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14) Proof. 1) we introduce the norm ω f D = |α| + |β| + |f1 (t)| dt. 2) that for some c Lm+1 D ≤ cm Lm Hence, the series ≤ cm+1 m!. D ∞ B(x, λ) = (iλ)m Lm+1 m! m=0 converges for |λ| < c−1 . We also see that B(x, λ) ∈ D and SB(x, λ) = eiλx . 6). 17) x where f (x) ∈ D, and g(x, t) is a continuous function of x and t (0 ≤ x, t ≤ ω). 17) holds for f (x) = δ(x) and f (x) = δ(ω − x). 17) holds also for f (x) ∈ L(0, ω). 16) by the method of successive approximations. This completes the proof of our assertion.

Constructing the inverse operator 11 that is, μ(t) = const = μ = μ. 53) is true. 52) is not only necessary but sufficient as well. 53) in the form M (x) − N (x) = μ, μ = μ. 54) Since the kernel s(x) of S is determined to a constant, we may assume, without loss of generality, that μ = 0. 55) M (x) = N (x). 56) are true. 6. We discuss separately triangular operators with difference kernels: x d Sf = dx s(x − t)f (t) dt, f ∈ L2 (0, ω). 12. 58) dx 0 where N2 (x) = S −1 1. Proof. A direct calculation shows that N1 (x) = 1.

3, S would be invertible. Thus, dim SA∗ HS = 1. Hence, the subspace A∗ HS has a common part HS1 of dimension n − 1 with HS . Similarly we derive that A∗ HS = HS1 , Putting HS2 ∗ =A HS1 dim SA∗ HS1 ≤ 1. 36) that dim HS2 ≥ n − 2. 37). Repeating this process we obtain subspaces HSk = A∗ HSk−1 ∩ HS (2 ≤ k ≤ n − 1), and HS ⊃ HS1 ⊃ · · · ⊃ HSn−1 , dim HSk = n − k. 38) Thus, there exists a function f0 ∈ HS such that fk = A∗k f0 ∈ HSk (1 ≤ k ≤ n − 1), f0 p = 0. 39) The system of functions f1 , f2 , . .

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