Current reading and homework assignments

Due Monday, 26 Aprili as exam preparation

Reading:
• Arveson, sections 2.4, 2.6-2.7
  • Review Prof. Harrell's lecture from 1 April.
  • Review Prof. Harrell's lecture from 6 April.
  • Review Prof. Harrell's lecture from 7 April.
  • Review Prof. Harrell's lecture from 8 April.
  • Review Prof. Harrell's lecture from 13 April.
  • Review Prof. Harrell's lecture from 15 April.
  • Review Prof. Harrell's lecture from 22 April.
  • Review Prof. Harrell's lecture from 24 April.
  Exercises
  1. Arevson, p. 49, 1-4.
  2. Arevson, p. 56, 3-5
  3. Arevson, p. 67, 3.
  4. Use the spectral theorem to prove Temple's inequality:
     Let A be a self-adjoint operator with a spectrum (A) contained in {₁} U [L, ). Here ₁ is an eigenvalue and L > ₁. Show that if u D(A) with ||u||=1, and <u, A u> < L, then

     ₁ <u, A u> - (||A u||² - <u, A u>²)/ (L - <u, A u>).

  5. Prove that a self-adjoint operator on a finite-dimensional Hilbert space has a cyclic vector iff all of its eigenvalues are simple.
  6. Let A and B be bounded self-adjoint operators. Show that the following are equivalent:
     1. A and B commute
     2. For all nonreal z and w, (A - z 1)^-1 commutes with (B - w 1)^-1.
     If you wish, you may assume A and B act on a Hilbert space.
  7. Consider T = - d²/dx² as an operator on L²(0,) with domain C_c. Is T essentially self-adjoint? Categorize all self-adjoint extensions of T.
  8. Consider the unbounded self-adjoint operator A on L²(-, ) defined as follows:
     1. Define i d/dx on the minimal domain of smoooth functions of compact support (as in class)
     2. Let P be the operator whose graph is the closure of the graph of the operator in step 1.
     3. A = f(P), defined by the functional calculus for f(x)= x² -|x| .
     Use the Fourier transform to specifically identify the following. (By "specifically," I mean that your answer should be self-contained and crystal clear to a mathematically literate reader without opening Arveson or consulting any other materials You may assume that the reader knows what the Fourier transform is, and normalizes it as in the lecture of 22 April.)
     1. Give a formula for A that diagonalizes it (as in Arveson section 2.4).
     2. Give a formula for the spectral projector associated with = [0,1]
     3. Give a formula for the spectral measure associated with the function u whose Fourier transform is exp(-k²/2).
     4. Determine the spectrum of A.
  9. An operator B on a Hilbert space is said to be compact relative to a self-adjoint operator A iff D(B) contains D(A) and B(A + iI)^-1 is compact. Show that for all > 0, and all u in D(A), ||Bu|| < ||A u|| + b ||u||.
     (The constant b will depend on .)
  10. If A and B are C* algebras with units and is an algebra homomorphism from A to B such that ||(a)|| ||a|| for all a in A, then is a *-homomorphism.
  11. Prove that if P and Q are orthogonal projectors on a Hilbert space and ||P - Q|| < 1, thenxi
      1. dim(Ran(P)) = dim(Ran(Q))
      2. dim(Ran(1 - P)) = dim(Ran(1 -Q))
      3. There exists a unitary operator U such that Q U = U P

Past homework assignments

Reading:
• Review your Real Analysis text for basic information about Hilbert and Banach spaces, L^p, and l^p.
• Profs. Andrew and Green's Spectral theory of operators in Hilbert space, Part I, Sections 1-4.
• Prof. Harrell's A short history of operator theory
• Review Prof. Harrell's lecture from 8 January (in part).
• Review Prof. Harrell's lecture from 13 January and the proof that L(B₁, B₂) is a Banach space with the operator norm.
• Review Prof. Harrell's lecture from 15 January and the proof of the Riesz lemma.
Exercises
1. Make a Venn diagram (set diagram) for the following types of operators: normal, self-adjoint, positive, unitary, projection.
2. Show that for Mⁿⁿ (the n by n matrices) the following topologies are equivalent:
   • The l² topology
   • The topology of the operator norm
   • The topology of the supremum norm
3. Andrew-Green, Section 2, Exercise 11. Note: absolute value signs are missing around the inner product.

