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MTH 510: Operator Theory and Operator Algebras Semester 2, 2014-2015

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MTH 510: Operator Theory and Operator Algebras Semester 2, 2014-2015 MTH 510: Operator Theory and Operator Algebras Semester 2, 2014-2015 Dr. Prahlad Vaidyanathan Contents Introduction3 I. Banach Algebras5 1. Definition and Examples...........................5 2. Invertible Elements..............................8 3. Spectrum of an Element........................... 10 4. Unital Commutative Banach Algebras................... 15 5. Examples of the Gelfand Spectrum..................... 18 6. Spectral Permanence Theorem........................ 20 II. C∗ Algebras 23 1. Operators on Hilbert Spaces......................... 23 2. C∗ Algebras.................................. 27 3. Spectrum of an Element........................... 31 4. Unital Commutative C∗ algebras...................... 34 5. Ideals and Quotients of C∗ algebras..................... 39 III.Diagonalization of Normal Operators 44 1. The Finite Dimensional Case........................ 44 2. Compact Normal Operators......................... 46 3. Multiplication Operators........................... 48 4. The Riesz Representation Theorem..................... 54 5. The Spectral Theorem............................ 57 IV.Spectral Measures 61 1. Complex Measures.............................. 61 2. Representations of C(X)........................... 67 3. Borel Functional Calculus.......................... 70 4. Spectral Measures............................... 80 5. Equivalence of the two Spectral Theorems................. 84 V. Applications of the Spectral Theorem 87 1. Ideals of B(H)................................. 87 2. Classification of Normal Operators..................... 90 VI.Instructor Notes 97 2 Introduction 0.1. Fundamental Problem of Linear Algebra: Given an n × n matrix of complex num- n n bers (ai;j) and an n-tuple (g1; g2; : : : ; gn) 2 C , we want to find (f1; f2; : : : ; fn) 2 C such that a1;1f1 + a1;2f2 + ::: + a1;nfn = g1 a f + a f + ::: + a f = g 2;1 1 2;2 2 2;n n 2 (.1) ::: = ::: an;1f1 + an;2f2 + ::: + an;nfn = gn The solution of this question leads to three fundamental concepts in Linear Alge- bra. We want to understand these concepts and apply them to the case where Cn is replaced by an infinite dimensional Banach space. 0.2. Invertibility: (i) We write T = (ai;j), and prove that this system has a solution iff T is sur- jective. Since Cn is finite dimensional, this is equivalent to saying that T is injective. Since Cn is finite dimensional, this is equivalent to saying that T is invertible in Mn(C) - ie. 9S 2 Mn(C) such that ST = TS = I (ii) (HW) Lemma: If T 2 B(X), then TFAE: (a) For every y 2 X; 9 unique x 2 X such that T x = y (b) 9S 2 B(X) such that TS = ST = I. (iii) Definition: We say that T 2 B(X) is invertible if 9S 2 B(X) such that TS = ST = I 0.3. Eigen-Values: (i) λ 2 C is an eigen value of T 2 B(Cn) iff 90 6= f 2 Cn such that T f = λf. Write n Vλ := ff 2 C : T f = λfg n Then, Vλ is a subspace of C . (ii) λ 2 C is an eigen value iff it is a root of the polynomial f(t) = det(T − λI). Hence, (a) T has an eigen-value (by the Fundamental Theorem of Algebra) (b) T has only finitely many eigen-values (c) λ is an eigen-value iff (T − λI) is not invertible. 3 (iii) Definition: For T 2 B(X), the spectrum of T is σ(T ) := fλ 2 C :(T − λI) is not invertibleg (iv) Example: Let X = C[0; 1] with the supremum norm, and T 2 B(X) be given by Z x T f(x) = f(t)dt 0 Then (a) T is not surjective since g 2 R(T ) iff g is continuously differentiable and g(0) = 0 (b) Hence, T is not invertible, so 0 2 σ(T ) (c) 0 is not an eigen-value of T . n n (v) Now suppose the eigen spaces fVλ1 ;Vλ2 ;:::;Vλk g span C , then every g 2 C can be decomposed as g = g1 + g2 + ::: + gk with gi 2 Vλi . Then, a solution to Equation .1 is given by −1 −1 −1 f = λ1 g1 + λ2 g2 + ::: + λk gk (Note that λi 6= 0 for all i since T is invertible) 0.4. Diagonalizability: n (i) Question: When do the fVλi : 1 ≤ i ≤ ng span C ? This is equivalent to saying that T is diagonalizable. (ii) Definition: We say a matrix T 2 B(Cn) is symmetric if t T = T (iii) Note: If T is a symmetric matrix, then T is diagonalizable iff n C = Vλ1 ⊕ Vλ2 ⊕ ::: ⊕ Vλk In other words, distinct eigen-values correspond to orthogonal eigen-spaces (iv) Spectral Theorem: If T 2 B(Cn) is a symmetric matrix, then T is diagonal- izable. 