Intuitive meaning of the members of the spectrum of a linear operator which are not eigenvalues Let $A$ be a (maybe unbounded) linear operator on a Hilbert space $\mathcal{H}$. If $A-\lambda$ is not injective, $\lambda$ is an eigenvalue. It simply means that $A$ does not change the "direction" of some vectors and so the vectors form an invariant subspace. The rest of the spectrum of $A$ consists of $\lambda$ such that $A - \lambda$ is injective but not surjective or it is bijective but the inverse is not bounded. Spectral theorems require not only eigenvalues but such $\lambda$s. Is there some intuitive or educational meaning for the existence of such $\lambda$?
 A: Fredholm was the first to define a general linear operator in the latter part of the 19th century. He converted partial differential equations to integral equations, and he studied these operators through what is now known as a resolvent equation. Fredholm connected the singularities of the resolvent $(L-\lambda I)^{-1}$ with the classical eigenfunctions of the differential operators, and this led a more general complex analysis of the resolvent $(L-\lambda I)^{-1}$. The resolvent is defined only if $L-\lambda I$ is a continuous bijection, and $\lambda \mapsto (L-\lambda I)^{-1}$ is a holomorphic operator function on this open set of $\lambda$, which is the resolvent set. As you might expect from classical holomorphic function theory, the resolvent is characterized in some sense by its singularities, which is known as the spectrum of $L$. You can classify the spectrum according to the type of defect preventing the continuous inversion of $(L-\lambda I)$, and the end result of that classification is what you're studying.
The idea of a spectral respresentation is that you form a contour integral to represent $A$ and all of its powers by integrating around the spectrum:
$$
           A^n = \frac{1}{2\pi i}\oint_{C} \lambda^n \frac{1}{\lambda I-A}d\lambda
$$
Then you can shrink the contour down around the spectrum. You may or may not get a nice representation. At least you do for a selfadjoint operator $L$. And you get a completeness result for nice operators when you can trade all of the singularities in the finite place for a singular residue at $\infty$, which at least happens for selfadjoint operators:
$$
        \lim_{\lambda\rightarrow\infty}\lambda\frac{1}{\lambda I-A}=I.
$$
It's quite a remarkable generalization of Complex Analysis where complex numbers are replaced with operators over a complex Banach or Hilbert space. These techniques were later studied by Mathematicians in the early 20th century in order to prove the convergence of a variety of spectral Fourier expansions associated with equations of Math-Physics. The early genesis of such ideas seems to trace back to Cauchy's analysis of Fourier expansions.
One of my favorite quotes is Abel's assessment of Cauchy: 
Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done.
As you might guess, Complex Analysis was used to first show that bounded operators on a Complex Banach space must have spectrum because the resolvent $(\lambda I-A)^{-1}$ is holomorphic in $\lambda$ and vanishes at $\infty$; so there must be singularities somewhere in the finite plane.
A: You can think of them in many ways. One way is to view your operator $A$ as a pointwise limit of operators $A_n$ with discrete spectrum. As $n$ gets larger the eigenvalues get denser in the parts of continuous spectrum and the associated eigenspaces split up and spread themselves out over this part. 
A good example of this perspective is the operator "multiplication with the variable" on $L^2([0,1])$ (commonly denoted by $x$ or $x\cdot$). If you want you can consider $L^2(\Bbb R)$ instead to have an unbounded operator.
Divide $[0,1]$ or $\Bbb R$ into intervals $(I_{n,k})_{k}$ of length $1/n$ and let $E_{n,k}$ be the characteristic function of $I_{n,k}$ and $\lambda_{n,k}$ be the midpoint of the interval $I_{n,k}$. Now let $A_n$ be the operator that multiplies a function with
$$\sum_{k}\lambda_{n,k}\ E_{n,k}(x).$$
This function is a step-function approximation of $x$. My suggestion is that you draw a picture of what $A_n$ does for low $n$, then let $n$ get larger. You will be able to see the above remarks manifest themselves in this process.
