Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hermitian operators and one-parameter groups of isometries in Hardy spaces

Authors: Earl Berkson and Horacio Porta
Journal: Trans. Amer. Math. Soc. 185 (1973), 331-344
MSC: Primary 47D10; Secondary 47B37
MathSciNet review: 0338833
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Call an operator A with domain and range in a complex Banach space X hermitian if and only if iA generates a strongly continuous one-parameter group of isometries on X. Hermitian operators in the Hardy spaces of the disc $ ({H^p},1 \leq p \leq \infty )$ are investigated, and the following results, in particular, are obtained. For $ 1 \leq p \leq \infty ,p \ne 2$, the bounded hermitian operators on $ {H^p}$ are precisely the trivial ones--i.e., the real scalar multiples of the identity operator. Furthermore, as pointed out to the authors by L. A. Rubel, there are no unbounded hermitian operators in $ {H^\infty }$. To each unbounded hermitian operator in the space $ {H^p},1 \leq p < \infty ,p \ne 2$, there corresponds a uniquely determined one-parameter group of conformal maps of the open unit disc onto itself. Such unbounded operators are classified into three mutually exclusive types, an operator's type depending on the nature of the common fixed points of the associated group of conformal maps. The hermitian operators falling into the classification termed ``type (i)'' have compact resolvent function and one-dimensional eigenmanifolds which collectively span a dense linear manifold in $ {H^p}$.

References [Enhancements On Off] (What's this?)

  • [1] E. Coddington, An introduction to ordinary differential equations, Prentice-Hall, Englewood Cliffs, N. J., 1961. MR 23 #A3869. MR 0126573 (23:A3869)
  • [2] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [3] P. Duren, Theory of $ {H^p}$ spaces, Pure and Appl. Math., vol. 38, Academic Press, New York, 1970. MR 42 #3552. MR 0268655 (42:3552)
  • [4] F. Forelli, The isometries of $ {H^p}$, Canad. J. Math. 16 (1964), 721-728. MR 29 #6336. MR 0169081 (29:6336)
  • [5] M. Heins, Complex function theory, Pure and Appl. Math., vol. 28, Academic Press, New York, 1968. MR 39 #413. MR 0239054 (39:413)
  • [6] S. Helgason, Differential geometry and symmetric spaces, Pure and Appl. Math., vol. 12, Academic Press, New York, 1962. MR 26 #2986. MR 0145455 (26:2986)
  • [7] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR 19, 664. MR 0089373 (19:664d)
  • [8] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR 24 #A2844. MR 0133008 (24:A2844)
  • [9] K. de Leeuw, W. Rudin and J. Wermer, The isometries of some function spaces, Proc. Amer. Math. Soc. 11 (1960), 694-698. MR 22 #12380. MR 0121646 (22:12380)
  • [10] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29-43. MR 24 #A2860. MR 0133024 (24:A2860)
  • [11] G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679-698. MR 24 #A2248. MR 0132403 (24:A2248)
  • [12] A. Shields, On fixed points of commuting analytic functions, Proc. Amer. Math. Soc. 15 (1964), 703-706. MR 29 #2790. MR 0165508 (29:2790)
  • [13] I. Vidav, Eine metrische Kennzeichnung der selbstadjungierten Operatoren, Math. Z. 66 (1956), 121-128. MR 18, 912. MR 0084733 (18:912a)
  • [14] A. Zygmund, Trigonometric series. Vol. I, 2nd rev. ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498. MR 0107776 (21:6498)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47D10, 47B37

Retrieve articles in all journals with MSC: 47D10, 47B37

Additional Information

Keywords: Hermitian operator, isometry, $ {H^p}$, Möbius transformation, one-parameter group
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society