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Mandatory course for the master study programme Mathematical
analysis. Recommended for the first year of master studies. Continuation
of the course NMMA401. Devoted to advanced topics in functional analysis -
spectral theory in Banach algebras, Gelfand transform, spectral theory of
bounded and unbounded operators.
Last update: Pyrih Pavel, doc. RNDr., CSc. (12.05.2022)
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The rules for 2023/2024:
The course is finished by a credit and an exam. Before passing the exam it is necessary to gain the credit.
The credit will be awarded after complete and correct solution of two homeworks and presenting a correct solution of one problem during the classes.
If the submitted solution of a homework is not complete and correct, a correction should be provided. The number of iterations is not a priori limited.
Detailed rules will be specified at the webpage of the lecturer. Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (05.02.2024)
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Rudin, W.: Functional analysis. Second edition, McGraw-Hill, Inc., New York, 1991 Meise R. and Vogt D. : Introduction to functional analysis, Oxford University Press, New York, 1997
Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (01.02.2024)
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The exam is oral with the possibility of a written preparation. Mainly knowledge and understanding of the notions and theorems explained during the semester will be tested. In addition, solving selected problems using the methods explained during the course will be a part of the exam. The lectures are the main source of materials for the exam. Last update: Cúth Marek, doc. Mgr., Ph.D. (03.02.2023)
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1. Banach algebras
Definition, examples, adding a unit, renorming Invertible elements, Neumann series Spectrum and its properties, spektral radius C*-algebra, self-adjoint and normal elements Holomorphic calkulus 2. Gelfand transform Complex homomorphisms and maximal ideals in commutative Banach algebras Gelfand transform and its properties Applications for commutative C*-algebras - Gelfand-Neimark theorem Applications for non-commutative C*-algebras - continuous funkction calkulus 3. Operators on a Hilbert space Self-adjoint operators, normal operators, positive operators, unitary operators, projections Continuous and measurable calkulus, spectral measure and integral with respect to it, spectral decomposition of a normal operator Polar decomposition, positive and negative part 4. Unbounded operators Unbounded operators on Banach spaces, closed operators, densely defined operators, spectrum Unbounded operators on Hilbert spaces, adjoint operator, symmetric and self-adjoint operators Cayley transform, deficiency indices Integral of an unbounded function with respect to a spectral measure Spectral decomposition of a self-adjoint operator Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (09.05.2022)
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Continuation of the course NMMA401. Devoted to advanced topics in functional analysis. Expected knowledge includes elements of functional analysis (the content of courses NMMA331 and NMMA401), complex analysis (Cauchy theorem, Cauchy formula) and measure and integration.
Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (05.02.2024)
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