An additional structure can be added to B H an involution More precisely for

# An additional structure can be added to b h an

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An additional structure can be added to B ( H ): an involution. More precisely, for any B B ( H ) and any f, g ∈ H one sets B * f, g := f, Bg . (1.12) Exercise 1.3.4. For any B B ( H ) show that (i) B * is uniquely defined by (1.12) and satisfies B * B ( H ) with B * = B , (ii) ( B * ) * = B , (iii) B * B = B 2 , (iv) If A B ( H ) , then ( AB ) * = B * A * . 12 CHAPTER 1. HILBERT SPACE AND BOUNDED LINEAR OPERATORS The operator B * is called the adjoint of B , and the proof the unicity in (i) involves the Riesz Lemma. A complete normed algebra endowed with an involution for which the property (iii) holds is called a C * -algebra. In particular B ( H ) is a C * -algebra. Such algebras have a well-developed and deep theory, see for example [Mur]. However, we shall not go further in this direction in this course. We have already considered two distinct topologies on H , namely the strong and the weak topology. On B ( H ) there exist several topologies, but we shall consider only three of them. Definition 1.3.5. A sequence { B n } n N B ( H ) is uniformly convergent to B B ( H ) if lim n →∞ B n - B = 0 , is strongly convergent to B B ( H ) if for any f ∈ H one has lim n →∞ B n f - B f = 0 , or is weakly convergent to B B ( H ) if for any f, g ∈ H one has lim n →∞ f, B n g - B g = 0 . In these cases, one writes respectively u - lim n →∞ B n = B , s - lim n →∞ B n = B and w - lim n →∞ B n = B . Clearly, uniform convergence implies strong convergence, and strong convergence implies weak convergence. The reverse statements are not true. Note that if { B n } n N B ( H ) is weakly convergent, then the sequence { B * n } n N of its adjoint operators is also weakly convergent. However, the same statement does not hold for a strongly convergent sequence. Finally, we shall not prove but often use that B ( H ) is also weakly and strongly closed. In other words, any weakly (or strongly) Cauchy sequence in B ( H ) converges in B ( H ). Exercise 1.3.6. Let { A n } n N B ( H ) and { B n } n N B ( H ) be two strongly conver- gent sequence in B ( H ) , with limits A and B respectively. Show that the sequence { A n B n } n N is strongly convergent to the element A B . Let us close this section with some information about the inverse of a bounded operator. Additional information on the inverse in relation with unbounded operators will be provided in the sequel. Definition 1.3.7. An operator B B ( H ) is invertible if the equation Bf = 0 only admits the solution f = 0 . In such a case, there exists a linear map B - 1 : Ran ( B ) → H which satisfies B - 1 Bf = f for any f ∈ H , and BB - 1 g = g for any g Ran ( B ) . If B is invertible and Ran ( B ) = H , then B - 1 B ( H ) and B is said to be invertible in B ( H ) (or boundedly invertible ). Note that the two conditions B invertible and Ran ( B ) = H imply B - 1 B ( H ) is a consequence of the Closed graph Theorem. In the sequel, we shall use the notation 1 B ( H ) for the operator defined on any f ∈ H by 1 f = f , and 0 B ( H ) for the operator defined by 0 f = 0.  • • • 