  Orthogonal polynomials $P_{n}(\lambda)$ are oscillating functions of $n$ as
$n\to\infty$ for $\lambda$ in the absolutely continuous spectrum of the
corresponding Jacobi operator $J$. We show that, irrespective of any specific
assumptions on coefficients of the operator $J$, amplitude and phase factors in
asymptotic formulas for $P_{n}(\lambda)$ are linked by certain universal
relations found in the paper.
  Our approach relies on a study of operators diagonalizing Jacobi operators.
Diagonalizing operators are constructed in terms of orthogonal polynomials
$P_{n}(\lambda)$. They act from the space $L^2 (\Bbb R)$ of functions into the
space $\ell^2 ({\Bbb Z}_{+})$ of sequences. We consider such operators in a
rather general setting and find necessary and sufficient conditions of their
boundedness.
