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Extra resources for An Operator Semigroup in Mathematical Genetics
Y N ∈ M and scalars α1 , . . , α N such that N αi xi ⊗ yi < . g.  or ). In symbols: M = l 1 ⊗ l 1 . 8) is true: it may be checked directly that for any m = ξi, j i, j∈I we have ξi, j ei ⊗ e j . m= i, j∈I Also, it follows that vectors ei ⊗ e j , i, j ≥ 1 form a Schauder basis for M. 1 Banach Spaces l 1 and M = l 1 ⊗ l 1 31 We also consider Ms , the subspace of M composed of symmetric matrices m = ξi, j i, j∈I with ξi, j = ξ j,i . Introducing x y = x ⊗ y + y ⊗ x ∈ Ms for x, y ∈ l 1 , x = y and x x = x ⊗ x, we see that Ms is a Banach space with Schauder basis ei e j .
The statement of the theorem does not change if we replace the phrase ‘Markov operator’ with ‘sub-Markov operator’ throughout. Before stating this key result, we did not want to intimidate the reader by mentioning that this is in fact a simple version of the celebrated Hille–Yosida theorem. But we cannot hide this information any longer, as it is fair to say that this is probably one of the most profound theorems discussed in this book. For the proof we need the following lemma. Lemma 1 Let P be a Markov operator and λ > 0 be given.
It is an exercise to check that the map A → A satisfies the properties listed in the definition of the normed space. As an example, we will show that all operators related to stochastic matrices have norm one. 15) does not exceed 1. To prove that it is in fact equal 1, consider x = e1 . Clearly x = 1. The value of P x is the vector equal to the first row in the matrix P. Since P is a stochastic matrix, the coordinates of this vector are non-negative and add up to 1. 15) is at least 1, proving our claim.