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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. mathphyshass.tex; September 11, 2013; 17:08; p. 18 20 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 . mathphyshass.tex; September 11, 2013; 17:08; p. 19 2.2. OPERATOR ALGEBRA 21 • 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. mathphyshass.tex; September 11, 2013; 17:08; p. 20 22 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 mathphyshass.tex; September 11, 2013; 17:08; p. 21 2.2. OPERATOR ALGEBRA 23 • 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) mathphyshass.tex; September 11, 2013; 17:08; p. 22 24 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 mathphyshass.tex; September 11, 2013; 17:08; p. 23 2.2. OPERATOR ALGEBRA 25 • 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) mathphyshass.tex; September 11, 2013; 17:08; p. 24 26 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 mathphyshass.tex; September 11, 2013; 17:08; p. 25 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. mathphyshass.tex; September 11, 2013; 17:08; p. 26 28 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) mathphyshass.tex; September 11, 2013; 17:08; p. 27 2.2. OPERATOR ALGEBRA 29 • 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 . mathphyshass.tex; September 11, 2013; 17:08; p. 28 30 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 ]. mathphyshass.tex; September 11, 2013; 17:08; p. 29 2.2. OPERATOR ALGEBRA 2.2.6 31 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 . mathphyshass.tex; September 11, 2013; 17:08; p. 30 32 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 . mathphyshass.tex; September 11, 2013; 17:08; p. 31 2.2. OPERATOR ALGEBRA 33 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. mathphyshass.tex; September 11, 2013; 17:08; p. 32 34 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 . mathphyshass.tex; September 11, 2013; 17:08; p. 33 2.2. OPERATOR ALGEBRA 35 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. mathphyshass.tex; September 11, 2013; 17:08; p. 34