Cyclic vectors in 𝐴^{-∞}

Brown, Leon; Korenblum, Boris

doi:10.1090/S0002-9939-1988-0915731-9

Cyclic vectors in $A^{-\infty }$
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by Leon Brown and Boris Korenblum

Proc. Amer. Math. Soc. 102 (1988), 137-138

DOI: https://doi.org/10.1090/S0002-9939-1988-0915731-9

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Abstract:

If $f$ is in ${A^{ - p}}$, then $f$ is cyclic in ${A^{ - \infty }}$ if and only if $f$ is cyclic in every ${A^{ - q}}(q{\text { > }}p)$. An analogous result holds for the Bergman spaces ${B^p}$. In this note we apply the theory developed in [2 and 3] to explain the relationship between cyclic vectors in ${A^{ - \infty }}$ and ${A^{ - p}}$ or ${B^p}$.

References

D. Aharonov, H. S. Shapiro, and A. L. Shields, Weakly invertible elements in the space of square-summable holomorphic functions, J. London Math. Soc. (2) 9 (1974/75), 183–192. MR 365150, DOI 10.1112/jlms/s2-9.1.183
Boris Korenblum, An extension of the Nevanlinna theory, Acta Math. 135 (1975), no. 3-4, 187–219. MR 425124, DOI 10.1007/BF02392019
Boris Korenblum, A Beurling-type theorem, Acta Math. 138 (1976), no. 3-4, 265–293. MR 447584, DOI 10.1007/BF02392318
G. Šapiro, Some observations concerning weighted polynomial approximation of holomorphic functions, Mat. Sb. (N.S.) 73 (115) (1967), 320–330 (Russian). MR 0217304