# A problem in the spectral theory of an ordinary differential by V. A. Tkachenko By V. A. Tkachenko

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This motivates the following definition. 4. Definition Let T ∈ Com(H) be a normal operator and f be a bounded function on σ(T ). Then N f (λn )Pn x + f (0)PKer (T) x, ∀x ∈ H. f (T )x := n=1 Note that if dim(H) < +∞ and 0 ∈ σ(T ), then f may be not defined at 0. The above definition, however, still makes sense because in this case PKer (T) = 0, since Ker (T) = 0, and we assume that f (0)PKer (T) = 0. The operator f (T ) is well defined because 2 N f (λn )Pn x + f (0)PKer (T) x = n=1 N |f (λn )(x, en )|2 + |f (0)|2 PKer (T) x 2 ≤ n=1 2 sup |f (λ)| x 2 < ∞, ∀x ∈ H, λ∈σ(T ) (cf.

6), the previous theorem is valid for self-adjoint operators. 18). 2 is known as the Hilbert–Schmidt theorem. e. Pn := (·, en )en and let PKer (T) be the orthogonal projection onto Ker(T ). 2), where the series are strongly convergent. ). Suppose a function f is analytic in some neighbourhood ∆f of σ(T ) and Ω is an admissible set such that σ(T ) ⊂ Ω ⊂ Cl(Ω) ⊂ ∆f . 11) and the Cauchy theorem f (T )x = − − 1 2πi 1 2πi f (λ)R(T ; λ)dλ x = ∂Ω N (λn − λ)−1 Pn x − λ−1 PKer (T) x dλ = f (λ) ∂Ω n=1 N − n=1 1 f (λ)(λn − λ)−1 dλ Pn x + 2πi ∂Ω 1 f (λ)λ−1 dλ PKer (T) x = 2πi ∂Ω N f (λn )Pn x + f (0)PKer (T) x, ∀x ∈ H.

1) is a linear functional and |f (x)| ≤ Bx y ≤ B x y =( B y ) x , ∀x ∈ H1 . So, f is a bounded linear functional on H1 and f ≤ B y . 34) there exists a unique z = z(B, y) ∈ H1 such that (Bx, y) = f (x) = (x, z), ∀x ∈ H1 , and z = f ≤ B y . ). e. B ∗ is bounded and B∗ ≤ B . 1) is satisfied and that the constructed operator B ∗ is the unique operator satisfying this equality. 2. Definition the adjoint of B. 3. Theorem Let H1 , H2 and H3 be Hilbert spaces, B, B1 , B2 ∈ B(H1 , H2 ) and T ∈ B(H2 , H3 ).