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.
Reference Book:
Reference Book HILBERT SPACE PROBLEM BOOK by P.R. HALMOS, Springer Verlag, New York
Text Book:
INVITATION TO LINEAR OPERATORS by TAKAYUKI FURUTA, Taylor and Francis, 2001. Unit I: Chapter 2: Sections: 2.1. Unit II: Chapter 2: Sections: 2.2 and 2.3. Unit III: Chapter 2: Sections: 2.4 and 2.5. Unit IV: Chapter2: Sections: 2.6 and 2.7. Unit V: Chapter 3: Sections 3.1 to 3.4.