Operators preserving orthogonality of polynomials
Let S be a degree preserving linear operator of ℝ[X] into itself. The question is if, preserving orthogonality of some orthogonal polynomial sequences, S must necessarily be an operator of composition with some affine function of ℝ. In  this problem was considered for S mapping sequences of Laguerre polynomials onto sequences of orthogonal polynomials. Here we improve substantially the theorems of  as well as disprove the conjecture proposed there. We also consider the same questions for polynomials orthogonal on the unit circle.