TITLE: Examples of (infinite dimensional) linear operators
QUESTION [7 upvotes]: I'm trying to familiarize myself with linear operators. In finite dimensions it is clear to me that they are matrices. No problem there. But then in infinite dimensions matters are not so clear to me. Of course the identity map is a linear operator. I also know that if the domain is a space of functions then the integration and differentiation operators are examples of linear operators. Furthermore I found the example of the shift operator (works on sequences and function spaces). But I feel that a few more examples would help me greatly in understanding linear operators better.
Now other than the ones I mentioned what are examples of linear operators $T: X \to Y$ where $X,Y$ are infinite dimensional normed linear spaces?

REPLY [4 votes]: The answers already given are nice examples but let me give some more just to emphasize the plethora of linear operators. Let $X$ be any set. Then we can create the Hilbert space with basis $X$, call it $\mathcal{H}_X$. Any permutation of $X$ give a linear (in fact unitary) operator on $\mathcal{H}_X$. Note that this will not be all unitary operators. 
Now given a locally compact group $G$ we can construct the Hilbert space $L^2(G)$, if $G$ is discrete this follows the contruction above. Then $G$ permutes $G$ by multiplying on the left (or right). Thus we get $G$ as a subgroup of the unitary operators. If $G$ is separable then  $L^2(G)$ is as well. So if we look at just one normed linear space (ie the unique infinite dimensional separable Hilbert space) there are enough (unitary) operators to have a copy of every separable locally compact group. ... That's ALOT of operators.