Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 3 1 2022 01 01 Dichotomy between operators acting on finite and infinite dimensional Hilbert spaces 33 41 171895 10.30504/jims.2023.392498.1104 EN Luis Bernal González Departamento de Análisis Matem\ático, Facultad de Matemáticas, Instituto de Matemáticas Antonio de Castro Brzezicki, Universidad de Sevilla, Avenida Reina Mercedes, Sevilla, 41080, Spain. M. S. Moslehian Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran. 0000-0001-7905-528X J. B. Seoane Sepúlveda Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid, 28040, Spain. Journal Article 2023 04 10 In this expository article, we give several examples showing how drastically different can be the behavior of operators acting on finite versus infinite dimensional Hilbert spaces. This essay is written as in such a friendly-reader to show that the situation in the infinite dimensional setting is trickier than the finite one. Hilbert space Matrix operator difference between finite and infinite dimension https://jims.ims.ir/article_171895_244ff9a14073747fc46bcc414a3bbb36.pdf