0.5. The goal of the course is to prove the spectral theorem for normal operators on a Hilbert space. (End of Day 1) 4 I. Banach Algebras 1. Definition and Examples Note: All vector spaces in this course will be over C. 1.1. Definition: (i) An algebra is a vector space A over C together with a bilinear multiplication under which A is a ring. In other words, for all α; β 2 C; a; b; c 2 A, we have (αa + βb)c = α(ac) + β(bc) and a(αb + βc) = α(ab) + β(ac) (ii) An algebra A is said to be a normed algebra if there is a norm on A such that (a)( A; k · k) is a normed linear space (b) For all a; b 2 A, we have kabk ≤ kakkbk (iii) A Banach algebra is a complete normed algebra. 1.2. Remark: (i) If X is a normed linear space, then kx + yk ≤ kxk + kyk. Hence, the map (x; y) 7! x + y is jointly continuous. ie. If xn ! x and yn ! y, then xn + yn ! x + y. (ii) Similarly, if A is a normed algebra, then the map (x; y) 7! xy is jointly continuous [Check!] 1.3. Examples: (i) A = C (ii) A = C[0; 1]. More generally, C(X) for X A compact, Hausdorff space. A = Cb(X), where X is a locally compact Hausdorff space. (iii) A = C0(X), where X is a locally compact Hausdorff space. [Check!] (iv) A = c0, the space of complex sequences converging to 0. Note: All the above examples are abelian. (v) A = Mn(C) for any n 2 N 5 (vi) A = B(X) for any Banach space X. (vii) A = L1(R) with multiplication given by convolution Proof. For f; g 2 A, we write Z f ∗ g(x) := f(t)g(x − t)dt R Now Z Z Z kf ∗ gkL1(R) = jf ∗ g(x)jdx ≤ jf(t)jjg(x − t)jdtdx = kfkkgk R R R by Fubini's theorem. The other axioms are easy to check. (viii) A = `1(Z) with multiplication given by convolution (proof is identical to the previous one). A is a commutative Banach algebra. 1.4. Definition: (i) Ideal (ii) Maximal ideal 1.5. Examples: (i) A = C[0; 1], then I = ff 2 C(X): f(1) = 0g is a maximal ideal. (ii) If A = Mn(C), then A has no non-trivial ideals Proof. Let f0g= 6 J C A, then choose 0 6= T 2 J, then 9Ti;j = a 6= 0. Let Ek;l be the permutation matrix obtained by switching the kth row of the identity matrix with the lth row. Then 0 T := E1;jTEi;1 2 J 0 th and T1;1 = a 6= 0. Now let F1;1 be the matrix with 1 in the (1; 1) entry and zero elsewhere. Then 1 F T 0F = F 2 J a 1;1 1;1 1;1 Similarly, F2;2;F3;3;:::;Fn;n 2 J. Adding them up, we have ICn 2 J and since J is an ideal, this means that J = A. (End of Day 2) (iii) If X is a locally compact Hausdorff space, then C0(X) is an ideal in Cb(X), the space of bounded continuous functions on X (iv) Let X be a Banach space and A = B(X), then set F(X) = fT 2 B(X): T has finite rankg Then F(X) is an ideal in A. 6 (v) If A = B(X), then the set K(X) of compact operators on X is a closed ideal in A. In fact, if H is a Hilbert space, then K(H) = F(H) 1.6. Theorem: If A is a Banach algebra, and I C A is a proper closed ideal, then A=I is a Banach algebra with the quotient norm ka + Ik = inffka + bk : b 2 Ig Proof. (i) Clearly, A=I is an algebra. (ii) Now we check that the axioms of the norm hold : (a) If ka + Ik = 0, then 9bn 2 I such that ka + bnk ! 0. Since I is closed, this means that a 2 I and hence a + I = 0 in A=I (b) Clearly, ka + Ik ≥ 0. (c) If a; b 2 A, then for any c; d 2 I ka + b + Ik ≤ ka + b + c + dk ≤ ka + ck + kb + dk This is true for any c; d 2 I, so taking infimum gives ka + b + Ik ≤ ka + Ik + kb + Ik (iii) We now check that A=I is a Banach algebra: If a; b 2 A, then for any c; d 2 I we have (a + c)(b + d) = ab + cb + ad + dc where x := cb + ad + dc 2 I. Hence kab + Ik ≤ kab + xk ≤ ka + ckkb + dk This is true for any c; d 2 I, so taking infimum gives kab+Ik ≤ ka+Ikkb+Ik (iv) A is complete (See [Conway, Theorem III.4.2]) 1.7. Definition: Let A and B be Banach algebras. (i) A map ' : A ! B is called a homomorphism of Banach algebras if (a) ' : A ! B is a continuous linear transformation of normed linear spaces (b) '(ab) = '(a)'(b) for all a; b 2 A.
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