Ondřej Kalenda, KMA MFF UK

Faculty of Mathematics and Physics

Functional Analysis 2

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Content of the course, expected knowledge
and connections to other courses

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Content of the lectures and classes

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Lecture notes to the course Functional Analysis 2

Summer semester 2023/2023

Lecture notes to the preceeding courses

Introduction to Functional Analysis (2022/2023) (only in Czech)

Functional Analysis 1 (2023/2024)

X. Banach algebras and Gelfand transform

X.1 Banach algebras - basic notions and properties -

A proof of Proposition X.2

A proof of Lemma X.6

A proof of Theorem X.7

X.2 Spectrum and its properties -

A proof of Remark (3)

A proof of Proposition X.8

Proofs of Theorem X.9, the following remark and Theorem X.10

Proofs of Lemmata X.11 and X.12

A proof of Theorem X.13

A proof of Proposition X.15

X.3 Holomorphic functional calculus -

Proofs of Proposition X.17 and the following remarks

Holomorphic calculus, proofs of Thm X.18 and the following remarks

X.4 Ideals, complex homomorphisms
and Gelfand transform

- Czech, English

A proof of Proposition X.22

A proof of Proposition X.23

A proof of Proposition X.24

A proof of Theorem X.25

Problems to Chapter X -

XI. C*-algebras and continuous functional calculus

XI.1 Algebras with involution and C*-algebras -

A proof of Proposition XI.5

XI.2 *-homomorphisms and Gelfand transform -

A proof of Example XI.7

A proof of Proposition XI.8

Proofs of Theorem XI.9 and Corollary XI.10

A proof of Corollary XI.11

XI.3 Continuous functional calculus for C*-algebras -

A proof of Proposition XI.12

A proof of Theorem XI.13

A proof of Theorem XI.14

A proof of Theorem XI.15

XI.4 Distinguished elements of C*-algebras -

A proof of Proposition XI.18

A proof of Proposition XI.20

A proof of Proposition XI.21(a,c)

Problems to Chapter XI -

XII. Bounded and unbounded operators on a Hilbert space

XII.1 More on bounded operators and their spectra -

A proof of Proposition XII.1

Proofs of Lemma XII.2 and Proposition XII.3

A proof of Proposition XII.4

A proof of Proposition XII.5

A proof of Theorem XII.6

XII.2 Unbounded operators between Banach spaces -

Several counterexamples

Explanation of the final remark

XII.3 Spectrum of an unbounded operator -

On variants of the definition of the resolvent set

Proofs of Proposition XII.14 and Lemma XII.15

XII.4 Operators on a Hilbert space -

Proofs of Proposition XII.18 - Proposition XII.21

Proofs of Lemma XII.22 and Proposition XII.23

Proofs of Lemma XII.24 - Corollary XII.26

XII.5 Symmetric operators and Cayley transform -

A proof of Theorem XII.27

Proofs of Lemma XII.28-Theorem XII.30

Remarks and questions on deficiency indices

Problems to Chapter XII -

Several examples of differential operators

Construction of self-adjoint Laplace operators

XIII. Spectral measures and spectral decompositions 55

XIII.1 Measurable calculus
for bounded normal operators

- Czech, English

Construction of meas. calc. and spectral measure, to Lemma XIII.3

Proof of Theorem XIII.4

XIII.2 Integral with respect to a spectral measure -

A proof of Lemma XIII.5

A proof of Lemma XIII.6

A proof of Theorem XIII.8

Proofs of Lemma XIII.9 and Corollary XIII.10

A proof of Theorem XIII.11

A proof of Theorem XIII.12

A proof of Proposition XIII.13

XIII.3 Spectral decomposition of a selfadj. operator -

Proofs of Lemma XIII.14-Corollary XIII.18

An analysis of operators of multiplication

Sections XIII.4 and XIII.5 were not addressed during the lectures (it was not planned to deal with them in detail), they are included here for completeness and for interested students. The relevant proofs may be found at the lecture notes to the course Functional Analysis 2 in 2021/2022 here (Sections VI.4 and VI.5).

XIII.4 Unbounded normal operators

- Czech, English

XIII.5 Complements to unbounded operators -

Problems to Chapter XIII -