# One-dimensional graph perturbations of selfadjoint relations

Seppo Hassi, H de Snoo

40 Citations (Scopus)

## Abstract

Let A be a selfadjoint operator (or a selfadjoint relation) in a Hilbert space h, let Z be a one-dimensional subspace of h(2) such that A boolean AND Z = {0, 0} and define S = A boolean AND Z*. Then S is a closed, symmetric operator (or relation) with defect numbers (1, 1) and, conversely, each such S and a selfadjoint extension A are related in this way. This allows us to interpret the selfadjoint extensions of S in h as one-dimensional graph perturbations of A. If Z = span {phi, psi}, then the function Q(l) = l[phi,phi] + [(A - l)(-1)(l phi - psi), (l) over bar phi - psi], generated by A and the pair {phi, psi}, is a Q-function of S = A boolean AND Z* and A. It belongs to the class N of Nevanlinna functions and essentially determines S and A. Calculation of the corresponding resolvent operators in the perturbation formula leads to Krein's description of (the resolvents of) the selfadjoint extensions of S. The class N of Nevanlinna functions has subclasses N-1 superset of N-0 superset of N--1 superset of N--3, each defined in terms of function-theoretic properties. We characterize the Q-functions belonging to each of these classes in terms of the pair {phi,psi}. If Q(I) belongs to the subclass N-k, k = 1,0, -1, -2, then all but one of the selfadjoint extensions of S have a Q-function in the same class, while the exceptional extension has a Q-function in N\N-1. In particular, if S is semi-bounded, the exceptional selfadjoint extension is precisely the Friedrichs extension. We consider our perturbation formula in the case where the Q-function Q(I) belongs to the subclass Nk, k = 1, 0, -1, -2, or if it is an exceptional function associated with this subclass. The resulting perturbation formulas are made explicit for the case that A or its orthogonal operator part is the multiplication operator in a Hilbert space L(2)(dp).

Original language English 123-164 42 Annales academiae scientiarum fennicae-Mathematica 22 1 Published - 1997

## Keywords

• RANK-ONE PERTURBATIONS
• EXTENSIONS
• OPERATORS

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