Bounded linear operators on a Hilbert space: 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. (Chapter 2 – Section: 2.1)

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. (Chapter 2 – Sections: 2.2 and 2.3)

Two kinds of classification of spectrum – Spectral mapping theorem Numerical range of an operator: Numericla 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. (Chapter 2 – Sections : 2.4 and 2.5)

Paranormal operators.Characterizations of convexoid operators: Some examples related to hyponormal, paranormal, normaloid and convexoid operators – Relations among several non-normal operators (Chapter2 Sections:2.6 and 2.7)

Young inequality and Holder- McCarthy inequality – Aluthge transformation on p_hyponormal operators and log - hypo normal operators. Chapter 3 – Sections 3.1 to 3.4)