Let AQ be the self-adjoint operator defined by the Q-function Q: z 7→ Qz through the Kreın-like resolvent formula (−AQ + z)−1 = (−A0 + z)−1 + GzWQ−z1V G∗z¯ , z ∈ ZQ , where V and W are bounded operators and ZQ := {z ∈ ρ(A0): Qz and Qz¯ have a bounded inverse}. We show that ZQ 6= ∅ = ZQ = ρ(A0) ∩ ρ(AQ) . We do not suppose that Q is represented in terms of a uniformly strict, operator-valued Nevanlinna function (equivalently, we do not assume that Q is associated to an ordinary boundary triplet), thus our result extends previously known ones. The proof relies on simple algebraic computations stemming from the first resolvent identity.
On inverses of Kreın’s Q-functions / C. Cacciapuoti, D. Fermi, A. Posilicano. - In: RENDICONTI DI MATEMATICA E DELLE SUE APPLICAZIONI. - ISSN 1120-7183. - 39:2(2018), pp. 229-240.
On inverses of Kreın’s Q-functions
D. Fermi;
2018
Abstract
Let AQ be the self-adjoint operator defined by the Q-function Q: z 7→ Qz through the Kreın-like resolvent formula (−AQ + z)−1 = (−A0 + z)−1 + GzWQ−z1V G∗z¯ , z ∈ ZQ , where V and W are bounded operators and ZQ := {z ∈ ρ(A0): Qz and Qz¯ have a bounded inverse}. We show that ZQ 6= ∅ = ZQ = ρ(A0) ∩ ρ(AQ) . We do not suppose that Q is represented in terms of a uniformly strict, operator-valued Nevanlinna function (equivalently, we do not assume that Q is associated to an ordinary boundary triplet), thus our result extends previously known ones. The proof relies on simple algebraic computations stemming from the first resolvent identity.File | Dimensione | Formato | |
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