Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 :)

share|cite|improve this question
A real Banach space? Consider a rotation in $\mathbb{R}^2$. – Martin Feb 4 '13 at 20:44
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 $T$).
share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.