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 offers fresh leads to Lerer’s study niche, along with Toeplitz, Toeplitz plus Hankel, and Wiener-Hopf operators, Bezout equations, inertia style effects, matrix polynomials, and similar parts in operator and matrix conception. Biographical fabric and Lerer's checklist of courses whole the volume.

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

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3) and where the parameter ???? is an arbitrary contraction from Ran(???? − ????1∗ ????1 ) into Ran(???? − ????2∗ ????2 ). 2. 1) if and only if ????1 is isometric on ???? or ????2 is isometric on ????. 1) (see [6, Section 2]). The left tangential Nevanlinna–Pick interpolation problem for the reproducing kernel Hilbert space ℋ(????) can be formulated as follows. We are given vectors ????1 , . . , ???????? ∈ ???? and points ????1 , . . , ???????? ∈ ???? along with numbers ????1 , . . , ???????? ∈ ℂ and seek ???? ∈ ℋ(????) (possibly also with ∥???? ∥ℋ(????) ≤ 1) satisfying the left tangential Nevanlinna–Pick interpolation conditions ⟨???? (???????? ), ???????? ⟩ = ???????? for ???? = 1, .

2. 1 (i) and has the minimal possible norm ∥????min∥ℋ(????????,???? ) = ∥???????? 2 x∥. 3. A function ℎ ∈ ℋ(????????,???? ) satisfies (???? ∗ ℎ)∧???? (???? ∗ ) = 0 if and only if ℎ is in the ˜ ????,???? ) with reproducing kernel reproducing kernel Hilbert space ℋ(???? ∗ ˜ ????,???? (????, ????) = ???????? − ????(????)????(????) − ???? ???? (????)???? −1 ???? ???? (????)∗ . A. Ball and V. 3. 4. 8 below under the stronger assumption ????1 > 0. 7) this condition implies that ???????? > 0 for all ???? = 1, . . , ????. , [8]) that the operator ???? is strongly stable in the sense that ???? ???? converge to zero in the strong operator topology.

6) as well as the inequalities ???????? ≥ ????????−1 ≥ 0 for ???? = 2, . . , ????. 7) Proof. 8) ????=0 Note that the latter series converges weakly since the pair (????, ???? ) is ????-output stable and ????ℎ ∈ ???????? (????). 8), after taking the inner product against a fixed vector ???? ∈ ???? , converges absolutely. We may then rearrange the series to have the form ∞ ∑ ∗ ???????????? ℎ???? , ????⟩ = ⟨????????,????,???? ⟨????????+???? (???? ∗ )????+???? ???? ∗ ???????? ℎ???? , ????⟩. ????,????=0 We may then invoke Abel’s theorem to take the limit as ???? ↗ 1 (justified by the facts that (????, ???? ) is ????-output stable and that ????ℎ ∈ ???????? (????)) to get ∗ ???????? ℎ ????????,????,???? ∗ ∧???? = (???? ????ℎ) ∗ (???? ) = ∞ ∑ ????,????=0 (???? ∗ )????+???? ???? ∗ ???????? ℎ???? .

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