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

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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**

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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