Frames and generalized operator orbits

Abstract

AbstractThis short note is concerned with various operator representations of frames in a Hilbert space $${{{\mathcal {H}}}}.$$ H . While it is known that only very special frames can be represented in the form $$\{ T^n \varphi \}_{n=0}^\infty $$ { T n φ } n = 0 ∞ for a bounded operator T,  we prove that every frame (actually, every Bessel sequence) has a representation of the form $$\{UT^k \varphi \}_{k=0}^\infty $$ { U T k φ } k = 0 ∞ for certain bounded operators U and T. We also provide a lifting procedure that allows to represent any given Bessel sequence in the form $$\{PT^k \varphi \}_{k=0}^\infty ,$$ { P T k φ } k = 0 ∞ , where T is a bounded operator on an ambient Hilbert space and P denotes the orthogonal projection onto the given Hilbert space $${{{\mathcal {H}}}}.$$ H . In particular, this implies that for any frame, the frame coefficients of $$f\in {{{\mathcal {H}}}}$$ f ∈ H can be calculated as inner products between f and a system of functions of the form $$\{T^k \varphi \}_{k=0}^\infty ,$$ { T k φ } k = 0 ∞ , for a bounded operator T on an ambient Hilbert space.