Spectral Synthesis and Common Cyclic Vectors

Paul S. Bourdon and Joel H. Shapiro

Michigan Math. J. 37 (19900, 71--90
Abstract: We adapt the notion of spectral synthesis, originally introduced into operator theory by John Wermer, to show that on any Hilbert space of holomorphic functions for which all point eval-uation functionals are continuous, the collection of nonscalar adjoint multiplication op-erators has a common cyclic vector. Our work extends previous results of Warren Wogen and K.C. Chan to their most natural setting. Our methods also produce common cyclic vectors for the collection non-scalar "translation-invariant" operators on the Fréchet space of entire functions, this generalizing another theorem of Chan.
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