Functional Analysis And Operator Theory

Abstract

This thesis is mainly about The Spectral Theorem for Normal Operators, Hyponormal Operators, Berger-Shaw Theorem, and an important corollary of it, Putnam's Inequality. We do have spectral theorem for Compact Normal Operators, the poof of which is not hard. But to relax the requirement of compactness and still get a similar result for a Normal Operator is quite a challenge. But it can be done and the countable sum in the compact case is replaced by an integral. Hyponormal operators share a remarkable number of properties with normal operators. But what is even more remarkable is that for hyponormal operators which are multicyclic we have the Berger-Shaw theorem. If an Operator is purely hyponormal, meaning it is hyponormal and the only reducing subspace of it where it is normal is the trivial space, then the real and the imaginary part of it is absolutely continuous w.r.t the Lebesgue Measure on its spectrum. The Putnam inequality tells us that the norm of the commutator of a Hyponormal Operator is bounded by the two-dimensional Lebesgue Measure of the spectrum of the operator.