Left Shift Semigroup and Its Infinitesimal Generator

Left shift operator

Throughout we consider the Hilbert space L2=L2(R), the space of all complex-valued functions with real variable such that f∈L2 if and only if ‖f‖22=∫∞−∞|f(t)|2dm(t)<∞ where m denotes the ordinary Lebesgue measure (in fact it's legitimate to consider Riemann integral in this context).

For each t≥0, we assign an bounded linear operator Q(t) such that (Q(t)f)(s)=f(s+t). This is indeed bounded since we have ‖Q(t)f‖2=‖f‖2 as the Lebesgue measure is translate-invariant. This is a left translation operator with a single step t.

Properties of Q(t)

In Hilbert space

The inner product in L2 is defined by (f,g)=∫∞−∞f(s)¯g(s)dm(s),f,g∈L2. If we apply Q(t) on f, we see (Q(t)f,g)=∫∞−∞f(s+t)¯g(s)dm(s)=∫∞−∞f(u)¯g(u−t)dm(u)(u=s+t)=(f,Q(t)∗g) where Q(t)∗ is the adjoint of Q(t), which happens to be a left translation operator with a single step t. Clearly we have Q(t)Q(t)∗=Q(t)∗Q(t)=I, which indicates that Q(t) is unitary. Also we can check in a more manual way: (Q(t)f,Q(t)g)=∫∞−∞f(s+t)¯g(s+t)dm(s)=∫∞−∞f(s+t)¯g(s+t)dm(s+t)=(f,g). By operator theory, since Q(t) is unitary and bounded, the spectrum of Q(t) lies in the unit circle S1.

As a semigroup

Note Q(0)=I and Q(t+u)f(s)=f(s+t+u)=f[(s+t)+u]=Q(u)f(s+t)=Q(t)Q(u)f(s) for all f∈L2, which is to say that Q(t+u)=Q(t)Q(u). Therefore we say {Q(t)} is a semigroup. But what's more important is that it satisfies strong continuity near the origin: limt→0‖Q(t)f−f‖2=0. This is not too hard to verify. It suffices to prove that limt→0∫∞−∞|f(s+t)−f(s)|2dm(s)=0. Note Cc(R) (continuous function with compact support) is dense in L2, and for f∈Cc(R), it follows immediately from properties of continuous functions. Next pick f∈L2. Then for ε>0 there exists some f1∈Cc(R) such that ‖f−f1‖2<ε4 and ‖f1(s+t)−f1(s)‖2<ε2 for t small enough. If we put f2=f−f1 we get ‖f(s+t)−f(s)‖2≤‖f1(s+t)−f1(s)‖2+‖f2(s+t)−f2(s)‖<ε2+2‖f2(s)‖<ε. The limit follows as ε→0.

Infinitesimal generator of Q(t)

Recall that the infinitesimal generator of Q(t) is defined to be A=limt→01t[Q(t)−I] which is inspired by ddtetA=A (thanks to von Neumann). Note if f∈L2 is differentiable, then Af(s)=limt→0f(s+t)−f(s)t=f′(s). The infinitesimal generator of Q(t) being differentiation operator is quite intuitive. But we need to clarify it in L2 which is much larger. So what is the domain D(A)? We don't know yet but we can guess. When talking about differentiation in Lp space, it makes sense to extend our differentiation to absolute continuity. Also we need to make sure that Af∈L2, hence we put D={f∈L2:f absolutely continuous, f′∈L2}. For every x∈D(A) and any fixed t we already have ddtQ(t)f(s)=f′(s+t)=Af(s+t) hence Af=f′ for every x∈D(A) and it follows that D(A)⊂D. In fact, A is the restriction of the differential operator on D(A). Conversely, By Hille-Yosida theorem, we see 1∈ρ(A) and also one can show that 1∈ρ(ddx). Therefore (I−ddx)D(A)=(I−A)D(A)=L2. But we also have D=(I−ddx)−1L2. Thus D=(1−ddx)−1(1−ddx)D(A)=D(A). The fact that (I−ddx)D=L2 can be realised by the equation f−f′=g, where the existence of solution can be proved using Fourier transform. Note ^f′(y)=iyˆf(y), with some knowledge of distribution, the result can also be given by D(A)={f∈L2:∫∞−∞|yˆf(y)|2dy<∞}.

Spectrum of the generator

By the Hille-Yosida theorem, the half plane {z:ℜz>0}⊂ρ(A). But we can give a more precise result of it.

Pick any f∈D(A). It is directly verified that (A−λI)f=f′−λf. Put g=(A−λI)f then ˆg(y)=iyˆf(y)−λˆf(y). Therefore ˆf(y)=ˆg(y)iy−λ∈L2. Conversely, suppose h(y)=ˆg(y)iy−λ∈L2, then ˆg(y)=iyh(y)−λh(y). If we take its Fourier inverse, we see g∈R(A−λI).

If g∈L2, then clearly ˆg∈L2. It remains to discuss ˆg(y)/(iy−λ). Note iy is on the imaginary axis, hence if λ is not purely imaginary, then ˆg(y)/(iy−λ)∈L2. If λ is purely imaginary however, then we may have ˆg(y)/(iy−λ)∉L2. For example, we can take ˆg=χ[s−1,s+1] where λ=is. Hence if λ is purely imaginary, R(A−λI) is a proper subspace of L2. Therefore we conclude: σ(A)={z∈C:ℜz=0}. This is an exercise on W. Rudin's Functional Analysis. You can find related theorems in Chapter 13.