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i) Let $X$ be a Banach space and $A \in \mathcal{L}(X) $ such that $0 \in \sigma(A) $ Show that if $0 \in \sigma_c(A) \bigcup \sigma_r(A) $ then is the linear map $A$ is not open.

ii) Examples of maps such that $0 \in \sigma_p(A) $ which are open and not open.

i)From here I can deduce that $ran(A)$ is a proper dense subset in $X$ and that its closure is properly contained in $X$ and that $A$ is injective . I am trying to show that not any open set is open , but don't know how to continue.

Vegan Maths
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  • Hint: Show that $\mathrm{ran}(A)$ is not open (in addition to what you proved, use that it is a subspace). – MaoWao Jul 09 '20 at 11:24
  • Yes,I thought of doing something like this . If I take the whole space $X$ then it is open , but I am not sure on how to deduce that the image is not open, I only know that it is dense. – Vegan Maths Jul 09 '20 at 11:29
  • No proper subspace of a normed space is open. If you don't manage to prove that yourself, you can look here: https://math.stackexchange.com/questions/148850/every-proper-subspace-of-a-normed-vector-space-has-empty-interior – MaoWao Jul 09 '20 at 11:31
  • Okay , thank you. What about ii)? – Vegan Maths Jul 09 '20 at 11:35
  • By the open mapping theorem and the remark above, an operator on a Banach space is open if and only if it is surjective. This should help you to find examples of both kinds. – MaoWao Jul 09 '20 at 11:41

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As @MaoWao said, $A$ is an open map if and only if $A$ is surjective. Therefore the problems can be formulates as follows:

i) If $A$ is injective but not invertible, show that it is not surjective (this is clear).

ii) Find examples of a noninjective surjective linear map and noninjective nonsurjective linear map. For this consider operators $\ell^2 \to \ell^2$ given by $$(x_n)_n \mapsto (x_2, x_3, x_4, \ldots), \qquad (x_n)_n \mapsto (0, x_2, x_3, \ldots).$$

mechanodroid
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