Reading:
• Profs. Andrew and Green's Spectral theory of operators in Hilbert space, Part I.
• Review Prof. Harrell's lecture from 20 January
• Review Prof. Harrell's lecture from 22 January
• Review Prof. Harrell's lecture from 27 January
• Review Prof. Harrell's lecture from 29 January
Exercises
1. Use the closed graph theorem to prove:
   Theorem. Suppose that T is defined on (all of) a Hilbert space H, and that for all x,y H,
   <x,Ty> = <Tx, y>. Then T is bounded.
2. Andrew-Green, p. 12, Exercise 3.
3. Andrew-Green, p. 19, Exercise 3.

Reading:
• Profs. Andrew and Green's Spectral theory of operators in Hilbert space, Part I.
• Review Prof. Harrell's lecture from 3 February
• Review Prof. Harrell's lecture from 5 February
Exercises
1. Andrew-Green, p. 19, Exercise 1,2.

Reading:
• Profs. Andrew and Green's /TB Part I.
• Arveson, Chapter I.1-I.4
• Review Prof. Harrell's lecture from 10 February
Exercises
1. deferred because of Comprehensive Exams

Reading:
• Profs. Andrew and Green's Spectral theory of operators in Hilbert space, Part II.
• Arveson, Chapter II.8
• Arveson, Chapter I.5-I.9
• Review Prof. Harrell's lecture from 12 February. Sorry, the pictures are blurry. If someone has nice notes from that day, please let me have copies to replace these pictures.
• Review Prof. Harrell's lecture from 19 February.
• Review Prof. Harrell's lecture from 20 February.
• Review Prof. Harrell's lecture from 24 February.
Exercises
1. Arveson, p. 74, #1.
2. Andrew-Green, p. 24, #4.

Reading:
• Profs. Andrew and Green's Spectral theory of operators in Hilbert space, Sections 7-9.
• Arveson, Chapter III.3
• Arveson, Chapter I.10-I.12
• Review Prof. Harrell's lecture from 25 February.
• Review Prof. Harrell's lecture from 26February.
• Review Prof. Harrell's lecture from 2 March.
• Review Prof. Harrell's lecture from 3 March.
• Review Prof. Harrell's lecture from 4 March.
Exercises
Note: due to late posting, these will be due on Wednesday the 17th. Note: if your homework is late, you may e-mail it to Prof. Harrell this week. One of the public copy machines can scan and e-mail documents, for instance.
1. Arveson, p. 24, #4.
2. Suppose that F is a linear functional on a unital Banach algebra A, with the property that F(ab) = F(a) F(b) for all a,b. Show that F is automatically continuous.

Reading:
• Arveson, sections 2.1-2.3
  • Review Prof. Harrell's lecture from 16 March.
  • Review Prof. Harrell's lecture from 17 March.
  • Review Prof. Harrell's lecture from 30 March.
  Exercises
  1. Work out in detail the structure of the Gel'fand spectrum of the commuting algebra of 3 by 3 matrices generated by

                [2   0   0 ]
                [          ]
          A :=  [0   1   1 ]
                [          ]
                [0   1   1 ]
     
     

     Note that this matrix is diagonable and has only two eigenvalues, 0 and 2. Specifically identify the maximal ideals, the characters, and the Gel'fand transform.
  2. Let A be a unital Banach algebra contianing elements a and b. Let w be a nonzero number. Show that a b - w 1 is invertible iff b a - w 1 is invertible .
  3. Let A be a C* algebra, and a an element of A. Show that the spectrum of a*a is nonnegative. (Preferably, do this without assuming a Hilbert space on which a acts.)

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