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Ible, and let (Π ˜ By (Bn ) we denote the operator ˜n = A˜n (I − Π ˜ n ). 39) 38 Chapter 1. 6, the real spectrum of B ˜ tained in the set (1) ∪ [d, +∞), so the operator Bn is Moore-Penrose invertible, √ ˜n∗ B ˜ n )−1 B ˜ n+ ≤ 1/ d. Henceforth the sequence (B ˜n ) is ˜n + Π ˜n∗ and B ˜n+ = (B B ˜ ˜ asymptotically Moore-Penrose invertible. In addition, we have An Πn → 0 as n → ∞ (cf. 8). 39) we obtain ˜n + G ˜n A˜n = B ˜ n → 0 as n → ∞. with G The proof of the suﬃciency is evident. 8 Approximation Methods in Para-Algebras Let X and Y be Banach spaces.

A0 en + ib0 en ) 30 Chapter 1. Complex and Real Algebras and Pn := (Wn )2 . It is clear that (Pn )2 = Pn , and Pn , n ∈ N are orthogonal projections such that on the space HR the sequence (Pn ) strongly converges to the identity operator. Moreover, the sequence (Wn ) weakly converges to zero. Notice that, as mappings, the operators Pn , Wn in HR coincide with the operators Pn , Wn in H. 5 remains valid in this situation. Consider now the bounded and linear operator M : HR → HR which is deﬁned on the basis elements ek , iek by M (ek ) = e−k , M (iek ) := −ie−k , k ∈ Z.

Let us recall some necessary notions. Consider a system M of four abelian groups N1 , S1 , S2 , N2 . Each group operation is called addition and is denoted by ‘+’. The system M is called a para-group and denoted by   S1 N2  M =  N1 S2 or by M = (N1 , S1 , S2 , N2 ) if it possesses a second operation called multiplication, and satisﬁes the following conditions: 1. If n1 , n1 ∈ N1 , n2 , n2 ∈ N2 , s1 ∈ S1 , s2 ∈ S2 , then the following products exist and belong to the designated groups of the system M : n1 n1 ∈ N1 , s1 n 1 ∈ S 1 , n2 n2 ∈ N2 , s1 s2 ∈ N2 , n 1 s2 ∈ S 2 , n 2 s1 ∈ S 1 , s2 s1 ∈ N1 , s2 n 2 ∈ S 2 . 