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Neste revisjon 		Forrige revisjon
tma4230:2019v:lectures_log [2019-03-25]
yura 		tma4230:2019v:lectures_log [2019-04-03]
yura
Linje 38: Linje 38:
  
   * **Lecture 20. 18.03** Definition of spectrum. Examples. Classification of points of spectrum. Resolvent function. Neumann series, Spectrum is bounded. Perturbation of invertible operator. Spectrum is closed.\\   * **Lecture 20. 18.03** Definition of spectrum. Examples. Classification of points of spectrum. Resolvent function. Neumann series, Spectrum is bounded. Perturbation of invertible operator. Spectrum is closed.\\
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 +  * **Lecture 21. 21.03** Analytic operator functions. Reminder: the main theorem of algebra, Liouville theorem. Spectrum is not empty. Spectral radius. Spectrum of  operator of integration.\\
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 +  * **Lecture 22. 21.03** Spectrum of compact operators. Invariant subspaces. Hilbert-Schmidt operators. \\ 
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 +  * **Lecture 23. 25.03** Integral operators as Hilbert-Schmidt operators. Unitary operators, their spectrum properties. Polar form (//reminder//). Selfadjoint operators, definition. Exponent of operator. Positive and negative operators. Polar form (with no proof)\\
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 +  * **Lecture 24. 28.03** Selfadjoint operators: Point spectrum is real. No residual spectrum.  Estimate from below. Continuous spectrum is real.  Spectral interval. Operator norm is defined by special interval. Orthogonality of eigenvectors, corresponding different eigenvalues. Spectral theorem for compact selfadjoint operators. Hint about the general spectral theorem for selfadjoint operators.\\
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 +  * **Lecture 25. 29.03** Functions of selfadjoint operators: polynomials of operators, arithmetic rules, spectral mapping theorem. Reminder: Weierstrass  theorem, Muntz theorem (sketch of the proof). Continuous functions of operator. Convergence, spectral mapping theorem.\\
    
-  * **Lecture . 25.03** Integral operators as Hilbert-Schmidt operators. Unitary operators, their spectrum properties. Polar form (//reminder//). Selfadjoint operators, definition. Exponent of operator. Positive and negative operators. Polar form (with no proof)+  * **Lecture 26. 01.04** Dunford calculus. Functions analytic in a  vicinity of spectrum. Definition of function of operator. Arithmetic rules: addition, multiplication. Spectral mapping theorem. Composition of functions. Adjoint operator. 
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 +  * **Lecture 27. 04.04** Summary. 
2019-04-03, Yurii Lyubarskii