Are FT and LT both isomorphic?

Question

As the following diagram:(from a textbook)

Note:
1. L2: L2 space, H2: H2 space
2. The upper one is in t-domain; the lower one, f-domain
3. : the Laplas transform operator
: the fourier tansform operator
4. The left one is for t < 0; the right one is for t > 0

The book says:

My question:

Is the Laplace transform is also an isomorphism? i.e. L2[0, inf) is isomorphic to H2?

Answer 1

Yes, this is sometimes referred to as the Paley-Wiener Theorem. The Laplace transform is just the Fourier transform of functions supported on $[0,\infty)$, viewed through a 90-degree rotation in the plane. For example, if $f \in L^{2}[0,\infty)$, then, for real $x$, $$ \mathscr{L}\{f\}(ix) = \int_{0}^{\infty}f(t)e^{-ixt}\,dt = C\mathscr{F}\{f\}(x) $$ where $C$ is your favorite Fourier transform constant, and $\mathscr{F}$ is the classical $L^{2}$ Fourier transform.