The backward shift on the space of Cauchy transforms

Cima, Joseph; Matheson, Alec; Ross, William

doi:10.1090/S0002-9939-03-07103-X

The backward shift on the space of Cauchy transforms
HTML articles powered by AMS MathViewer

by Joseph A. Cima, Alec Matheson and William T. Ross PDF

Proc. Amer. Math. Soc. 132 (2004), 745-754 Request permission

Abstract:

This note examines the subspaces of the space of Cauchy transforms of measures on the unit circle that are invariant under the backward shift operator $f \to z^{-1}(f - f(0))$. We examine this question when the space of Cauchy transforms is endowed with both the norm and weak${}^*$ topologies.

References

A. B. Aleksandrov, Invariant subspaces of shift operators. An axiomatic approach, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 7–26, 264 (Russian, with English summary). Investigations on linear operators and the theory of functions, XI. MR 629832
Alexandru Aleman and William T. Ross, The backward shift on weighted Bergman spaces, Michigan Math. J. 43 (1996), no. 2, 291–319. MR 1398156, DOI 10.1307/mmj/1029005464
Sheldon Axler, Bergman spaces and their operators, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 1–50. MR 958569
Paul Bourdon and Joseph A. Cima, On integrals of Cauchy-Stieltjes type, Houston J. Math. 14 (1988), no. 4, 465–474. MR 998448
J. A. Cima, A. Matheson, and W. T. Ross, The Cauchy transform, preprint.
Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, Mathematical Surveys and Monographs, vol. 79, American Mathematical Society, Providence, RI, 2000. MR 1761913, DOI 10.1090/surv/079
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
R. G. Douglas, H. S. Shapiro, and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 1, 37–76 (English, with French summary). MR 270196, DOI 10.5802/aif.338
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Kenneth Hoffman, Banach spaces of analytic functions, Dover Publications, Inc., New York, 1988. Reprint of the 1962 original. MR 1102893
Paul Koosis, Introduction to $H_p$ spaces, 2nd ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. MR 1669574
B. I. Kornbljum, Closed ideals of the ring $A^{n}$, Funkcional. Anal. i Priložen. 6 (1972), no. 3, 38–52 (Russian). MR 0324424
Stefan Richter and Carl Sundberg, Invariant subspaces of the Dirichlet shift and pseudocontinuations, Trans. Amer. Math. Soc. 341 (1994), no. 2, 863–879. MR 1145733, DOI 10.1090/S0002-9947-1994-1145733-9
William T. Ross and Harold S. Shapiro, Generalized analytic continuation, University Lecture Series, vol. 25, American Mathematical Society, Providence, RI, 2002. MR 1895624, DOI 10.1090/ulect/025
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043