Frames Associated with C₀-Semigroups

A central question in dynamical frame theory asks when the orbit \Λⁿ f₊ of a vector f under a bounded operator $Λ$ on a separable infinite-dimensional Hilbert space H is a frame. Christensen, Hasannasab and Philipp answered it, showing in particular that such an operator must be similar to the compression of the unilateral shift on H²(T) to a coinvariant subspace. We prove a continuous analogue: if a C₀-semigroup (t)≥0⊂ B(H) generates a continuous frame (t)φ≥0, then the operators $T(t)$ are simultaneously similar to the compressions of the shift semigroup to a fixed coinvariant subspace. Moreover, (t)φ≥0 is a continuous frame if and only if ⁿ(I-A)⁻¹φ₊ is a frame, where A is the infinitesimal generator and C=(I+A)(I-A)⁻¹ the cogenerator; the frame bounds of the two systems differ by a factor $2$. We also derive necessary conditions for the existence of frame vectors and discuss several examples. Finally, we show that the compressed shift on a model space K_θ is the cogenerator of a C₀-semigroup admitting frame vectors if and only if the inner function $θ$ has a finite angular derivative at $1$; in that case we describe all frame vectors, and the boundary reproducing kernel at $1$ generates a tight continuous frame.