2.2. OPERATOR ALGEBRA 19. If S is a subset of E, then the set
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1 2.2. OPERATOR ALGEBRA 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 is the inverse image of the set A. L 1 (S ) = {v E L(v) S } 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 For any operator L the sets Im(L) and Ker (L) are vector subspaces. mathphyshass.tex; September 11, 2013; 17:08; p. 18
2 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 is called the rank of the operator L. rank (L) = dim Ker (L) Theorem 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 (L 1 + L 2 )(x) = L 1 (x) + L 2 (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. A 0 = I A 1 = A, A 2 = 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 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
3 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 A 2 = I idempotent if A 2 = A, and nilpotent if for some integer k A k = 0. Two operators A and B are equal if for any u V Au = Bu If Ae i = Be i for all basis vectors in V then A = B. Operators are uniquely determined by their action on a basis. Theorem An operator A is equal to zero if and only if for any u, v V (u, Av) = 0 Theorem 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 The inverse of an automorphism is unique. mathphyshass.tex; September 11, 2013; 17:08; p. 20
4 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. where I is the identity operator. P n (T) = a n T n + a 1 T + a 0 I, Commutator of two operators A and B is an operator [A, B] defined by [A, B] = AB BA Theorem Properties of commutators. Anti-symmetry [A, B] = [B, A] linearity [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, A m ] = 0 [A, A 1 ] = 0 [A, f (A)] = 0 mathphyshass.tex; September 11, 2013; 17:08; p. 21
5 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 (x) = k=0 f (k) (x 0 ) (x x 0 ) k k! Then for an operator T f (T) = k=0 f (k) (x 0 ) (T x 0 I) k k! Exponential Example. exp(t) = k=0 1 k! T k 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 : R 2 R 2 A(x, y) = ( y, x) mathphyshass.tex; September 11, 2013; 17:08; p. 22
6 24 CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES Note that so Therefore A 2n = ( 1) n I, which is a rotation by the angle t A 2 = I A 2n+1 = ( 1) n A exp(ta) = cos ti + sin ta 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 dt Rules of the differentiation Example. Exponential of the adjoint. Let It satisfies the equation with initial condition Let Ad A be defined by Then = lim h 0 H(t + h) H(t) h d da (AB) = dt dt B + AdB dt d exp(ta) = A exp(ta) dt X(t) = e ta Be ta d X = [A, B] dt X(0) = I Ad A B = [A, B] X(t) = exp(tad A )B = k=0 t k [A, [A, B] k! } {{ } k mathphyshass.tex; September 11, 2013; 17:08; p. 23
7 2.2. OPERATOR ALGEBRA 25 Duhamel s Formula. Proof. Let Then and We compute So, Therefore d exp[h(t)] = exp[h(t)] dt 1 0 exp[ sh(t)] dh(t) dt Y(s) = e sh d dt esh Y(0) = 0 d exp[h] = exp[h]y(1) dt dy ds Y(1) = exp( Ad H ) which can be written in the form Y(1) = = [H, Y] + H ( s + Ad H )Y = H exp(sad H )H ds e (1 s)h H e (1 s)h ds exp[sh(t)]ds By changing the variable s (1 s) we get the desired result. Particular case. If H commutes with H then Campbell-Hausdorff Formula. t e H(t) = e H t H exp A exp B = exp[c(a, B)] Consider U(s) = e A e sb = e C(s) mathphyshass.tex; September 11, 2013; 17:08; p. 24
8 26 CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES Of course We have Also, Let Then Therefore Now, let Note that Then s U = U e Ad C 1 0 F(z) = C(0) = A d ds U = UB exp[ τc] s C exp[τc] dτ 1 0 e τz dτ = 1 e z z s U = UF(Ad C ) s C Ψ(z) = F(Ad C ) s C = B 1 F(log z) = z log z z 1 = Ade C = Ade A Ade sb = e Ad A e sad B Ψ(e Ad A e sad B )F(Ad C ) = I Therefore, we get a differential equation with initial condition Therefore, C(1) = A + s C = Ψ(e Ad A e sad B )B C(0) = A 1 This gives a power series in Ad A, Ad B. 0 Ψ(e Ad A e sad B )Bds Particular case. If [A, B] commutes with both A and B then ( ) 1 e A e B = e A+B exp [A, B] 2 mathphyshass.tex; September 11, 2013; 17:08; p. 25
9 2.2. OPERATOR ALGEBRA Self-Adjoint and Unitary Operators The adjoint A of an operator A is defined by (Au, v) = (u, A v), u, v E. Theorem 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 and anti-selfadjoint if A = A A = A Every operator A can be decomposed as the sum A = A S + A A of its selfadjoint part A S and its anti-selfadjoint part A A A S = 1 2 (A + A ), A A = 1 2 (A A ). Theorem 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 only for u = 0. (u, Hu) = 0 mathphyshass.tex; September 11, 2013; 17:08; p. 26
