NTMSCI 6, No. 4, 8-15 (2018) 8 New Trends in Mathematical Sciences http://dx.doi.org/10.20852/ntmsci.2018.310 A new class of operator ideals and approximation numbers Pınar Zengin Alp and Emrah Evren Kara Department of Mathematics, Faculty of Science and Arts, D¨ uzce University, D¨ uzce, Turkey Received: 25 July 2017, Accepted: 28 September 2018 Published online: 4 October 2018. Abstract: In this study, we introduce the class of generalized Stolz mappings by generalized approximation numbers . Also we prove that the class of ℓ α p −type mappings are included in the class of generalized Stolz mappings by generalized approximation numbers and we define a new quasinorm equivalent with ‖T ‖ α φ ( p) . Further we give a new class of operator ideals by using generalized approximation numbers and symmetric norming function and we show that this class is an operator ideal. Keywords: Operator ideal, s-numbers, symmetric norming function. 1 Introduction The operator ideal theory has a special importance in functional analysis. One of the most important methods to construct operator ideals is via s− numbers. Pietsch defined the approximation numbers of a bounded linear operator in Banach spaces, in 1963 [13]. Later on, the other examples of s−numbers, namely Kolmogorov numbers, Weyl numbers, etc. are introduced to the Banach space setting. In this paper, by N and R + we denote the set of all natural numbers and non-negative real numbers, respectively. A bounded linear operator whose dimension of the range space is finite is called a finite rank operator [9]. Let E and F be real or complex Banach spaces and L (E , F ) denotes the space of all bounded linear operators from E to F and L denotes the space of all bounded linear operators between any two arbitrary Banach spaces. A map s =(s n ) : L → R + assigning to every operator T ∈ L a non-negative scalar sequence (s n (T )) n∈N is called an s−number sequence if the following conditions are satisfied for all Banach spaces E , F, E 0 and F 0 : (S1) ‖T ‖ = s 1 (T ) ≥ s 2 (T ) ≥ ... ≥ 0 for every T ∈ L (E , F ) , (S2) s m+n−1 (S + T ) ≤ s m (S)+ s n (T ) for every S, T ∈ L (E , F ) and m, n ∈ N, (S3) s n (RST ) ≤‖R‖ s n (S) ‖T ‖ for some R ∈ L (F, F 0 ) , S ∈ L (E , F ) and T ∈ L (E 0 , E ) , where E 0 , F 0 are arbitrary Banach spaces, c  2018 BISKA Bilisim Technology ∗ Corresponding author e-mail: karaeevren@gmail.com