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45) Here, the series on the right-hand side is the limit of Sn (t) = nk=0 (Sn (t))n≥0 being the Cauchy sequence because of the estimate m Sm (t) − Sn (t) = k=n+1 t k Qk k! m ≤ k=n+1 t k Qk k! t k Qk k! , n m ≤ k=n+1 tk Q k! ≥ 0 with k , holding for m > n. e. defines the exponential function t → et Q , for any bounded linear operator Q and, in fact, for any (not necessarily non-negative) t ∈ R. Moreover, for commuting operators Q 1 and Q 2 , we have et (Q 1 +Q 2 ) = et Q 1 et Q 2 . 34), as follows.

Since Y is complete, (An x)n≥1 has a limit, say Ax. e. that A is linear. Passing with m → ∞ in the estimate above, we obtain that for all x and all sufficiently large n, An x − Ax ≤ x . This implies both that A is bounded ( Ax ≤ Ax − An x + An x An ) x ), implying A ∈ L(X, Y), and that (An )n≥1 converges to A. ≤ ( + 36 4 Mathematical Tools The fact that the space of bounded linear operators with values in a Banach space is a Banach space itself is of profound significance. 20) has an inverse (the operator I defined by I x = x is called the identity operator).

35) do exist but the qi ’s may be infinite. 30). Not in the sense of operator norm convergence, anyway. e. that the time the Markov chain spends at each point (conditional on reaching this point) is positive. g. 65]. 36) for granted. 30) does not want to be satisfied, we need to take a more modest approach. 37) exists for all x ∈ D(Q). 37). 37) speaks of norm convergence in l 1 . 37) exists merely for some, and not for all, x ∈ l 1 . 37) Qx is merely a notation for the limit (as yet, see further on).