10 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. U = exp(a), A = A Unitary operators preserve the inner product. Theorem 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 Trace and Determinant The trace of an operator A is defined by tr A = n (e i, Ae i ) 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
11 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 dt det(i + ta) t=0 = tr A det(i + ta) = I + ttr A + O(t 2 ) ( ) d 1 da det A = det A tr A dt dt Note that tr I = n, det I = 1. Theorem Let A be a self-adjoint operator. Then det exp A = e tr 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 The zeta-functions has the properties ζ(0) = n, and ζ (0) = log det A. mathphyshass.tex; September 11, 2013; 17:08; p. 28
12 30 CHAPTER 2. FINITE-DIMENSIONAL VECTOR SPACES Finite Difference Operators Let e i be an orthonormal basis. The shift operator E is defined by Ee 1 = 0, Ee i = e i 1, i = 1,..., n, that is, or E f = n 1 i=1 f i+1 e i (E f ) i = f i+1 Let Δ = E I = I E 1 Next, define an operator D by E = exp(hd) that is, D = 1 h log E = 1 h log(i + Δ) = 1 log(i ) h Also, define an operator J by J = ΔD 1 Then Δ f i = f i+1 f i f i = f i f i 1 Problem. Compute U(t) = exp[td 2 ]. mathphyshass.tex; September 11, 2013; 17:08; p. 29
13 2.2. OPERATOR ALGEBRA Projection Operators A Hermitian operator P is a projection if P 2 = P Two projections P 1, P 2 are orthogonal if P 1 P 2 = P 2 P 1 = 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 is the projection operator onto S. P u = w The operators P and P are called complementary projections. They have the properties: P = P, (P ) = P, P + P = I, P 2 = P, (P ) 2 = P, PP = P P = 0. More generally, a collection of projections {P 1,..., P k } is a complete orthogonal system of complimentary projections if P i P k = 0 if i k and k P i = P P k = I. i=1 mathphyshass.tex; September 11, 2013; 17:08; p. 30
14 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 An operator P is a projection if and only if P is idempotent and self-adjoint. Theorem 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 { e i } m i=1 be an orthonormal set and S = span 1 i m { e i }. Then the operator m P = e i e i is the projection onto S. If e i is an orthonormal basis then the projections Examples for a complete orthogonal set. i=1 P i = e i e i Let u be a unit vector and P u be the projection onto the one-dimensional subspace (line) S u spanned by u defined by P u v = u(u, v). The orthogonal complement S u is the hyperplane with the normal u. The operator J u defined by J u = I 2P u 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 J u = J u, J 2 u = I. mathphyshass.tex; September 11, 2013; 17:08; p. 31
15 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 u 1 and u 2 be an orthonormal system of two vectors and P u1,u 2 be the projection operator onto the two-dimensional space (plane) S u1,u 2 spanned by u 1 and u 2 P u1,u2v = u 1 (u 1, v) + u 2 (u 2, v). Let N u1,u 2 be an operator defined by Then and N u1,u 2 v = u 1 (u 2, v) u 2 (u 1, v). N u1,u 2 P u1,u 2 = P u1,u 2 N u1,u 2 = N u1,u 2 N 2 u 1,u 2 = P u1,u 2. A rotation operator R u1,u 2 (θ) with the angle θ in the plane S u1,u 2 is defined by R u1,u 2 (θ) = I P u1,u 2 + cos θ P u1,u 2 + sin θ N u1,u 2. The rotation operator is unitary, that is, it satisfies the equation Exercises R u 1,u 2 R u1,u 2 = I. 1. Prove that the range and the kernel of any operator are vector spaces. 2. Show that (aa + bb) = aa + bb a, b R, (A ) = A (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
16 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 = 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 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 = 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
17 2.2. OPERATOR ALGEBRA Prove that an operator is projection if and only if it is idempotent and selfadjoint. 21. Give an example of an idempotent operator in R 2 which is not a projection. 22. Show that any projection operator P is positive. Moreover, show that v E (Pv, v) = Pv Prove that the sum P = P 1 +P 2 of two projections P 1 and P 2 is a projection operator if and only if P 1 and P 2 are orthogonal. 24. Prove that the product P = P 1 P 2 of two projections P 1 and P 2 is a projection operator if and only if P 1 and P 2 commute. mathphyshass.tex; September 11, 2013; 17:08; p. 34
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