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Homework Help: Spectrum of a linear operator on a Banach space

  1. Nov 21, 2012 #1
    1. The problem statement, all variables and given/known data
    I have a number of problems, to be completed in the next day or so (!) that I am pretty stuck with where to begin. They involve calculating the spectra of various different linear operators.

    2. Relevant equations
    The first was:
    Let X be the space of complex-valued continuous functions on Ω a closed bounded subset of ℂ, with supremum norm. Define for x [itex]\in[/itex] X, t [itex]\in[/itex] Ω
    (Tx)(t) = tx(t)

    I have found the spectrum of this by showing ker(λI - T) is zero and λI - T is onto for all λ not in Ω. So the spectrum of T is Ω.

    I am pretty happy with this- I later found the same example in a book online and they agree with my answer.

    I now have various ones involving sequences and I'm a lot more confused with these:
    i) Let X be space of cts functions converging to zero with sup norm, & define:
    T((aj)) = ((j+1)-1aj+1)
    I think norm of T is 1/2 and this gives a radius bound for possible λ but have no idea really where to go from here- I think the kernel is trivial for all non zero λ but not getting very far with showing whether its surjective or not.

    ii) Let S be the bounded linear operator on l1 defined by
    T((aj)) = (aj - 2aj+1 + aj+2)

    Show the spectrum of T is a cardioid.
    Again I can find a general bound for λ using the norm of T but get stuck after this.
    3. The attempt at a solution

    Above.. any ideas on what to try next, or advice in general on methods of calculating spectra would be very welcome. Thanks in advance!
  2. jcsd
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