Definition of spectrum of an operator The spectrum of a bounded operator is the eigenvalues for the corresponding matrix. 
Consider the following wiki link:
http://en.wikipedia.org/wiki/Multiplication_operator 
In the example, it says "Its spectrum will be the interval $[0, 9]$ (the range of the function $x\mapsto x^2$ defined on $[−1, 3]$)."   
My question is, what is the definition of spectrum in this example? Is it related to eigenvalues?  

The first definition only holds on finite-dimensional spaces. In a more general context, it is called the point spectrum $\sigma_p(A)$.
The more general definition of the spectrum is the one found in books on functional analysis, it contains the point spectrum.
It is defined as follows: let $X$ be $\mathbf{C}$-Banach space, and $A:D(A)\subset X\rightarrow X$ a linear operator on $X$. The spectrum $\sigma(A)$ is the complement $\mathbf{C}\setminus\rho(A)$, where $\rho(A)$ is the resolvent set.
