Every Frame Is a Sum of Three (but Not Two) Orthonormal Bases - and Other Frame Representations

We show that every frame for a Hilbert space H can be written as a (multiple of a) sum of three orthonormal bases for H. We next show that this result is best possible by including a result of N.J. Kalton: A frame can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis. We further show that every frame can be written as a (multiple of a) sum of two tight frames with frame bounds one or a sum of an orthonormal basis and a Riesz basis for H. Finally, every frame can be written as a (multiple of a) average of two orthonormal bases for a larger Hilbert space. 1.Frames as Operators If H is a Hilbert space, we denote the set of all bounded operators T : H → H by B(H). We will always use (en) to denote an orthonormal basis on H. Recall that a sequence (xn) in a Hilbert space H is called a frame for H if there are constants 0 < A ≤ B so that for all x ∈ H we have A‖x‖ ≤ ∑ n | < x, xn > | ≤ B‖x‖. We call A, B the frame bounds for the frame and if A = B, we call this a tight frame. The frame definition has many equivalent forms. We will work here with frames thought of as operators on H. That is, a sequence (xn) is a frame on H if and only if there is an operator T : H → H so that Ten = xn and T is an onto 1991 Mathematics Subject Classification. 46C05, 47A05, 47B65.