By Didenko V., Silbermann B.

This publication offers with numerical research for definite sessions of additive operators and similar equations, together with singular crucial operators with conjugation, the Riemann-Hilbert challenge, Mellin operators with conjugation, double layer strength equation, and the Muskhelishvili equation. The authors suggest a unified method of the research of the approximation tools into consideration in keeping with specified genuine extensions of advanced C*-algebras. The checklist of the tools thought of contains spline Galerkin, spline collocation, qualocation, and quadrature equipment. The publication is self-contained and available to graduate scholars.

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**Extra info for Approximation of additive convolution-like operators: Real C-star-algebra approach**

**Example text**

Suppose that the element a ˜ is Moore-Penrose invertible in A, + ˜. By B we denote the element Ψ(˜ a+ ). let a ˜ be the Moore-Penrose inverse for a 2×2 ∗ −1 Evidently, B belongs to EA˜ , so for all real λ such that 0 < |λ| < ||BB || , the 2×2 element E − λBB ∗ is invertible in EA ˜ . 25) 2×2 also belongs to EA ˜ . ) ˜ . (Note that E denotes the identity element of A 2×2 −1 ∗ ˆ A belongs to E 2×2 , then B = (A) Suﬃciency. If Aˆ is invertible in E ˜ A ˜ A and is the Moore-Penrose inverse for A = Ψ(˜ a).

A careful consideration of its proof shows that this lemma is also true for real C ∗ -algebras. More precisely, we have the following lemma. 7. Let B be a real C ∗ -algebra with identity, and let b ∈ B be a selfadjoint element with ||b2 −b|| < 1/4. Then there exists a self-adjoint element g ∈ B such that b + g is a projection and ||g|| ≤ 2||b2 − b||. To verify this result, one can exploit the isometrical isomorphism between the real algebras R(b) and CR (spr R(b)) (cf. 3) and follow the corresponding proof for complex algebras, since this proof uses only real-valued functions.

B) p˜ = 12 (e + em), where e is the identity element in A. Proof. If the element g is invertible from the left, then relation iv) implies f = g. Using the equality g = g ∗ , one obtains g = e/2. 6 Operator Sequences: Stability Let X, Y be real or complex Banach spaces, and let (PnX ), (PnY ) be sequences of projection operators such that these sequences and the sequences ((PnX )∗ ), ((PnY )∗ ) converge strongly to the identity operators in the corresponding Banach spaces. Consider the operator equation Ax = y, x ∈ X, y ∈ Y, A ∈ L(X, Y ).