# Advances in Structured Operator Theory and Related Areas: by M. A. Kaashoek (auth.), Marinus A. Kaashoek, Leiba Rodman,

By M. A. Kaashoek (auth.), Marinus A. Kaashoek, Leiba Rodman, Hugo J. Woerdeman (eds.)

This quantity is devoted to Leonid Lerer at the party of his 70th birthday. the most half provides fresh ends up in Lerer’s learn niche, inclusive of Toeplitz, Toeplitz plus Hankel, and Wiener-Hopf operators, Bezout equations, inertia sort effects, matrix polynomials, and similar components in operator and matrix idea. Biographical fabric and Lerer's record of courses entire the volume.

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**Extra info for Advances in Structured Operator Theory and Related Areas: The Leonid Lerer Anniversary Volume**

**Example text**

Ball and V. 1 amounts to the following Lagrange– Sylvester interpolation problem: LSP: Given a scalar Schur-class function ???? ∈ ????, distinct points ????1 , . . , ???????? ∈ ???? and a collection {???????????? } of complex numbers, ﬁnd all functions ???? ∈ ℋ(????????,???? ) such that ???? (????) (???????? )/????! = ???????????? for ???? = 0, . . , ???????? − 1; ???? = 1, . . , ????. 17). ????! ,???????? −1 ????,????=1 which in turn is known to be positive deﬁnite unless ???? is a Blaschke product of degree ???? < n := ????1 + ⋅ ⋅ ⋅ + ???????? , in which case ????1 is positive semideﬁnite and rank ????1 = ????.

For this choice, an alternative formula for Ψ???? is the following 1 − 12 −1 (???????? − ???? ∗ )(???????? − ???? ∗ )−1 ????????+1 (???? − ???????? )−1 ???????? Ψ???? (????) = ????????2 (???? − ???????? )−1????????+1 . 6. Let Ψ1 , . . 25). Then ????−1 ∑ ????=1 − 12 (???? − ???????? )−????+1 ???????? Ψ???? (????)Ψ???? (????)∗ − 12 ¯ ∗ −????+1 ¯ ????−???? ???????? (???? − ???????? ) (1 − ???? ????) ????1−1 −????+1 −1 = ???????? (???? − ???????? ∗ )−????+1 . 28) Proof. 26) and can be written equivalently as − 12 (???? − ???????? )−????+1 ???????? Ψ???? (????)Ψ???? (????)∗ − 12 ???????? (???? − ???????? ∗ )−????+1 (1 − ????????)????−???? = (???? − ???????? )−????+1 ????????−1 ∗ −????+1 ¯ ????−???? (???? − ???????? ) (1 − ???? ????) − (???? − ???????? )−???? −1 ????????+1 ∗ −???? ¯ ????−????−1 (???? − ???????? ) .

6) and x ∈ ℂ???? , ﬁnd ???? ∈ ℋ(????) (possibly also with ∥???? ∥ℋ(????) ≤ 1) so that ????????∗ ???? = x. We now formulate our abstract left tangential Nevanlinna–Pick interpolation problem as follows. Let ????(????, ????) be an ℒ(????)-valued positive kernel on a Cartesian product set Ω×Ω and let ℋ(????) be the associated reproducing kernel Hilbert space, that is, the unique inner product space of ????-valued functions on Ω that contains the functions ???? → ???????? (????) := ????(????, ????)???? for all ﬁxed ???? ∈ Ω and ???? ∈ ???? which in turn have the reproducing property for ℋ(????): ⟨????, ???????? ????⟩ℋ(????) = ⟨???? (????), ????⟩???? for all ???? ∈ ℋ(????).