Linear Operators And Spectral Theory

Linear Operators and Spectral Theory Applied Mathematics Seminar - V.I. Math 488, Section 1, WS2003 Regular Participants: V. Batchenko V. Borovyk R. Cascaval D. Cramer F. Gesztesy O. Mesta K. Shin M. Zinchenko Additional Participants: C. Ahlbrandt Y. Latushkin K. Makarov Coordinated by F. Gesztesy 1 Contents V. Borovyk: Topics in the Theory of Linear Operators in Hilbert Spaces O. Mesta: Von Neumann’s Theory of Self-Adjoint Extensions of Symmetric Operators and some of its Refinements due to Friedrichs and Krein D. Cramer: Trace Ideals and (Modified) Fredholm Determinants F. Gesztesy and K. A. Makarov: (Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited M. Zinchenko: Spectral and Inverse Spectral Theory of Second-Order Difference (Jacobi) Operators on N and on Z K. Shin: Floquet and Spectral Theory for Second-Order Periodic Differential Equations K. Shin: On Half-Line Spectra for a Class of Non-Self-Adjoint Hill Operators V. Batchenko and F. Gesztesy: On the Spectrum of Quasi-Periodic Algebro-Geometric KdV Potentials 2 Topics in the Theory of Linear Operators in Hilbert Spaces Vita Borovyk Math 488, Section 1 Applied Math Seminar - V.I., WS 2003 February, 2003 - The spectral theorem for bounded and unbounded self-adjoint operators - Characterizations of the spectrum , point spectrum, essential spectrum, and discrete spectrum of a self-adjoint operator - Stone’s theorem for unitary groups - Singular values of compact operators, trace class and Hilbert–Schmidt operators 1 1 Preliminaries For simplicity we will always assume that the Hilbert spaces considered in this manuscript are separable and complex (although most results extend to nonseparable complex Hilbert spaces). Let H1 , H2 be separable Hilbert spaces and A be a linear operator A : D(A) ⊂ H1 → H2 . We denote by B(H1 ,