By Jussi Behrndt, Karl-Heinz Förster, Carsten Trunk

The current booklet is a memorial quantity dedicated to Peter Jonas. It incorporates a number of fresh unique learn papers written via famous experts within the box of operator idea in Hilbert and Krein areas. The papers include new effects for difficulties as regards to the world of study of Peter Jonas: Spectral, perturbation and scattering thought for linear operators and the research of comparable sessions of capabilities.

**Read Online or Download Recent Advances in Operator Theory in Hilbert and Krein Spaces (Operator Theory: Advances and Applications) PDF**

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**Additional info for Recent Advances in Operator Theory in Hilbert and Krein Spaces (Operator Theory: Advances and Applications)**

**Example text**

2 there exists a unit vector eδ ∈ Cn such that e − eδ < δ and the function (Lδ (λ)−1 eδ , eδ ) has a pole in each eigenvalue of Lδ . Consider the matrix function Lδ s,t;eδ (λ) = Lδ (λ) + (λs + t)( · , eδ ) eδ . We proved in part (a) that the eigenvalues of Lδ s,t;e (−, +)-interlace the eigenδ value of Lδ . The (−, +)-interlacing of the ordered eigenvalues of Ls,t;e and L is obtained by taking the limit as δ goes to 0. 6) holds for Ls,t+τ ;e. Then we apply (b) and take the limit as τ goes to 0 to obtain the (−, +)-interlacing stated in the theorem.

Let L(λ) = λ2 + λB + C be a hyperbolic n × n matrix polynomial and let e be a unit vector in Cn . Assume that s , and (i) L has only simple eigenvalues: {α±j }nj=1 ∈ T2n −1 (ii) the function (L(λ) e, e) has a pole at each of these eigenvalues. Then the ordered eigenvalues of L∞;e = Pe L|ran Pe strictly block-interlace the ordered eigenvalues {α±j }nj=1 of L. Proof. Let c ∈ (α−1 , α1 ). Then −(λ − c)(L(λ)−1 e, e) is a Nevanlinna function, which has a pole at each of the points α±j , j = 1, . . , n, and tends to 0 as λ → ∞.

1. Introduction Before to state the problem to be studied, let us recall some basic notions of Operator Theory in spaces with indeﬁnite metrics. The details one can see, for example, in the monograph [4], where also priority questions are considered. Let H be a Hilbert space endowed with a scalar product (·, ·), and P ± be 2 ∗ orthogonal projections (P ± = P ± = P ± ) such that P + + P − = I. So we have the following decomposition of H: H = H+ ⊕ H− , Denote J = P + −P − H± = P ± H. 1) and deﬁne a bilinear form (J-metric) in H as follows: [x, y] = (Jx, y).