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

MathSciNet review:
1425127

Full-text PDF Free Access

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**0007173**, 10.1090/S0002-9904-1942-07797-4**2.**S. V. Hruščëv, N. K. Nikol′skiĭ, and B. S. Pavlov,*Unconditional bases of exponentials and of reproducing kernels*, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 214–335. MR**643384****3.**Frigyes Riesz and Béla Sz.-Nagy,*Functional analysis*, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR**0071727****4.**Ivan Singer,*Bases in Banach spaces. I*, Springer-Verlag, New York-Berlin, 1970. Die Grundlehren der mathematischen Wissenschaften, Band 154. MR**0298399****5.**Robert M. Young,*An introduction to nonharmonic Fourier series*, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**591684**

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:
http://dx.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