Content of the lectures and classes
Lecture 1 - 20.2.2017
Introductory information - a brief content of the course, requirements to gain the credit.
Beginning of Section X.1 (unbounded operators between Banach spaces) - till Proposition 4.
Lecture 2 and Classes 1 - 21.2.2017
Continuation of Section X.1 - Proposition 5 and Proposition 7, beginning of Section X.3 (spectrum of an
unbounded operator) - definition of resolvent set, resolvent function and spectrum.
Discussion of conditions from Proposition 7 for nonclosed operators, including examples, applications
to comparison of different versions of the definition of the resolvent set, an example illustrating that
it may happen T(R+S)⫌TR+TS.
Lecture 3 - 27.2.2017
Completion of Section X.2 - Proposition 8 and Lemma 9. Section X.3 (operators on a Hilbert space) - till
Proposition 11.
Classes 2 - 28.2.2017
Operators of multiplication on c0 and on lp for p∈[1,∞) - they are closed, densely defined,
spectrum and characterization of their boundedness. Unbounded operators of multiplication on l∞ are not densely defined.
Application of the operators of multiplication to constructions of counterexamples
(ST need not have a closed extension if S is bounded and T is closed and densely
defined; S+T need not have a closed extension if S and T are densely defined
and closed operators with the same domain). Operators of multiplication on
Lp(μ) for p∈[1,∞) - they are closed, densely defined,
characterization of their boundedness and spectrum (to be completed).
Lecture 4 - 6.3.2017
Continuation of Section X.3 - from Proposition 12 till Proposition 17.
Lecture 5 and Classes 3 - 7.3.2017
Completion of Section X.3 - from Lemma 18 till the end of the section.
Spectrum of the multiplication operator - completion. Operators Tj:f↦f' on different subspaces of
L2(0,1) - D(T1)={f∈AC[0,1]; f'∈L2(0,1)},
D(T2)={f∈D(T1); f(0)=0}, D(T3)={f∈D(T1); f(1)=0},
D(T4)={f∈D(T1); f(0)=f(1)=0}, D(T5)={f∈D(T1); f(0)=f(1)}
- computation of the adjoint operators.
Lecture 6 - 6.3.2017
Section X.4 (symmetric operators and Cayley transform) - till Theorem 24.
Lecture 7 and Classes 4 - 14.3.2017
Completion of Section X.4 - final remarks (1) and (2); the remark (3) only briefly mentioned.
Section X.5 (integral with respect to a spectral measure) - till the remark following Lemma 25.
Eigenvalues and the spectrum of the operators Tj:f↦f' from the previous class.
Lecture 8 - 20.3.2017
Continuing Section X.5 - from Proposition 26 to Theorem 28 including the first part of the proof
(D(Φ(f)) is a dense subspace).
Lecture 9 and Classes 5 - 21.3.2017
Continuing Section X.5 - completion of the proof of Theorem 28 and Theorem 29(a,b).
Computation of adjoint operators to the operators T1,2:f↦f', where
D(T1)={f∈ACloc[0,∞); f,f'∈L2(0,∞)},
D(T2)={f∈D(T1); f(0)=0} (almost finished).
Lecture 10 - 27.3.2017
Completion of Section X.5 - Theorem 29(c,d,e) and Proposition 30. Section X.6 (spectral decomposition of a selfadjoint operator) - till Theorem 33.
The construction of the measurable calculus and of the spectral measures for a bounded normal operator (including Proposition 31) was only briefly recalled.
Lecture 11 and Classes 6 - 28.3.2017
Completion of Section X.6 - from Lemma 34 to the end of the section.
Completion of the computation of adjoint operators to the operators T1,2
from the last week, computation of eigenvalues and spectra.
Lecture 12 - 3.4.2017
Section X.7 (Unbounded normal operators)- Lemmata 38 and 39 with a proof; Theorem 40 and a basic scheme
of its proof, its corollaries.
Section X.8 (Complements to the theory of unbounded operators) - Theorem 44 and a basic scheme
of a proof of Proposition 43.
Lecture 13 and Classes 7 - 4.4.2017
Completion of Section X.8 - Proposition 45 and Theorem 46 with a basic scheme of a proof.
Computation of the adjoint operator to the operator T:f↦f', where
D(T)={f∈ACloc(R); f,f'∈L2(R)},
spectrum and beginning of diagonalization.
Lecture 14 - 10.4.2017
Chapter XI (More on locally convex topologies), Section XI.1 (Lattice of locally convex
topologies and admissible topologies) - till the definition of the Mackeyho topology.
Classes 8 - 11.4.2017
Spectral measure of the multiplication operator; diagonalization of the differentiation operator on L2(R) using the
Fourier transform, its spectral measure; diagonalization of the differentitation operator on L2(0,2π)
using Fourier series for diffent boundary conditions, spectral measures of these operators.
Classes 9 - 18.4.2017
Laplace operator on L2(Ω) defined on the space of test functions on Ω and on the space of restrictions of test
functions on Rd - description of the adjoint operators using Sobolev-type spaces, operators are closable; selfadjoint Laplace operator
on L2(Rd); construction of a selfadjoint Laplace operator on L2(Ω).
Lecture 15 - 24.4.2017
Continuing Section XI.1 - from Lemma 4 to Corollary 8.
Lecture 16 - 25.4.2017
Completion of Section XI.1 - Example 9. Section XI.2 (bw*-topology and Krein-mulyan theorem)
- to Theorem 12.
Lecture 17 and Classes 10 - 2.5.2017
Completion of Section XI.2 - from Corollary 13 to the end of the section.
Topology of uniform convergence on elements of a given family of bounded subsets of a normed space
and its dual; applications to the topology of uniform convergence on bounded countable sets.
Classes 11 and Lecture 18 - 9.5.2017
Spaces c0(Γ) and lp(Γ) for p∈[1,∞)
with topology of pointwise convergence - description of their duals, coincidence of the weak and Mackey topologies,
the topology of uniform convergence on weak* compact sets is strictly stronger than the Mackey topology.
Section XI.3 (Compact convex sets) - to Lemma 17.
Lecture 19 - 15.5.2017
Continuing Section XI.3 - from Theorem 18 to Proposition 22.
Lecture 20 and Classes 12 - 16.5.2017
Completion of Section XI.3 - from Theorem 23 to the end of the section.
Compact convex sets in R3 with a non-closed or even non-Fσ
set of extreme points; comparison of bw* and w* topologies
on lq for q∈(1,∞] using multiples of canonical vectors.
Lecture 21 - 22.5.2017
Section XI.4 (Weakly compact sets and operators in Banach spaces) - to Lemma 25,
further Lemma 27 and the proof of Theorem 26 using Lemma 27, Theorem 28 and Proposition 29.
Lecture 22 - 23.5.2017
Completion of Section XI.4 - from Theorem 28 to the end of the section.
