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 contemporary ends up in Lerer’s study niche, such as Toeplitz, Toeplitz plus Hankel, and Wiener-Hopf operators, Bezout equations, inertia sort effects, matrix polynomials, and similar parts in operator and matrix concept. Biographical fabric and Lerer's record of courses whole the volume.

**Read Online or Download Advances in Structured Operator Theory and Related Areas: The Leonid Lerer Anniversary Volume PDF**

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**Sample text**

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 = ????. ,???????? −1 ⎤???? ℓ+???? 1 − ????(????)????(????) ⎥ ⎢ 1 ∂ ⎦ . ????! 1. If a Schur-class function is not a unimodular constant, then the matrices ???????? are positive deﬁnite for ???? ≥ 2.

13) is equal to the operator √ ????ℎ of multiplication by a function 1 ˜ □ ℎ ∈ ℋ(????) such that ∥ℎ∥ ˜ = ∥ℎ∥ℋ(????) ≤ 1 − ∥???? − 2 x∥2 . 6. 2)) as follows. 3 (ii) if and only if ⎡ ⎤ 1 x∗ ????????∗ P=⎣ x ???? ????????∗ ⎦ ≥ 0. ???????? ???????? ????ℋ(????) The latter condition is equivalent to the positivity on Ω× Ω of the following kernel: ⎤ ⎡ ???? (????)∗ 1 x∗ ???? ???? (????)∗ ⎦ ર 0. 14) K(????, ????) = ⎣ x ???? (????) ???? (????) ????(????, ????) This equivalence is justiﬁed by the fact that the set of all vectors of the form ⎡ ⎤ ???? ???????? ∑ ⎣ ⎦ ???????? ????(????) = (???????? ∈ ℂ, ???????? ∈ ????, ???????? ∈ ???? , ???????? ∈ Ω) ????=1 ????(⋅, ???????? )???????? is dense in ℂ ⊕ ???? ⊕ ℋ(????) and since for every such vector, ???? 〈 [ ????ℓ ] [ ???????? ]〉 ∑ ⟨P????, ????⟩ℂ⊕???? ⊕ℋ(???????? ) = K(???????? , ????ℓ ) ????????ℓ , ???????????? ????,ℓ=1 ℓ ???? ℂ⊕???? ⊕???? .

Then: Interpolation in Sub-Bergman Spaces 33 1. 32) ????=1 for some choice of ℎ???? ∈ ℋ(????ℰ,???? ) and ℎ???? ∈ ???????? (???? ) for ???? = 1, . . , ???? − 1. 2. 32) for some ℎ???? ∈ ℋ(????ℰ,???? ) and ℎ???? ∈ ???????? (???? ) for ???? = 1, . . , ???? − 1 such that ∥ℎ???? ∥2ℋ(????????,???? ) + ????−1 ∑ ????=1 1 ∥ℎ???? ∥2???????? (???? ) ≤ 1 − ∥???? − 2 x∥2 . 1 In particular, such solutions exist if and only if ∥???? − 2 x∥2 ≤ 1. If ???? ≡ 0, then the space ℋ(????????,???? ) amounts to the Bergman space ???????? (????). 3) that ???? = 0 and subsequently, ???????? = ????????,????,???? for ???? = 1, .