Algebras of operators on Banach spaces and operator ideals - NMMA651

• Title: Algebry operátorů na Banachových prostorech a operátorové ideály
• Guaranteed by: Department of Mathematical Analysis (32-KMA)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2022
• Semester: winter
• E-Credits: 3
• Hours per week, examination: winter s.:2/0, Ex [HT]
• Capacity: unlimited
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: not taught
• Language: English
• Teaching methods: full-time
• Teaching methods: full-time
• Guarantor: Bence Horváth; Tommaso Russo

Annotation

English

Last update: doc. RNDr. Pavel Pyrih, CSc. (17.07.2021)

The main aim of this course is to study Fredholm, Riesz, strictly singular, and inessential operators on Banach spaces. As a tool we additionally introduce and study Schauder bases in Banach spaces.

Czech

Last update: doc. RNDr. Pavel Pyrih, CSc. (17.07.2021)

Budeme se zabývat Fredholmovými, Rieszovými, striktně singulárními a neesenciálními operátory na Banachových prostorech. Navíc se budeme zabývat Schauderovými bázemi, které poslouží jako pomocný aparát.

Course completion requirements

Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (20.07.2021)

The course is finished by an exam.

Literature

Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (20.07.2021)

F. Albiac and N. J. Kalton, Topics in Banach space theory (Springer, 2006)

S. E. Caradus, W. E. Pfaffenberger and W. Yood, Calkin algebras and algebras of operators on Banach spaces (Marcel dekker, 1974)

H. G. Dales, Banach Algebras and Automatic Continuity (Oxford University Press, 2000)

R. E. Megginson, An introduction to Banach space theory (Grad. Texts Math. 183, Springer-Verlag, New York, 1998)

N. J. Laustsen, Lecture Notes on Banach spaces and their operators (Lecture notes, 2018)

A. Pietsch, Operator ideals (North Holland, 1980)

Requirements to the exam

Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (20.07.2021)

The students could choose between a seminar on a topic of self study or a standard oral examination from the topics presented during the semester.

Syllabus

Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (20.07.2021)

1. Fredholm operators: Definitions and basic properties, duality, Yood's Lemma and its consequences, Atkinson's Theorem, continuity of the Fredholm index, Riesz-Schauder operators.

2. Riesz operators: Definition, algebraic properties, Interlude (Banach algebras, spectral theory, holomorphic function calculus), the essential spectrum.

3. Inessential operators: Definition and basic properties, the Jacobson radical, Kleinecke's Theorem.

4. Strictly singular operators: Introduction, Kato's Lemma, the strictly singular operators form an operator ideal, the strictly singular operators contain the compact operators.

5. Schauder bases: Introduction, the basis projections and the coordinate functionals, basic sequences, duality, equivalence of bases and stability, block basic sequences and the Bessaga-Pelczynski Selection Principle.

6. The Gohberg-Markus-Feldman Theorem: Applying the B-P Selection Principle to c_0 and l_p, the proof of the G-M-F Theorem.

7. Separable C(K) spaces: Revision (dual space, extreme points, separability, Banach-Stone, Stone-Weierstrass, containment of c_0) the Cantor set, universality of C([0,1]), Miljutin theorem. Countable compacta: the dual is l_1, C(K) is c_0 saturated, hints at the classification, there are uncountably many non isomorphic C(K) spaces

8. Tentative: Eidelheit's Theorem, the structure of homomorphisms from B(X) and A(X).

Entry requirements

Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (20.07.2021)

A solid base in real, complex and basic functional analysis (Hahn{Banach Theorem, Open Mapping Theorem, Banach-Steinhaus Theorem, Banach-Alaoglu Theorem) and linear and abstract algebra. Some knowledge in point-set topology is helpful.