Examples of (infinite dimensional) linear operators

Question

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?

Answer 1

Some important examples are furnished by integral kernels. Basically if $k:\mathbb{R}^2 \to \mathbb{R}$ is some function then we can define an operator $K$ on a function space by setting $$(Kf)(x) = \int_\mathbb{R} \! k(x,y) f(y) \, dy.$$ Of course one has to be careful here that one restricts the functions $f$ and $k$ so that everything makes sense. One way to do this is by requiring that $k \in L^2(\mathbb{R}^2)$ and then defining the operator only for functions $f \in L^2(\mathbb{R})$. An important example of an operator given by an integral kernel is the Fourier transform $$(Ff)(x) = \int_\mathbb{R} \! e^{-ixy} f(y) \, dy.$$ However, formalizing this example requires on to go a slightly more complicated route because the function $e^{-ixy} \notin L^2(\mathbb{R}^2)$. Other examples are given by Multiplication Operators, Convolution Operators and Composition Operators. And there are many others.

Answer 2

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.

Answer 3

We have

1. The identity.
2. Finite rank operators. These are operators for which the image is finite dimensional.
3. In Hilbert spaces we could take limits of finite rank operators. This will give us the compact operators. Or consider the compact operators in general.
4. Take linear combinations of the above.

In general it might be hard to give other examples as there are spaces for which this is all there is. See here. This is why without further assumptions on the spaces it is hard for you to give more examples (there may be no more).

Answer 4

If you take $A$ to be a Banach algebra, say, then there are multiplication operators coming from the left regular representation of $A$ on itself. I.e. for $a \in A$ you define a linear operator $L_a : A \to A$ by $$ L_a(b) = ab \quad \text{ for all } b \in A. $$ Then there are related things like GNS representations.

Answer 5

Let $\Lambda$ be the set of infinitely generated supernumbers with respect to the $1$-norm of the coefficients. In particular, I'll take the coefficients of the Grassmann generators to be complex. A typical supernumber has the form: $$ z = z_0+z_{i}\eta^i+\frac{1}{2}z_{ij}\eta^i\eta^j+ \cdots $$ where the implied sums are taken over $\mathbb{N}$ and the norm of $z$ is given by: $$ |z| = |z_0|+\sum_i |z_i|+\sum_{i,j} |z_{ij}|+ \cdots $$ Fix a particular supernumber, say $w$ then define $$ L_w(z) = wz \qquad \text{and} \qquad R_w(z) = zw $$ for all $z \in \Lambda$. These are continuous linear maps. In fact, these are superdifferentiable.

I wouldn't say these are typical examples. You can read more in the pioneering papers on this area of superanalysis by Alice Rogers.