Spectral Theory

Prerequisites: Measures and Integration and Functional Analysis

Bounded and unbounded linear operators. Integral operators, multiplication operators and differential operators. The extension theorem for bounded operators. The Fourier transform in L1(Rn), S(Rn) and L2(Rn). L. Schwartz’ distributions, tempered distributions and compact support distributions. Sobolev Spaces Hs(Rn). Applications to linear and nonlinear evolution equations. Closed, closable, symmetric and self-adjoint operators. Resolvent and spectrum. Cayley’s transform. Differentiation of measures. Hahn’s decomposition. The Radon-Nikodyn decomposition theorem. The Riemann-Stieltjes and Lebesgue-Stieltjes integrals. The Spectral theorem for self-adjoint operators in spectral integral form, multiplication operator form, and functional calculus form. Stone’s theorem.

References:
HILLE, E. – Methods in Classical and Functional Analysis. Reading, Mass., Addison-Wesley Pub. Co., 1972.
KOLMOGOROV, A. N., FOMIN, S. V. – Introductory Real Analysis, Dover Publ., Inc. Translated from the seconde russian edition, 1970.
REED. M., BARRY, S. – Methods of Modern Mathematical Physics vols. I e II. New York : Academic Press, 1972-1978.
RIESZ, F., SZ. -NAGY, B. – Functional Analysis, Frederick Ungar Publ.Co. Translated from the second french edition, 1955.
RUDIN, W. – Real and Complex Analysis. New York, McGraw-Hill, 1966.
STONE, M. – Linear Transformations in Hilbert Space and their Applications to Analysis, Amer. Math. Soc. Colloq. Publ., vol. 15,  1932.
THAYER, J. – Operadores Auto-adjuntos e Equações Diferenciais Parciais. Rio de Janeiro, Projeto Euclides, IMPA, 1987.