1198 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF

Lemma 34.5.2 Let s < t. Then for u,Y satisfying 34.5.28

⟨Bu(t) ,u(t)⟩−⟨Bu(s) ,u(s)⟩+ ⟨(B(t)−B(s))u(s) ,u(t)⟩

+⟨(B(t)−B(s))u(s) ,u(t)−u(s)⟩= 2∫ t

s⟨Y (r) ,u(t)⟩dr

−⟨B(t)u(t)−B(t)u(s) ,u(t)−u(s)⟩ (34.5.30)

Proof: It follows from the following computations

B(t)u(t)−B(s)u(s) =∫ t

sY (r)dr

and so2∫ t

s⟨Y (r) ,u(t)⟩dr−⟨B(t)u(t)−B(s)u(s) ,u(t)−u(s)⟩

= 2⟨∫ t

sY (r)dr,u(t)

⟩−⟨B(t)u(t)−B(s)u(s) ,u(t)−u(s)⟩

= 2⟨B(t)u(t)−B(s)u(s) ,u(t)⟩−⟨B(t)u(t)−B(s)u(s) ,u(t)−u(s)⟩

= 2⟨B(t)u(t) ,u(t)⟩−2⟨B(s)u(s) ,u(t)⟩−⟨B(t)u(t) ,u(t)⟩+⟨B(t)u(t) ,u(s)⟩+ ⟨B(s)u(s) ,u(t)⟩−⟨B(s)u(s) ,u(s)⟩

= ⟨B(t)u(t) ,u(t)⟩−⟨B(s)u(s) ,u(s)⟩+[⟨B(t)u(t) ,u(s)⟩−⟨B(s)u(s) ,u(t)⟩]

= ⟨B(t)u(t) ,u(t)⟩−⟨B(s)u(s) ,u(s)⟩+⟨(B(t)−B(s))u(s) ,u(t)⟩

Thus⟨Bu(t) ,u(t)⟩−⟨Bu(s) ,u(s)⟩+ ⟨(B(t)−B(s))u(s) ,u(t)⟩

= 2∫ t

s⟨Y (r) ,u(t)⟩dr−⟨B(t)u(t)−B(s)u(s) ,u(t)−u(s)⟩

Now consider the last term. It equals

⟨B(t)u(t)− (B(s)−B(t)+B(t))u(s) ,u(t)−u(s)⟩

= ⟨B(t)u(t)− ((B(s)−B(t))u(s)+B(t)u(s)) ,u(t)−u(s)⟩

= ⟨B(t)u(t)−B(t)u(s) ,u(t)−u(s)⟩+ ⟨(B(t)−B(s))u(s) ,u(t)−u(s)⟩

It follows that

⟨Bu(t) ,u(t)⟩−⟨Bu(s) ,u(s)⟩+ ⟨(B(t)−B(s))u(s) ,u(t)⟩