Content of the lectures and classes
Lecture 1 - February 19, 2024
Introductory information - a brief content of the course, assumed knowledge.
Beginning of Chapter X (Banach algebras and Gelfand transform) and Section X.1 (Banach algebras - basic notions and properties) - till Proposition X.2.
Example X.1(8) was only briefly mentioned, Example X.1(9) was omitted.
Lecture 2 and Classes 1 - February 23, 2024
Information on the credit and on homeworks. Completing Section X.1 - from the remarks following Proposition X.2 to the end
of the section. Problem 1 to Chapter X.
Lecture 3 - February 26, 2024
Beginning of Section X.2 (Spectrum and its properties) - to Theorem X.13 including the first part of the proof. Remark (3) after the introductory definitions and
Lemma X.11 were not proved, proof are available at the lecture notes.
Lecture 4 and Classes 2 - March 1, 2024
Student presentation - Problems 4 and 6 to Chapter X.
Completion of Section X.2 - the remaining part of the proof of Theorem X.13, then continuation to the end of the section.
Lecture 5 - March 4, 2024
Beginning of Section X.3 (holomorphic functional calculus) - Proposition X.17, construction of the functional calculus, independence on the choice
of a cycle, a part of the proof of Theorem X.18 (assertions (b)-(e), a part of assertion (a) - linearity, multiplicativity in case the first factor
is a rational function).
Lecture 6 and classes 3 - March 8, 2024
Student presentation - Problems 20 and 24 to Chapter X. Completion of Section X.3 - the rest of the proof of Theorem X.18 (the remaining part of assertion (a), assertions (f)-(i)) and the following remarks.
Lecture 7 - March 11, 2024
Section X.4 (ideals, complex homomorphisms and Gelfand transform) - to Proposition X.22.
Classes 4 - March 15, 2024
Student presentation - Problems 26-28 to Chapter X. Application of the holomorphic calculus to some elements addressed in
these problems (i.e., a part of Problem 35 to Chapter X), Problems 22, 36 and a variant of Problem 25 to Chapter X.
Lecture 8 - March 18, 2024
Completion of Section X.4 - from Proposition X.23 to the end of the section.
Beginning of Chapter XI (C*-algebras and continuous function calculus), more specifically Section XI.1 (Algebras with involutions and C*-algebras - basic properties) - introductory definitions, remarks (1) and (4), Examples XI.1(1),(2).
Lecture 9 and classes 5 - March 22, 2024
Student presentation - Problems 31-33 to Chapter X. Continuation of Section XI.1 - remarks (2) and (3), then from Example XI.3 to Proposition XI.5(a).
Lecture 10 - March 25, 2024
Completion of Section XI.1 - the remaining part of the proof of Proposition XI.5 and the following remark. Beginning of Section XI.2 (*-homomorphisms and
Gelfand transform of C*-algebras - the introductory definition, Proposition XI.6, Proposition XI.8(a,c), Theorem XI.9.
Lecture 11 and classes 6 - April 5, 2024
Student presentation - Problems 40 and 42 to Chapter X. Continuation of Section XI.2 - completing the proof of Theorem XI.9,
Corollary XI.10, Example XI.7 and Corollary XI.11.
Lecture 12 - April 8, 2024
Completion of Section XI.2 - a proof of Proposition XI.8(b). Beginning of Section XI.3 (Continuous functional calculus in C*-algebras) - Proposition XI.12, Theorem XI.13 (sketch of the proof, a detailed proof is available at the lecture notes), Theorem XI.14 and a basic scheme of Theorem XI.15.
Lecture 13 and Classes 7 - April 12, 2024
Student presentation - Problem 1 to Chapter XI. A brief information on connection of the Gelfand transform on L1(Rd)
and the Fourier transform. Completion of Section XI.3 - comments to Theorem XI.15. Beginning of Section XI.4 (Distinguished elements of C*-algebras)
- to Proposition XI.18.
Lecture 14 - April 15, 2024
Completion of Section XI.4 - from the definition of a projection to the end of the section. Beginning of Chapter XII (Operators on Hilbert spaces),
namely of Section XII.1 (More on bounded operators and their spectra) - to Proposition XII.1(c).
Lecture 15 - April 19, 2024
Completion of Section XII.1 - from Proposition XII.1(d) to the end of the section.
Lecture 16 - April 22, 2024
Section XII.2 (Notion of an unbounded operator between Banach spaces) - the whole section. Examples XII.12 were skipped and the final remark
was only briefly mentioned (details of both of them may be found at the lecture notes). Definition of the spectrum of an unbounded operator.
Classes 8 and Lecture 16 - April 26, 2024
Student presentations - problems 7,8,9,15,25 to Chapter XII. Beginning of Section XII.3 (Spectrum of an unbounded
operator) - to Proposition XII.14(c).
Lecture 18 - April 29, 2024
Completion of Section XII.3 - Proposition XII.14(d) and Lemma XII.15. Beginning of Section XII.4 (Operators on a Hilbert space) - to Proposition XII.21.
Classes 9 and Lecture 19 - May 3, 2024
Student presentation - problem 19 to Chapter XII. Problem 28(1) to Chapter XII. Continuation of Section XII.4
- from the definition of a symmetric operator to Lemma XII.24.
Lecture 20 - May 6, 2024
Completion of Section XII.4 - Theorem XII.25 and Corollary XII.26. Beginning of Section XII.5 (Symmetric operators and Cayley transform) - to Theorem XII.29.
Lecture 21 and Classes 10 - May 10, 2024
Completion of Section XII.5 - Theorem XII.30 and a brief information on deficiency indices. Analysis of operators from Problems 33 and 35 to Chapter XII
(adjoint operators, eigenvalues and spectrum).
Lecture 22 - May 13, 2024
Beginning of Chapter XIII (Spectral measures and spectral decompositions), namely Section XIII.1 (Measurable calculus and spectral measure for normal bounded operators) - to Theorem XIII.4(a). A proof of Proposition XIII.1 was only briefly sketched.
Lecture 23 and Classes 11- May 17, 2024
Student presentation - a part of Problem 22 to Chapter XII (to be completed next time).
Completion of Section XIII.1 - Theorem XIII.4(b-f). Beginning of Section XIII.2 (Integral with respect to a spectral measure) - to Lemma XIII.6.
Lecture 24 - May 20, 2024
Continuation of Section XIII.2 - from the remark following Lemma XIII.6 to Theorem XIII.11 (including a proof of the existence of the required
operator - the uniqueness is easy an its proof is available at the lecture notes).
Lecture 25 and classes 12 - May 24, 2024
Student presentation - Problem 22 to Chapter XII. Completion of Section XIII.2 (Theorem XIII.12 was only briefly mentioned,
its proof is available at the lecture notes). Section XIII.3 (Spectral decomposition of unbounded self-adjoint operator) - whole section.
