Reading:
• Berberian, Chapter 8.
• Akhiezer-Glazman, Chapter 5.
• Review the lecture of 30 November.
• Review the lecture of 2 December.
• Review the lecture of 4 December.
Study problems (do not hand in):
• Berberian, p. 148, all (these appear not to have been assigned yet)
• Berberian, P. 187, #1,2,4,5,8.
• Berberian, all other problems not done earlier
• Let |A| := (A*A)^1/2 (the positive square root). Show that
  1. The operator norm of |A| and the operator norm of A are equal.
  2. Show |A| = A iff A 0.
  3. Show that if A_n tends to A in the operator norm, then |A_n| tends to |A| in the operator norm.
  4. It is not necessarily true that |A + B| |A| + |B|. (Use 2 by 2 matrices.)
  5. It is not necessarily true that |A B| = |A| |B|. (Use 2 by 2 matrices.)
• Consider the mapping A -> A*. Establish whether this map is continuous in the three topologies for operators (uniform = operator norm, strong, weak).
• Show that a bounded operator T is CC iff for every orthonormal set {e_k} in the orthogonal complement of N(T), T e_k -> 0.
• Suppose that S and T are operators and U is a unitary operator such that T = U S U^-1.
  1. Show that S is continuous iff T is continuous.
  2. Show that S and T have the same eigenvalues, and that their eigenspaces have the same dimensions.
  3. If S is normal, then so is T.
  4. If a projector P reduces S, then there is a projector that reduces T. What is the formula for that projector?
• Suppose that 0 is an eigenvalue of T=T*, with infinitely many independent eigenvectors. Suppose that C is a CC operator. Show that 0 is in the spectrum of T+C.
• Let T_k be a sequence of bounded self-adjoint operators that converges in operator norm to T.
  1. Show that T is self-adjoint.
  2. Let z be a non-real number. Show that (T_k - z)^-1 -> (T - z)^-1.
• Show that every closed subspace in the range of a CC operator is finite-dimensional.
• Consider the Volterra operator V on L²(0,1) defined so that [V f](x) is the indefinite integral from 0 to x of f(y) dy. Discuss in as much detail as possible the operators V, V*, and V+V*. Consider how they are classified and their spectra.
• Consider the integral operator K on L²(0,1) with integral kernel K(x,y) = 1 - 2 sin(x) sin(y). Discuss K, K*, KK*, and exp(K)
• Same question with K(x,y) = xy + 2 sin(x) sin(y) .
• Check in later in case I think of even more good study problems!

Past homework assignments

Reading:
• Berberian, Sections 1.1-2.4
• Review the lecture of 17 August
• Review the lecture of 19 August
Exercises:
• Berberian, P. 6, #2; P. 12, #1-2; P. 16, #1,4,12.
• Berberian, P. 20, #3; There is a minor misstatement in this problem. Find it.
• Berberian, P. 23, #3,5.

Reading:
• Berberian, Chapter 2
• Review the lecture of 21 August
• Review the lecture of 24 August
• Optional: Read Chapters 1-2 of Harrell-Herod
Exercises:
• Berberian, P. 27, #3
• Berberian, P. 32, #2,3; optional: 8.
• Berberian, P. 38, #1,2,4,5

Reading:
• Berberian, Chapter 2, Sections 3.1-3.1
• Review the lecture of 31 August
• Review the lecture of 2 Sepember
• Review the lecture of 4 Sepember
• Optional: Read Chapters 1-2 of Harrell-Herod
Exercises:
• Berberian, P. 42, #2,4
• Berberian, P. 48, #2,4,5
• Berberian, P. 50, #1,4
• Berberian, P. 54, #3
• Berberian, P. 56, #2
• Due Monday, 14 September.
  There was exam on this date.
  Reading:
  • Berberian, Sections 1.1-3.3.
  • Optional: Read Chapters 1-3 of Harrell-Herod.
  • Optional: Read Chapters 7-8 of the on-line text by James Herod.
  • Review the lecture of 11 September
  Study problems:
  • Review the previously assigned homework, and exercises from Berberian, particularly the ones that are more concrete or constructive.
  • Berberian, P. 61, #3-6, 8
  • Berberian, P. 64, #1-3,7
  • Let f(t) = t for -1 <= x <= 1. Find the best L² (= r.m.s.) approximation for f(t)
    1. in the span of the set of functions {exp(i k x}, for -N <= k <= N. (This is the truncated full Fourier series for f(t).)
    2. in the span of the set of functions {sin(k x/2 }, for -N <= k <= N. (This is one of the variants of the truncated Fourier sine series for f(t) on this interval.)
    3. among all polynomials of degree N.
  • Work the problems in the first three chapters of Harrell-Herod
  • Work the old tests labeled T1 etc. in the bank of old tests (but ignore the question from 1999 about special kinds of matrices).

Reading:
• Berberian, Chapter 3
• Look at the grader's solutions to selected parts of HW 3.
• Look at the articles on Legendre polynomials and the Legendre Differential Equation in Wolfram MathWorld.
• Review the lecture of 16 September
• Review the lecture of 18 September
Exercises:
• Consider the following partial differential equation for a function u(t,x) for -1 <= x <= 0, 0 > 0:
  
  We impose boundary conditions that the solution is nonsingular at x= -1 and x = 1. (Some discussion of this equation is to be found in the lecture of 16 September. Note that if P_m(x) is the m-th Legendre polynomial, then L P_m(x) = m (m+1) P_m(x).) If at time 0, u(0,x) = x + 2 x². Solve for u(t,x) when t > 0.
• Berberian, P. 69, #1-3, 6. (The point of #6 is to verify the statements.)
• Let P be a projector, which means that P = P². Find
  1. exp(t P), as defined by the Taylor series for exp(x). Simplify the answer so that there is no infinite series.
  2. A specific formula for exp(t P) f, where P is the orthogonal projector onto the subspace of even functions within the space L²[-1,1].
  3. A specific formula for exp(t P) f, where P is the orthogonal projector onto the span of (1,0,0,1) and (1,1,0,0) within the space C⁴.

