Introduction .................................................... 5
Summary of the main results .................................... 11
1. Extension theorem. The case of the strong operator
topology .................................................... 12
1.1. Key lemma .............................................. 12
1.2. Extension theorem for strong operator integral
equalities ............................................. 19
1.3. Generalization of the Newton-Leibniz formula ........... 26
1.4. Application to resolvents of unbounded scalar type
spectral operators in a Banach space G ................. 35
2. Extension theorem. The case of the topology σ(B(G),Ν) ....... 38
2.1. Introduction ........................................... 38
2.2. Existence of the weak integral with respect to the
σ(B(G),Ν)-topology ..................................... 38
2.3. Commutation and restriction properties ................. 44
2.4. Extension theorem for integral equalities with
respect to the σ(B(G),Ν)-topology ...................... 52
2.5. Generalization of the Newton-Leibniz formula ........... 57
References ..................................................... 59
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