A representation theorem for Schauder bases
in Hilbert space
Authors:
Stephane Jaffard and Robert M. Young
Journal:
Proc. Amer. Math. Soc. 126 (1998), 553-560
MSC (1991):
Primary 46B15; Secondary 47A55
DOI:
https://doi.org/10.1090/S0002-9939-98-04168-9
MathSciNet review:
1425127
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: A sequence of vectors in a separable Hilbert space
is said to be a Schauder basis for
if every element
has a unique norm-convergent expansion
If, in addition, there exist positive constants and
such that
then we call a Riesz basis. In the first half of this paper, we show that every Schauder basis for
can be obtained from an orthonormal basis by means of a (possibly unbounded) one-to-one positive self adjoint operator. In the second half, we use this result to extend and clarify a remarkable theorem due to Duffin and Eachus characterizing the class of Riesz bases in Hilbert space.
- 1. R. J. Duffin and J. J. Eachus, Some notes on an expansion theorem of Paley and Wiener, Bull. Amer. Math. Soc. 48 (1942), 850-855. MR 4:97e
- 2. S. V. Khrushchev, N. K. Nikolskii and B. S. Pavlov, Unconditional bases of exponentials and reproducing kernels, in ``Complex Analysis and Spectral Theory'' (V. P. Havin and N. K. Nikolskii, editors), pp. 214-335, Lecture Notes in Mathematics, vol. 864, Springer-Verlag, Berlin/Heidelberg (1981). MR 84k:46019
- 3. F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar Publ. Co., New York (1955). MR 17:175i
- 4. I. Singer, Bases in Banach spaces I, Springer-Verlag, New York/Heidelberg (1970). MR 45:7451
- 5. R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York (1980). MR 81m:42027
Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46B15, 47A55
Retrieve articles in all journals with MSC (1991): 46B15, 47A55
Additional Information
Stephane Jaffard
Affiliation:
Centre de Mathématiques et Leurs Applications, 61 Avenue du Président Wilson, 94235 Cachan cedex, France
Robert M. Young
Affiliation:
Department of Mathematics, Oberlin College, Oberlin, Ohio 44074
DOI:
https://doi.org/10.1090/S0002-9939-98-04168-9
Keywords:
Schauder basis,
Riesz basis.
Received by editor(s):
April 17, 1996
Received by editor(s) in revised form:
August 22, 1996
Communicated by:
Dale Alspach
Article copyright:
© Copyright 1998
American Mathematical Society