We prove the existence of tight frames whose elements
lie on an arbitrary ellipsoidal surface within a real or complex
separable Hilbert space $\mathcal{H}\, $, and we analyze the set of
attainable frame bounds. In the case where $\mathcal{H}\,$ is real
and has finite dimension, we give an algorithmic proof. Our main tool
in the infinite dimensional case is a result we have proven which
concerns the decomposition of a positive invertible operator into a
strongly converging sum of (not necessarily mutually orthogonal)
self-adjoint projections. This decomposition result implies the
existence of tight frames in the ellipsoidal surface determined by
the positive operator. In the real or complex finite dimensional
case, this provides an alternate (but not algorithmic) proof that
every such surface contains tight frames with every prescribed length
at least as large as $\dim\mathcal{H}\, $. A corollary in both finite
and infinite dimensions is that every positive invertible operator is
the frame operator for a spherical frame.