Abstract
Some improvements are made in the spectral theory of transformations which are nearly selfadjoint. Consider a densely defined transformation T, with domain and range in a Hilbert space 
, such that the adjoint T* of T has the same domain as T and T — T* is (the restriction of) a completely continuous transformation. The existence of invariant subspaces is not known for (the resolvents of) T if no further hypothesis is made on T — T*. But if T — T* belongs to the class of completely continuous operators introduced by Macaev [1], then invariant subspaces exist which cleave the spectrum of the transformation. The hypotheses of the Brodskii expansion [2] are satisfied. A generalization of the Fourier transformation results when the expansion is formulated in terms of the nodal Hilbert spaces of analytic functions of a Livšič-Brodskii node [3].
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© 1984 Springer Basel AG
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de Branges, L. (1984). The Expansion Theorem for Hilbert Spaces of Analytic Functions. In: Dym, H., Gohberg, I. (eds) Topics in Operator Theory Systems and Networks. OT 12: Operator Theory: Advances and Applications, vol 12. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5425-2_3
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DOI: https://doi.org/10.1007/978-3-0348-5425-2_3
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