Convolution operators preserving the set of totally positive sequences

A real sequence $(a_k{)}_{k=0}^{\mathrm{\infty }}$ is called {\it totally positive} if all minors of the infinite Toeplitz matrix ${\parallel a_{j-i}\parallel }_{i,j=0}^{\mathrm{\infty }}$ are nonnegative (here $a_k=0$ for $k<0$). In this paper, which continues our earlier work \cite{kv}, we investigate the set of real sequences $(b_k{)}_{k=0}^{\mathrm{\infty }}$ with the property that for every totally positive sequence $(a_k{)}_{k=0}^{\mathrm{\infty }},$ the sequense of termwise products $(a_k{b}_k{)}_{k=0}^{\mathrm{\infty }}$ is also totally positive. In particular, we show that for every totally positive sequence $(a_k{)}_{k=0}^{\mathrm{\infty }}$ the sequence ${\left(a_k{a}^{-k(k-1)}\right)}_{k=0}^{\mathrm{\infty }}$ is totally positive whenever $a^2\ge \mathrm{3.503.}$ We also propose several open problems concerning convolution operators that preserve total positivity.