Invitation to Linear Operators

Takayuki Furuta

HILBERT SPACES: Inner Product Spaces and Hilbert Spaces. Jordan-Neuman Theorem. Orthogonal Decomposition of Hilbert Space. Gram-Schmidt Orthonormal Procedure and its Applications. FUNDAMENTAL PROPERTIES OF BOUNDED LINEAR OPERATIONS: Bounded Linear Operations on Hilbert Space. Partial Isometry Operator and Polar Decomposition of an Operator. Polar Decomposition of an Operator and its Applications. Spectrum of an Operator. Numerical Range of an Operator. Relations Among Several Classes of Non-normal Operators. Charactorizations of Convexoid Operators and Related Examples. FURTHER DEVELOPMENT OF BOUNDED LINEAR OPERATORS: Young Imequality and Holder-McCarthy Inequality. Lowner-Heinz Inequality and Furuta Inequality. Chaotic Order and the Relative Operator Entropy. Aluthge Transformation on P-Hyponormal Operators and Log-Hyponormal Operators. A Subclass of Paranormal Operators Including Loh-Hyponormal Operators and Several Related Classes. Operator Inequalities Associated With Kantorovich Inequality and Holder-McCarthy Inequality. Some Properties on Partial Isometry, Quasinormality and Paranormality. Weighted Mixed Schwarz Inequality and Generalized Schwarz Inequality. Selberg Inequality. An Extension of Heinz-Kato Inequality. Norm Inequalities Equivalent to Lower-Heinz Inequality. Norm Inequalities Equivalent to Heinz Inequality. Bibliography. Index.