Banach Spaces

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  • What is a nuclear operator? (Coleman)
  • Give an example of an integral operator which is nuclear. (Coleman)
  • What can you say about the specturm of a nuclear operator? (Coleman)
    • Could it be the empty set? (Arveson)
  • Give an example of an operator on a real Banach space with no specturm. (Arveson)
  • Does the sum of the elements of the spectrum of a nuclear operator converge? (Coleman)
  • What is a trace class operator? (Coleman)
  • What is a Hilbert-Schmidt operator? Can you give an example over {\mathcal L}^2 of the unit interval? (Coleman)
  • Can [0,1] be the spectrum of a compact operator? (Arveson)
  • What is the spectrum of M_{e^{2\pi it}}? How could you know that it is invertible? What is the inverse? (Arveson)
  • If T is an operator on a Banach space, what is cos2T + sin2T? (Arveson)
  • What is cosT (Arveson)
  • If f is an entire function, what is fT? (Arveson)
  • List the properties of the functional calculus. (Arveson)
  • Consider {\mathcal C}([0,1],\R). Is there a natural topology on this space? (Arveson)
  • Let S=\{f\in {\mathcal C}[0,1]\,|\, |f(x)-f(y)|\leq |x-y|\}. What properties does it have (e{.}g{.} closed, complete, bounded compact)? (Arveson)
  • Let S_0=\{f\in S\,|\, f(0)=0\}. What properties does it have (e.g. closed, complete, bounded, compact) ? (Arveson)
  • What is the Riesz theory of compact operators?
  • What is a Fredholm operator? Can any Fredholm operator be written as the sum of an invertible operator with a compact operator? What is the Fredholm index? What are its properties? How can you obtain an isomorphism between the abstract index group and the integers?
  • Suppose you have an operator x on a Hilbert space such that x − x2 is compact. What can you tell me about it?
  • In the previous question, you had a projection in the Calkin algebra, and you showed that it can be lifted to B(H). Can you do the same for a unitary?
  • What is the polar decomposition? What can you say about it?

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