By Daniel Alpay (Editor), Victor Vinnikov (Editor)

This quantity comprises six peer-refereed articles written at the social gathering of the workshop Operator thought, process thought and scattering thought: multidimensional generalizations and comparable issues, held on the division of arithmetic of the Ben-Gurion collage of the Negev in June, 2005. The ebook will curiosity a large viewers of natural and utilized mathematicians, electric engineers and theoretical physicists.

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6); in the present paper we are interested only in this situation. 16) at z1 ∈ T with the properties (1), (2), and (3) we denote by Szκ1 ;2p (Sz1 ;2p , respectively). 11): 1 − s(z)s(w)∗ , z, w ∈ hol (s), 1 − zw∗ the reproducing kernel Pontryagin space P(Ks ) is denoted by P(s). Ks (z, w) = 22 D. Alpay, A. Dijksma and H. 18) with a complex number γ: γ = s(z1 ), a bounded operator T in some Pontryagin space (P, · , · ), and elements u and v ∈ P. 18) we form the operator matrix V= T ·, v u γ : P C → P .

Dijksma and H. 18) with a complex number γ: γ = s(z1 ), a bounded operator T in some Pontryagin space (P, · , · ), and elements u and v ∈ P. 18) we form the operator matrix V= T ·, v u γ : P C → P . 19) Then the following statements are equivalent, see [19]: (a) s(z) ∈ Sz1 . 19) P is isometric in and closely innerconnected, that is, C P = span T j v j = 0, 1, . . 19) P is coisometric (that is, its adjoint is isometric) in and closely outerconC nected, which means that P = span T ∗i v i = 0, 1, .

Then β00 = 1, βi,−1 = β−1,j = 0, i, j = 0, 1, . . 8). We have j d∗ki dkj = βij = k=0 i, j = 0, 1, . . , i + j > 0. 8) k=0 i ∗i−k j j−k , z z k 1 k 1 and it is to be shown that this expression equals j k=0 i−1 j j−k + (z1∗ )i−k z k k 1 j−1 + 1 − |z1 |2 k=0 j−1 k=0 i j − 1 j−k (z1∗ )i−k z1 k k i−1 j − 1 j−1−k . (z1∗ )i−1−k z1 k k Comparing coeﬃcients of (z1∗ )i−k z1j−k it turns out that we have to prove the relation i−1 j i j−1 i−1 j−1 i−1 j−1 i j + + − = . 9) gives the desired result. To prove the last statement we use that a Hermitian matrix is positive if and only if the determinant of each of its principal submatrices is positive.