A lot of the classical Lie groups are defined as the subset of the general linear group known as the automorphism groups. In sum: they're the set of linear transformations on a vector space that preserve some sort of inner product, or similar. The vector spaces of square integrable functions have inner products, and linear transformations that preserve them. For example, the Fourier transform preserves the product: $$\langle f | g \rangle = \int f^\star(t)\, g(t) \operatorname{d}t.$$
Is there a developed theory of the set of linear transforms that preserve this sort of inner product?
This is the notion of a unitary operator on a Hilbert space. They are extremely well studied.