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tma4230:2019v:lectures_log [2019-03-25]
yura
tma4230:2019v:lectures_log [2019-04-03] (nåværende versjon)
yura
Linje 43: Linje 43:
   * **Lecture 22. 21.03** Spectrum of compact operators. Invariant subspaces. Hilbert-Schmidt operators. \\    * **Lecture 22. 21.03** Spectrum of compact operators. Invariant subspaces. Hilbert-Schmidt operators. \\ 
        
-  * **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)+  * **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)\\ 
 +   
 +  * **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.\\ 
 +    
 +  * **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 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. 
 +   
 +  * **Lecture 27. 04.04** Summary. ​
2019-04-03, Yurii Lyubarskii