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Journal of Operator Theory

Volume 51, Issue 2, Spring 2004  pp. 321-334.

Krein's resolvent formula and perturbation theory

Authors P. Kurasov (1) and S.T. Kuroda (2)
Author institution: (1) Department of Mathematics, Lund Institute of Technology, Box 118, 221 00 Lund, Sweden
(2) Department of Mathematics, Gakushuin University, 1-5-1 Mejiro Toshima-Ku, Tokyo, 171--8588, Japan

Summary:  The difference between the resolvents of two selfadjoint extensions of a certain symmetric operator $ A $ is described by Krein's resolvent formula. We will prove an analog of Krein's formula in a general framework, apply it to extensions theory, and give a straightforward proof of Krein's formula including the case that $ A$ is not necessarily densely defined. We will also present a modification of Krein's formula adjusted to perturbation theory and prove the corresponding resolvent estimate.

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