A Representation Theorem for Schauder Bases in Hilbert Space

A sequence of vectors {f1, f2, f3, . . . } in a separable Hilbert space H is said to be a Schauder basis for H if every element f ∈ H has a unique norm-convergent expansion f = ∑ cnfn. If, in addition, there exist positive constants A and B such that A ∑ |cn| ≤ ∥∥∥∑ cnfn∥∥∥2 ≤ B∑ |cn|, then we call {f1, f2, f3, . . . } 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. 1. The main theorem A sequence of vectors {f1, f2, f3, . . . } in a separable Hilbert space H is said to be a Schauder basis for H if every element f ∈ H has a unique norm-convergent expansion f = ∑ cnfn. If, in addition, there exist positive constants A and B such that A ∑ |cn| ≤ ∥∥∥∑ cnfn∥∥∥2 ≤ B∑ |cn|, then we call {f1, f2, f3, . . . } a Riesz basis. Riesz bases have been studied intensively ever since Paley and Wiener first recognized the possibility of nonharmonic Fourier expansions f = ∑ cne iλnt for functions f in L2(−π, π) (see, for example, [2] and [5] and the references therein). In the first half of this paper, we show that every Schauder basis for H can be obtained from some orthonormal basis by means of a (possibly unbounded) one-toone positive self adjoint operator. In the second half, we use this result to extend Received by the editors April 17, 1996 and, in revised form, August 22, 1996. 1991 Mathematics Subject Classification. Primary 46B15; Secondary 47A55.