Isometric equivalence of isometries on 𝐻^{𝑝}

Cima, Joseph; Wogen, Warren

doi:10.1090/proc/13106

Isometric equivalence of isometries on $H^p$
HTML articles powered by AMS MathViewer

by Joseph A. Cima and Warren R. Wogen PDF

Proc. Amer. Math. Soc. 144 (2016), 4887-4898 Request permission

Abstract:

We consider a natural notion of equivalence for bounded linear operators on $H^p,$ for $1 \leq p < \infty , p \neq 2.$ We study the structure of isometries on $H^p$ of finite codimension and we determine when two such isometries are equivalent. Among these isometries, we determine which operators $S$ satisfy $\bigcap _1^{\infty } S^n H^p=(0).$

References

Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026
Richard M. Crownover, Commutants of shifts on Banach spaces, Michigan Math. J. 19 (1972), 233–247. MR 361843
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Karel de Leeuw, Walter Rudin, and John Wermer, The isometries of some function spaces, Proc. Amer. Math. Soc. 11 (1960), 694–698. MR 121646, DOI 10.1090/S0002-9939-1960-0121646-9
Richard J. Fleming and James E. Jamison, Isometries on Banach spaces: function spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 129, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1957004
Frank Forelli, The isometries of $H^{p}$, Canadian J. Math. 16 (1964), 721–728. MR 169081, DOI 10.4153/CJM-1964-068-3
James E. Robinson, Crownover shift operators, J. Math. Anal. Appl. 130 (1988), no. 1, 30–38. MR 926826, DOI 10.1016/0022-247X(88)90384-8
Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7
Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190