• Nour Elimane, TRIKI2022-07-032022-07-032022universite of el ouedhttps://dspace.univ-eloued.dz/handle/123456789/12039mémoire master mathématiqueOur main goal in this thesis, is to improve or find some important inequalities related to the usual operator norm and the numerical radius by using the Ky−F an norm defined on K(H) space of compact linear operators on the complex Hilbert space, we study the most important properties of this norm, trying to get satisfactory results.Notre objectif principal dans cette m ́emoire, est d’am ́eliorer ou de trouver certaines in ́egalit ́es importantes li ́ees `a la norme de l’op ́erateur usuel et au rayon num ́erique en utilisant la norme Ky −F an. d ́efinie sur l’espace K(H) des op ́erateurs lin ́eaires compacts sur l’espace de Hilbert complexe, nous ́etudier les propri ́et ́es les plus importantes de cette norme, en essayant d’obtenir des r ́esultats satisfaisants.Our main goal in this thesis, is to improve or find some important inequalities related to the usual operator norm and the numerical radius by using the Ky−F an norm defined on K(H) space of compact linear operators on the complex Hilbert space, we study the most important properties of this norm, trying to get satisfactory results.frComplex Hilbert space, bounded linear operators, compact operators, spec- trum, numerical range, usual operator norm, spectral radius, numerical radius.Espace de Hilbert complexe, op ́erateurs lin ́eaires born ́es, op ́erateurs com- pacts, spectre, domaine num ́erique, norme usuelle des op ́erateurs, rayon spectral, rayon num ́erique, norme de Ky − F an.Master