$A$-Norm and $A$-numerical Radius Inequalities for Sums Of Operators in semi-Hilbertian spaces

Abstract: The aim of this study is to establish many new inequalities for the operator $A$-norm and $A$-numerical radius of sums of bounded linear operators in Hilbert spaces. In particular, two refinements are made to the generalized triangle inequality for operator norm. Additionally, we examine a number of intriguing uses for two bounded linear operators in the Cartesian decomposition of an operator. These disparities improve and generalize several earlier results in the literature. Some applications of our results are given. We also present some instances to support our results.