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

1. Nov 21, 2012

### Zoe-b

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 $\in$ X, t $\in$ 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!
Zoe