Operator representations of frames

The purpose of this paper is to consider representations of frames {f
_k
}
_k∈I
in a Hilbert space ℋ of the form {f
_k
}
_k∈I
= {T
^k
f
₀
}
_k∈I
for a linear operator T; here the index set I is either ℤ or ℒ
₀
. While a representation of this form is available under weak conditions on the frame, the analysis of the properties of the operator T requires more work. For example it is a delicate issue to obtain a representation with a bounded operator, and the availability of such a representation not only depends on the frame considered as a set, but also on the chosen indexing. Using results from operator theory we show that by embedding the Hilbert space ℋ into a larger Hilbert space, we can always represent a frame via iterations of a bounded operator, composed with the orthogonal projection onto ℋ. The paper closes with a discussion of an open problem concerning representations of Gabor frames via iterations of a bounded operator.