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On the point spectrum of selfadjoint extensions. (English) Zbl 0791.47005
Let H be a symmetric operator with spectral gap and infinite deficiency indices. The question is examined, to which extend the point spectrum of a selfadjoint extension H ^ of H within the set of regular points of H can be prescribed. Among others it is shown that within a spectral gap of H each kind of pure point can be generated by a selfadjoint extension of H. The question whether von Neumann’s theorem on the existence of selfadjoint extensions of C-real symmetric operators has a converse is answered (positively for deficiency (1,1), negatively for deficiency (n,n) with n>1).

MSC:
47A20Dilations, extensions and compressions of linear operators
47B25Symmetric and selfadjoint operators (unbounded)
47A10Spectrum and resolvent of linear operators
References:
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[3]Friedrichs, K.: Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren. Math. Ann.109, 465–487 (1934) · Zbl 0008.39203 · doi:10.1007/BF01449150
[4]Krein, M.G.: Theory of self-adjoint extensions of semi-bounded Hermitean operators and its application. I (in Russian). Mat. Sb.20 (no. 3), 431–490 (1947)
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[6]Derkach, V.A., Malamud, M.M.: Generalized resolvents and the boundary value problems for Hermitean operators with gaps. J. Funct. Anal.95, 1–95 (1991) · Zbl 0748.47004 · doi:10.1016/0022-1236(91)90024-Y