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In this talk, we discuss the classical measures of noncompactness and in particular Hausdorff measure of noncompactness. We discuss their applications in various problems of analysis such as differential equations, integral equations and characterization of compact matrix operators between sequence spaces. The most effective way in the characterization of compact operators between the Banach spaces is applying the Hausdorff measure of noncompactness. In this talk, we present some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators given by infinite matrices that map an arbitrary BK-space into classicla sequence spaces. Many linear compact operators may be represented as matrix operators in sequence spaces or integral operators in function spaces [2]. Recently the measures of noncompactness are applied in solving infinite system of differential equations [5], [4] and integral equations in sequence spaces [1, 3]. References: [1] R. Arab, M. Mursaleen and S.M.H. Rizvi, Positive solution of a quadratic integral equation using generalization of Darbo’s fixed point theorem, Numer. Funct. Anal. Optim., 40(10) (2019), 1150–1168. [2] J. Banas and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, 2014. [3] B. Hazarika, R. Arab and M. Mursaleen, Applications of measure of noncompactness and operator type contraction for existence of solution of functional integral equations, Complex Anal. Oper. Theory, DOI: 10.1007/s11785-019-00933-y. - 2 - [4] M. Mursaleen, E. Porhadi and R. Saadati, Solvability of infinite systems of secondorder differential equations with boundary conditions in ℓp, Quaest. Math., DOI: 10.2989/16073606.2019.1617800. [5] M. Mursaleen and S.M.H. Rizvi, Solvability of infinite system of second order differential equations in c0 and ℓ1 by Meir-Keeler condensing operator, Proc. Amer. Math. Soc., 144(10) (2016) 4279–4289.