Simultaneous Preservation of Orthogonality of Polynomials by Linear Operators Arising from Dilation of Orthogonal Polynomial Systems

For an orthogonal polynomial system \(p = \left( {p_n } \right)_{n \in N_0 } \)and a sequence \(d = \left( {d_n } \right)_{n \in N} 0\) of nonzero numbers,let \(S_{p,d} \) be the linear operator defined on the linear spaceof all polynomials via \(S_{p,d} p_n = d_n p_n \)for all \(n \in N_0 \).We investigate conditions on \(p\)and \(d\) under which\(S_{p,d} \) can simultaneously preserve the orthogonality ofdifferent polynomial systems. As an application, we get that for\(p = \left( {L_n^\alpha } \right)\), a generalized Laguerre polynomial system, no\(d\) can simultaneously preserve the orthogonality of twoadditional Laguerre systems, \(\left( {L_n^{\alpha + t_1 } } \right)\) and\(\left( {L_n^{\alpha + t_2 } } \right)\), where \(t_1 ,t_2 \ne 0\)and \(t_1 \ne t_2 \). On the other hand, for \(p = \left( {T_n } \right)\),the Chebyshev polynomial system and \(d = \left( {\left( { - 1} \right)^n } \right)\),\(S_{p,d} \) simultaneously preserves the orthogonality of uncountablymany kernel polynomial systems associated with p. We study manyother examples of this type.