Definition of spectrum of an operator

Question

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?

Answer 1

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.

Answer 2

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.