Coefficient multipliers of Bloch functions
HTML articles powered by AMS MathViewer
- by J. M. Anderson and A. L. Shields
- Trans. Amer. Math. Soc. 224 (1976), 255-265
- DOI: https://doi.org/10.1090/S0002-9947-1976-0419769-6
- PDF | Request permission
Abstract:
The class $\mathcal {B}$ of Bloch functions is the class of all those analytic functions in the open unit disc for which the maximum modulus is bounded by $c/(1 - r)$ on $|z| \leqslant r$. We study the absolute values of the Taylor coefficients of such functions. In particular, we find all coefficient multipliers from ${l^p}$ into $\mathcal {B}$ and from $\mathcal {B}$ into ${l^p}$. We find the second Köthe dual of $\mathcal {B}$ and show its relevance to the multiplier problem. We identify all power series $\sum {a_n}{z^n}$ such that $\sum {w_n}{a_n}{z^n}$ is a Bloch function for every choice of the bounded sequence $\{ {w_n}\}$. Analogous problems for ${H^p}$ spaces are discussed briefly.References
- J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. MR 361090
- Gregory F. Bachelis, On the ideal of unconditionally convergent Fourier series in $L_{p}\,(G)$, Proc. Amer. Math. Soc. 27 (1971), 309–312. MR 271640, DOI 10.1090/S0002-9939-1971-0271640-X
- Gregory F. Bachelis, On the upper and lower majorant properties in $L^{p}(G)$, Quart. J. Math. Oxford Ser. (2) 24 (1973), 119–128. MR 320636, DOI 10.1093/qmath/24.1.119
- P. L. Duren and A. L. Shields, Properties of $H^{p}$ $(0<p<1)$ and its continuing Banach space, Trans. Amer. Math. Soc. 141 (1969), 255–262. MR 244751, DOI 10.1090/S0002-9947-1969-0244751-8
- D. J. H. Garling, The $\beta$- and $\gamma$-duality of sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967), 963–981. MR 218881, DOI 10.1017/s0305004100041992
- D. J. H. Garling, On symmetric sequence spaces, Proc. London Math. Soc. (3) 16 (1966), 85–106. MR 192311, DOI 10.1112/plms/s3-16.1.85
- Edwin Hewitt, The ranges of certain convolution operators, Math. Scand. 15 (1964), 147–155. MR 187016, DOI 10.7146/math.scand.a-10738
- Jean-Pierre Kahane, Some random series of functions, D. C. Heath and Company Raytheon Education Company, Lexington, Mass., 1968. MR 0254888
- C. N. Kellogg, An extension of the Hausdorff-Young theorem, Michigan Math. J. 18 (1971), 121–127. MR 280995 G. Köthe, Topological vector spaces.I, Springer-Verlag, Berlin, 1960; English transl., Springer-Verlag, New York, 1969. MR 24 #A411; 40 #1750.
- Hugo Steinhaus, Additive und stetige Funktionaloperationen, Math. Z. 5 (1919), no. 3-4, 186–221 (German). MR 1544384, DOI 10.1007/BF01203518
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 224 (1976), 255-265
- MSC: Primary 30A78
- DOI: https://doi.org/10.1090/S0002-9947-1976-0419769-6
- MathSciNet review: 0419769