∗ A ≥ 0
• An operator A is called unitary if
AA∗ = A∗ A = I .
• An operator U is isometric if for any v ∈ E
||Uv|| = ||v||
• Example.
A∗ = −A
U = exp(A),
• Unitary operators preserve the inner product.
• Theorem 2.2.10 Let U be a unitary operator on a real vector space E.
Then there exists an anti-selfadjoint operator A such that
U = exp A .
• Recall that the operators U and A satisfy the equations
U ∗ = U −1 and A∗ = −A.
2.2.4
Trace and Determinant
• The trace of an operator A is defined by
tr A =
n
�
(ei , Aei )
i=1
• The determinant of a positive operator on a finite-dimensional space is
defined by
det A = exp(tr log A)
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• Properties
tr AB = tr BA
det AB = det A det B
tr (RAR−1 ) = tr A
det(RAR−1 ) = det A
• Theorem.
��
d
det(I + tA)�� = tr A
t=0
dt
det(I + tA) = I + ttr A + O(t2 )
�
�
d
−1 dA
det A = det A tr A
dt
dt
• Note that
tr I = n ,
det I = 1 .
• Theorem 2.2.11 Let A be a self-adjoint operator. Then
det exp A = etr A .
• Let A be a positive definite operator, A > 0. The zeta-function of the
operator A is defined by
ζ(s) = tr A
−s
1
=
Γ(s)
�
∞
0
dt t s−1 tr e−tA .
• Theorem 2.2.12 The zeta-functions has the properties
ζ(0) = n ,
and
ζ � (0) = − log det A .
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CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES
2.2.5
Finite Difference Operators
• Let ei be an orthonormal basis. The shift operator E is defined by
Ee1 = 0,
Eei = ei−1 ,
i = 1, . . . , n,
that is,
Ef =
n−1
�
fi+1 ei
i=1
or
(E f )i = fi+1
Let
Δ=E−I
∇ = I − E −1
Next, define an operator D by
E = exp(hD)
that is,
D=
1
1
1
log E = log(I + Δ) = − log(I − ∇)
h
h
h
Also, define an operator J by
J = ΔD−1
Then
Δ fi = fi+1 − fi
∇ fi = fi − fi−1
• Problem. Compute U(t) = exp[tD2 ].
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2.2. OPERATOR ALGEBRA
2.2.6
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Projection Operators
• A Hermitian operator P is a projection if
P2 = P
• Two projections P1 , P2 are orthogonal if
P1 P2 = P2 P1 = 0 .
• Let S be a subspace of E and E = S ⊕ S ⊥ . Then for any u ∈ E there exist
unique v ∈ S and w ∈ S ⊥ such that
u = v + w.
The vector v is called the projection of u onto S .
• The operator P on E defined by
Pu = v
is called the projection operator onto S .
• The operator P⊥ defined by
P⊥ u = w
is the projection operator onto S ⊥ .
• The operators P and P⊥ are called complementary projections. They have
the properties:
P∗ = P,
(P⊥ )∗ = P⊥ ,
P + P⊥ = I ,
P2 = P ,
(P⊥ )2 = P⊥ ,
PP⊥ = P⊥ P = 0 .
• More generally, a collection of projections {P1 , . . . , Pk } is a complete orthogonal system of complimentary projections if
Pi Pk = 0
and
k
�
i=1
if
i�k
Pi = P1 + · · · + Pk = I .
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CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES
• The trace of a projection P onto a vector subspace S is equal to its rank, or
the dimension of the vector subspace S ,
tr P = rank P = dim S .
• Theorem 2.2.13 An operator P is a projection if and only if P is idempotent
and self-adjoint.
• Theorem 2.2.14 The sum of projections is a projection if and only if they
are orthogonal.
• The projection onto a unit vector |e� has the form
P = |e��e|
• Let {|ei �}mi=1 be an orthonormal set and S = span 1≤i≤m {|ei �}. Then the operator
m
�
P=
|ei ��ei |
i=1
is the projection onto S .
• If |ei � is an orthonormal basis then the projections
Pi = |ei ��ei |
for a complete orthogonal set.
Examples
• Let u be a unit vector and Pu be the projection onto the one-dimensional
subspace (line) S u spanned by u defined by
Pu v = u(u, v) .
