eng Science and Education Publishing American Journal of Mathematical Analysis 2333-8431 2022-11-08 10 1 1 2 10.12691/ajma-10-1-1 AJMA20221011 article Bahman Mashood b_mashood@hotmail.com 1 66 Shipely Avenue, Daly City, CA 94015, US Let H be a separable Hilbert space and let B(H) be the set of all bounded operators acting on H. Given T¡Ê B(H), we show that T has a proper invariant subspace, i.e., there exists a proper Hilbert subspace LH such that T(L)L. This problem has only been solved for special cases so far and in this article we try to offer a solution that can take care of the most general cases. http://pubs.sciepub.com/ajma/10/1/1/ajma-10-1-1.pdf Hilbert space bounded operator invariant subspace polynomial orthogonal basis