A representation theorem for Schauder bases in Hilbert space

Abstract

A sequence of vectors { f 1 , f 2 , f 3 , … } \{f_1,f_2,f_3,\dotsc \} in a separable Hilbert space H H is said to be a Schauder basis for H H if every element f ∈ H f\in H has a unique norm-convergent expansion \[ f = ∑ c n f n . f=\sum c_nf_n. \] If, in addition, there exist positive constants A A and B B such that \[ A ∑ | c n | 2 ≤ ‖ ∑ c n f n ‖ 2 ≤ B ∑ | c n | 2 , A\sum |c_n|^2\le \left \|\sum c_nf_n\right \|^2\le B\sum |c_n|^2, \] then we call { f 1 , f 2 , f 3 , … } \{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 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.