The orthogonal complement S u⊥ is the hyperplane with the normal u. The
operator Ju defined by
Ju = I − 2Pu
is called the reflection operator with respect to the hyperplane S u⊥ . The
reflection operator is a self-adjoint involution, that is, it has the following
properties
Ju∗ = Ju ,
Ju2 = I .
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The reflection operator has the eigenvalue −1 with multiplicity 1 and the
eigenspace S u , and the eigenvalue +1 with multiplicity (n − 1) and with
eigenspace S u⊥ .
• Let u1 and u2 be an orthonormal system of two vectors and Pu1 ,u2 be the
projection operator onto the two-dimensional space (plane) S u1 ,u2 spanned
by u1 and u2
Pu1 ,u2 v = u1 (u1 , v) + u2 (u2 , v) .
Let Nu1 ,u2 be an operator defined by
Nu1 ,u2 v = u1 (u2 , v) − u2 (u1 , v) .
Then
Nu1 ,u2 Pu1 ,u2 = Pu1 ,u2 Nu1 ,u2 = Nu1 ,u2
and
Nu21 ,u2 = −Pu1 ,u2 .
A rotation operator Ru1 ,u2 (θ) with the angle θ in the plane S u1 ,u2 is defined
by
Ru1 ,u2 (θ) = I − Pu1 ,u2 + cos θ Pu1 ,u2 + sin θ Nu1 ,u2 .
The rotation operator is unitary, that is, it satisfies the equation
R∗u1 ,u2 Ru1 ,u2 = I .
2.2.7
Exercises
1. Prove that the range and the kernel of any operator are vector spaces.
2. Show that
(aA + bB)∗ = aA∗ + bB∗
∗ ∗
(A ) = A
∀a, b ∈ R ,
(AB)∗ = B∗ A∗
3. Show that for any operator A the operators AA∗ and A + A∗ are selfadjoint.
4. Show that the product of two selfadjoint operators is selfadjoint if and only if they
commute.
5. Show that a polynomial p(A) of a selfadjoint operator A is a selfadjoint operator.
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CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES
6. Prove that the inverse of an invertible operator is unique.
7. Prove that an operator A is invertible if and only if Ker A = {0}, that is, Av = 0
implies v = 0.
8. Prove that for an invertible operator A, Im(A) = E, that is, for any vector v ∈ E
there is a vector u ∈ E such that v = Au.
9. Show that if an operator A is invertible, then
(A−1 )−1 = A .
10. Show that the product AB of two invertible operators A and B is invertible and
(AB)−1 = B−1 A−1
11. Prove that the adjoint A∗ of any invertible operator A is invertible and
(A∗ )−1 = (A−1 )∗ .
12. Prove that the inverse A−1 of a selfadjoint invertible operator is selfadjoint.
13. An operator A on E is called isometric if ∀v ∈ E,
||Av|| = ||v|| .
Prove that an operator is unitary if and only if it is isometric.
14. Prove that unitary operators preserves inner product. That is, show that if A is a
unitary operator, then ∀u, v ∈ E
(Au, Av) = (u, v) .
15. Show that for every unitary operator A both A−1 and A∗ are unitary.
16. Show that for any operator A the operators AA∗ and A∗ A are positive.
17. What subspaces do the null operator 0 and the identity operator I project onto?
18. Show that for any two projection operators P and Q, PQ = 0 if and only if QP = 0.
19. Prove the following properties of orthogonal projections
P∗ = P,
(P⊥ )∗ = P⊥ ,
P⊥ + P = I,
PP⊥ = P⊥ P = 0 .
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2.2. OPERATOR ALGEBRA
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20. Prove that an operator is projection if and only if it is idempotent and selfadjoint.
21. Give an example of an idempotent operator in R2 which is not a projection.
22. Show that any projection operator P is positive. Moreover, show that ∀v ∈ E
(Pv, v) = ||Pv||2 .
23. Prove that the sum P = P1 +P2 of two projections P1 and P2 is a projection operator
if and only if P1 and P2 are orthogonal.
24. Prove that the product P = P1 P2 of two projections P1 and P2 is a projection
operator if and only if P1 and P2 commute.
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