Stud. Univ. Babe¸ s-Bolyai Math. 64(2019), No. 2, 225–237 DOI: 10.24193/subbmath.2019.2.08 Operator norms of Gauß-Weierstraß operators and their left quasi interpolants Ulrich Abel Abstract. The paper deals with the Gauß–Weierstraß operators Wn and their left quasi interpolants W [r] n . The quasi interpolants were defined by Paul Sablonni` ere in 2014. Recently, their asymptotic behaviour was studied by Octavian Agratini, Radu P˘ alt˘ anea and the author by presenting complete asymptotic expansions. In this paper we derive estimates for the operator norms of Wn and W [r] n when acting on various function spaces. Mathematics Subject Classification (2010): 41A36, 41A45, 47A30. Keywords: Approximation by positive operators, operator norm. 1. Introduction For 1 ≤ p ≤ +∞ and c> 0, let L p c (R) denote the space of all locally integrable functions f : R → R, such that the weighted norm ‖fw c ‖ L p (R) ‖f ‖ L p c (R) :=  ∞ −∞ |f (t)| p w c (t) dt  1/p (1 ≤ p< +∞) , ‖f ‖ L ∞ c (R) := ess sup t∈R |f (t)| w c (t) (p =+∞) is finite, where the weight function w c is given by w c (t) := e −ct 2 . In the particular case c = 0, we obtain the ordinary spaces L p 0 (R)= L p (R) and L ∞ 0 (R)= L ∞ (R), respectively. This paper has been presented at the fourth edition of the International Conference on Numerical Analysis and Approximation Theory (NAAT 2018), Cluj-Napoca, Romania, September 6-9, 2018.