Operator theory -brief plan of the course (2013)
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1. Brief review of basic fact and terminology related to complete normed spaces and linear functionals.

2. Finite, infinite and block matrices as linear operators. Schur test for boundedness. The adjoint and the hermitian adjoint of an operator.

3. Finite rank and compact linear operators. Examples of integral operators. Operators with compact resolvent. Application to some differential equations.

4. Hilbert-Schmidt operators. Examples of integral operators.

5. Spectral representation of compact selfadjoint operators in Hilbert spaces.

6. Applications of spectral Thorem for compact operators. Polar decomposition.

7. Fredholm alternative, application to integral equations.

8. Sesquilinear forms corresponding to Hilbert space operators. Lax-Milgram theorem.

9. Numerical range. Comparing the numerical and spectral radii of an operator

10. General Spectral Thorem for normal operators.

11. Examples of spectral measures and applications of the Spectral Theorem.

12. Elementary properties of closed unbounded operators. Examples of differential operators. Cayley transform of symmetric operators.

13. Spectral theorem for unbounded operators.

14. Spectra of functions of operators.

15 Some recent results and open problems in operator theory.

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tutorials on operator theory

Tutorials will discuss concrete examples and applications. The aim is to deepen understanding of the developed theory and to encourage students to try “hands on” approach to the problems.

Bibliography 
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1. N.I.Ahiezer. I.M. Glazman, Theory of Lienar Operators In Hilbert Spaces, Ungar, N.Y., 1961
2. J. B. Conway, Course in functional analysis, Springer-Verlag, New York, 1985.
3. G. K. Lax, Functional Analysis, Warszawa, Wiley-Interscience, 2002.
4. G. Pedersen, Analysis Now, Springer-Verlag, N.Y. 1989.
5. W. Rudin, Functional analysis, McGraw-Hill, 1973. 
6. Notes on the history of operator theory by Evens M.Harrell II:
 http://www.mathphysics.com/opthy/OpHistory.html