Reading:
• Berberian, Chapter 3, Sections 4.1-4.2
• Optional: Read Chapters 5-6 and 7-8 of the on-line text by James Herod.
• Review the lecture of 21 September
• Review the lecture of 23 September
• Review the lecture of 25 September
Exercises:
• Berberian, P. 72, #2,3.
• Berberian, P. 74, #1-3.
• Berberian, P. 75, #1,3,4
• Berberian, P. 81, #5,6

Reading:
• Berberian, Sections 4.1-4.8
• Review the lecture of 25 September
• Review the lecture of 28 September
• Review the lecture of 30 September
• Review the lecture of 2 October
Exercises:
• Berberian, P. 84, #4,10. In addition, briefly comment on the relationship between what is shown in #10 and the example of the right and left shift on l².
• Berberian, P. 86, #3,5.
• Berberian, P. 87, #1,2
• Berberian, P. 90, #2,3
• Berberian, P. 92, #1,2.

There was an exam on this date.
Reading:
• Berberian, Chapter 4 (and all earlier chapters)
• Review the lecture of 7 October
• Review the lecture of 9 October
• Review the lecture of 12 October
• Optional: Read Chapters 9 and 11 of the on-line text by James Herod. (There may be occasional references to concepts we have on developed yet. Ignore them.)
• Read the section of Harrell-Herod on Geometry and integral operators and this extract of the subsequent chapter.
Study problems:
• Berberian, Chapters 3-4, all exercises you haven't yet done.
• Work the exercises in the section of Harrell-Herod on Geometry and integral operators.
• Work the problems in the second test of Math 6580 in 2006.
• Review the scattered solutions from a test from many years ago.

Reading:
• Berberian, Chapter 4
• Review the lecture of 16 October
Exercises:
None! However, you are expected to hand in a short proposal of a term paper.

Reading:
• Berberian, Chapters 4-5
• Review the lecture of 19 October
• Review the lecture of 21 October
• Review the lecture of 23 October
Exercises: Hand in ten of the following. (Optional, but even better: all of them)
• Berberian, P. 98, #5-7
• Berberian, P. 102, #2
• Berberian, P. 104, #1,2
• Berberian, P. 107, #2,9,11
• Berberian, P. 110, #1,2,7

Reading:
• Berberian, Sections 5.7-8, 6.1-8.
• Akhiezer-Glazman, Sections 14-17, 22,23,30-32,35,36. (Much of this overlaps reading we have already done in Berberian. There are some differences in notation and terminology.)
• Review the lecture of 26 October
• Review the lecture of 28 October
• Review the lecture of 30 October
Exercises:
• Berberian, P. 134, #3,5,7,9
• Berberian, P. 141, #2,3
• Berberian, P. 143, #1
• Berberian, P. 144, #2,11
• Berberian, P. 152, #1,3,9

Reading:
• Berberian, Sections 6.8-7.3.
• Akhiezer-Glazman, Sections 38-42.
• Optional: Read Chapters 13-18 of the on-line text by James Herod.
• Review the lecture of 2 November
• Review the lecture of 4 November
• Review the lecture of 6 November
Exercises:
• Berberian, P. 152, #12,14
• Berberian, P. 154, #3,4,9 (except for (vi), which is challenging and optional),11
• Berberian, P. 160, #2,16,17.
• Berberian, P. 165, #3,4

Reading:
• Berberian, Chapters 7-8.
• Akhiezer-Glazman, Sections 27-29.
• Optional: Read Chapters 15-20 of the on-line text by James Herod.
• Review the lecture of 6 November
• Review the lecture of 9 November
• Review the lecture of 11 November
• Review the lecture of 13 November
Exercises:
• Berberian, P. 136, #11.
• Berberian, P. 165, #6,7.
• Berberian, P. 168, #1,2,4.
• Berberian, P. 170, #2,3,5,7. With Exercise 3, provide an explicit example of an isometric operator with an approximate eigenvalue ("AP value") that is not an eigenvalue.
• Optional: Berberian, P. 170, #12. We will discuss this one in class.

Only the term paper is due!

Reading:
• Berberian, Chapter 8.
• Akhiezer-Glazman, Sections 38-43,46.
• Optional: Read Chapters 19-22 of the on-line text by James Herod.
• Optional: Read the chapter Geometry and integral operators of Harrell-Herod.
• Review the lecture of 16 November.
• Review the lecture of 18 November.
• Review the lecture of 20 November.
• Review the lecture of 23 November.
• Review the lecture of 25 November.
Study problems (do not hand in):
• Berberian, P. 170, #8,10,12.
• Berberian, P. 174, #3 (which depends on 144 #2), 4, 5,13. Also show that if A 0 and A B with B a CC operator, then A is CC.
• Berberian, P. 180, #1,2.
• Let D(T) be the domain of definition of an operator T that is not assumed continuous.
  1. Show that D(T) is a normed linear space with the norm ||x||_T := ||x|| + ||T x||.
  2. Show that T is closed iff the space of part (a) is complete (i.e., a Banach space).

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