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 general definition of spectrum for a bounded operator $A$ is the set of complex numbers $\lambda$ such that $A - \lambda I$ is not invertible.
EDIT: To avoid potential confusion: "not invertible" means there is no bounded operator $B$ such that $(A-\lambda I)B = B (A - \lambda I) = I$.  See also my answer to another recent question.
