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A representation theorem for Schauder bases in Hilbert space

Author(s): Stephane Jaffard; Robert M. Young
Journal: Proc. Amer. Math. Soc. 126 (1998), 553-560.
MSC (1991): Primary 46B15; Secondary 47A55
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Abstract: A sequence of vectors $\{f_1,f_2,f_3,\dotsc\}$ in a separable Hilbert space $H$ is said to be a Schauder basis for $H$ if every element $f\in H$ has a unique norm-convergent expansion

$\begin{displaymath}f=\sum c_nf_n.\end{displaymath}$

If, in addition, there exist positive constants $A$ and $B$ such that

$\begin{displaymath}A\sum|c_n|^2\le\left\|\sum c_nf_n\right\|^2\le B\sum|c_n|^2,\end{displaymath}$

then we call $\{f_1,f_2,f_3,\dotsc\}$ a Riesz basis. In the first half of this paper, we show that every Schauder basis for $H$ 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.

References:

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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