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We use the notation C ∞ (E, F ) = {f : E → F | f smooth}. (i) There exist smooth mappings, which are not continuous! ) In particular for linear and multilinear mappings we have the following characterization (cf. 5). Proposition 2. A linear map l: E → F between locally convex spaces (lcvs) is smooth iff l is bounded. (ii) A bilinear map b: E × F → G of lcvs is smooth iff b is bounded. (i) One of the main questions at the starting point of the calculus is to provide convenient methods for testing smoothness of curves in order to check smoothness of more general maps.

8) 1 σ(H∞ ) ∩ [(−∞, −1) ∪ (1, ∞)] = σd (H∞ ) = { (κj + κ−1 j )}j∈N . 8) holds. Proof. 1. In the notation of [13], the Weyl m-functions associated with H∞ are given by m∞,± (z) = −f∞,+ (k ±1 , 1)[a∞ (0)f∞,+ (k ±1 , 0)]−1 , 1 z = (k + k −1 ), |k| < 1. 11) with uniform convergence for z in compact subsets of C\R. ) ∈ 2 ((a, ∞)), a ∈ Z. 12) D the operator HN in 2 ((−∞, 1)) with a Dirichlet boundary condition at n = 1) (HN,− together with strong resolvent convergence of HN to H∞ (cf. 5). 13) where we used the obvious notation, m± (λ + i0) = limε↓0 m± (λ + iε) in connection 1 with the branch cut k = z + (z 2 − 1) 2 , |k| ≤ 1 approaching |k| = 1 nontangentially from inside the unit k–disk.

6. 4. Spectral Properties of H∞ This section describes our principal results concerning spectral properties of the limit operator H∞ . As in Sect. 2), respectively. 1). 1. 1). 4) 1 1 −1 { (κj + κ−1 j )}j∈N ⊆ σp (H∞ ) ⊆ { (κj + κj )}j∈N . 5) The spectral multiplicity of H∞ on (−1, 1) is two while σ(H∞ )\[−1, 1] has multiplicity one. 7) in the weak sense. 8) 1 σ(H∞ ) ∩ [(−∞, −1) ∪ (1, ∞)] = σd (H∞ ) = { (κj + κ−1 j )}j∈N . 8) holds. Proof. 1. In the notation of [13], the Weyl m-functions associated with H∞ are given by m∞,± (z) = −f∞,+ (k ±1 , 1)[a∞ (0)f∞,+ (k ±1 , 0)]−1 , 1 z = (k + k −1 ), |k| < 1.

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