The paper deals with the Gau\ss --Weierstra\ss\ operators $W_{n}$ and their left quasi interpolants $W_{n}^{\left[ r\right] }$. The quasi interpolants were defined by Paul Sablonni\`{e}re in 2014. Recently, their asymptotic behaviour was studied by Octavian Agratini, Radu P\u{a}lt\u{a}nea and the author by presenting complete asymptotic expansions. In this paper we derive estimates for the operator norms of $W_{n}$ and $W_{n}^{\left[ r\right] }$ when acting on various function spaces.