Norm of bounded linear operators-Ad joint Operators-Generalized polarization identity and its applications- Several Properties on projection operators- Generalized Schwarz inequality and square root of positive operator-spectral representations of self ad joint operator.

Partial isometry operator and its characterization Polar decomposition of an operator: Invariant subspace and reducing subspace-Polar decomposition of non-normal operator – Hereditary property on the polar decomposition of an operator.

Two kinds of classification of spectrum – Spectral mapping theorem Numerical range of an operator: Numerical range is a convex set – Numerical radius is equivalent to operator norm – The closure of numerical range includes the spectrum – Normaloid operator and spectraloid operator.

Paranormal operators. Characterizations of convexoid operators: Some examples related to hyponormal, paranormal, normaloid and convexoid operators – Relations among several non-normal operators.

Young inequality and Holder- McCarthy inequality – Aluthge transformation on phyponormal operators and log - hypo normal operators.