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Let $X$ be a Banach real space and $T \in\mathcal{B}(X)$, where $T$ is an operator. Study if: $$\sigma(T)\neq \varnothing.$$

Can you help me please, thanks :)

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A real Banach space? Consider a rotation in $\mathbb{R}^2$. –  Martin Feb 4 '13 at 20:44

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up vote 2 down vote accepted

Not sure exactly how comprehensive of an answer you seek, but certainly there are examples of bounded linear operators on a Banach space and which have a nonempty spectrum. For example:

  1. $T:\mathbb{R}^n\to \mathbb{R}^n$, $Tx=Ax$, where $A_{n\times n}$ is the matrix representation of $T$. Then $\sigma(A)=\{\text{eigenvalues of }A\}\not=\varnothing$.
  2. $T:\ell^2 \to \ell^2$, $T\{x_1,x_2,\dots\}=\{0,x_1,x_2,\dots\}$. Then $0\in\sigma(T)$ (but note that $0$ is not an eigenvalue of $T$).
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