Content of the lectures and classes
Lecture 1 - 15.2.2022
Introductory information - a brief content of the course, requirements to gain the credit.
Beginning of Chapter V (Bounded and unbounded operators on Hilbert spaces) and Section V.1 (various types of bounded operators on
Hilbert spaces) - till Proposition 4(d).
Lecture 2 and Classes 1 - 21.2.2017
Continuation of Section V.1 - from Proposition 4(e) till Proposition 6.
Operator (x,y)↦(y,-x) on R2 is unitary and has zero numerical radius;
the numerical range of the operator given by the same formula on C2 is the segment connecting
i and -i; the numerical range of the operator (x,y)↦(0,x) on C2 is
the closed disc centered at 0 with radius 1/2. It follows that the constants in Proposition 4(a) are optimal.
Lecture 3 - 22.2.2022
Continuation of Section V.2 - from Proposition 7 to the beginning of the proof of Theorem 9 Lemma 9.
Lecture 4 - 24.2.2022
Completion of Section V.1 - from Theorem 9 to the end of the section, the proof of Proposition 10 was only sketched.
Beginning of Section V.2 - till Lemma 13.
Lecture 5 - 1.3.2022
Completion of Section V.2 - from Proposition 14 to the end of section. Examples 17 were only brielfly mentioned, the same
applies to the concluding remark. Section V.3 (Spectrum of an unbounded operator) - till Proposition 19(c).
Lecture 6 and classes 2 - 3.3.2022
Completion of Section V.3 (Proposition 19(d) and Lemma 20)
Properties and characterizations of partial isometries. Backward shift on l2 - it is a partial isometry,
its adjoint is the forward shift. For both operators computing of the spectrum (it is the closed unit disc), the point spectrum
(it is the open unit disc for the backward shift and the empty set of the forward shift), the approximate point spectrum (it is the
whole spectrum for the backward shift and the unit circle for the forward shift), the residual spectrum (it is empty
for the backward shift and the open unit disc for the forward shift) and the numerical range (it is the open unit disc).
Lecture 7 - 8.3.2022
Section V.4 (operators on a Hilbert spaces) - till Proposition 26(b).
Classes 3 - 10.3.2022
Proof of Proposition 26(c), definitions of self-adjoint and symmetric operators. Multiplication operators on l2
- boundedness, compactness, adjoint operator, normality, relationship to the Hilbert-Schmidt theorem, spectrum and point spectrum.
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)}
- they are densely defined, computation of the adjoint operators - the easy inclusions and T4*=-T1.
Lecture 8 - 15.3.2022
Completion of Section V.4 - from Lemma 27 to the end of the section.
Beginning of Section V.5 (Symmetric operators and the Cayley transform) - till Proposition 32(a).
Classes 4 - 17.3.2022
Completion of the example from the last week: T4*=-T1, T1*=-T4, T2*=-T3,
T3*=-T2, T5*=-T5. Eigenvalues and spectra of these operators -
σ(T1)=σp(T1)=C, σ(T2)=σ(T3)=∅,
σ(T4)=C, σp(T4)=∅, σ(T5)=σp(T4)={2kπi; k∈Z}.
Operators Tj:f↦f' on two subspaces of
L2(0,∞) - D(T1)={f∈ACloc[0,∞); f,f'∈L2(0,∞)},
D(T2)={f∈D(T1); f(0)=0}, T2*=-T1, T1*=-T2..
Lecture 9 - 22.3.2022
Completion of Section V.5 - from Theorem 32(b) to the end of the section. Final remarks were only briefly mentioned.
Lecture 10 - 24.3.2022
Beginning of Chapter VI (Spectral measures and spectral decompositions), namely Section VI.1 - to Lemma 3. The proof of Lemma 3 was only
briefly indicated.
Lecture 11 - 29.3.2022
Completion of Section VI.1 - Theorem 4. Beginning of Section VI.2 (Integral with respect to a spectral measure) - to Lemma 6(g).
Lecture 12 and Classes 5 - 31.3.2022
Continuing Section VI.2 - Lemma 6(h), Proposition 7 and the first part of Theorem 8 (the existence and uniqueness of
Φ0(f), if
f and g coincide except for a set from 𝒩, then Φ0(f)=Φ0(g)).
Operators of multiplication on L2(μ) for a finite measure - spectral measure and measurable calculus, the same
for operators of multiplication on ℓ2(Γ) and on L2(μ) for a decomposable measure.
Lecture 13 - 5.4.2022
Continuation of Section VI.2 - proof of Theorem 8, Lemma 9, Corollary 10 and a part of proof of Theorem 11.
Lecture 14 and classes 6 - 7.4.2022
Completion of Section VI.2 - the rest of the proof of Theorem 11, Theorem 12 and most of its proof, Proposition 13
and its proof.
Computation of the integral of an unbounded function with respect to the spectral measure of the multiplication
operator from the last week; spectral measure of the forward shift on ℓ2(Z) via the unitary equivalence
with a multiplication operator (using Fourier series).
Lecture 15 - 12.4.2022
Section VI.3 (Spectral decomposition of an unbounded self-adjoint operator). Beginning of Section IV.4 (Unbounded normal operators)
- Lemma 19 and most of the proof.
Lecture 16 and classes 7 - 14.4.2022
Completion of Section VI.4 - recalling Lemma 19 and its meaning, Lemma 20 and its proof, Theorem 21 - expanation of its
statement and of the basic scheme of a proof, comments to its corollaries. Section VI.5 (Complements to the theory of unbounded
operators) - Proposition 24 (explanation of the statement ant of the basic scheme of the proof), Theorem 25 as a consequence
of Proposition 24, Theorems 26 and 27 briefly commented.
Diagonalization of the differentiation operator on L2(0,2π).
Lecture 17 - 19.4.2022
Beginning of Chapter VII (More on locally convex topologies) - recalling some basic notions and facts, beginning of Section VII.1
(Lattice of locally convex topologies and toplogies agreeing with duality) - to Proposition 3.
Classes 8 - 21.4.2022
Diagonalization of the differentiation operators on L2(0,2π) with various boundary conditions.
Construction of a selfadjoint Laplace operator on L2(Ω).
Lecture 18 - 26.4.2022
Conitunation of Section VII.1 - to Proposition 7.
Classes 9 - 28.4.2022
Construction of a selfadjoint Laplace operator on L2(Ω) - continuation.
Diagonalization of the operator of differentiation on R.
Corollary 8 and Example 9 from Section VII.1.
Topology of uniform convergence on elements of a given family of bounded subsets of a normed space
and its dual.
Lecture 19 - 3.5.2022
Section VII.2 (bw*-topology and Krein-mulyan theorem) - to Corollary 14.
Lecture 20 and Classes 10 - 5.5.2022
Completing Section VII.2 - Theorem 15 and briefly the following remarks.
Application of the results from the last week to the topology of uniform convergence on bounded countable sets.
Spaces c0 and lp for p∈[1,∞)
with topology of pointwise convergence - description of their duals, coincidence of the weak and Mackey topologies.
Sketch of the same for spaces c0(Γ) and lp(Γ) for p∈[1,∞).
Lecture 21 - 10.5.2022
Section VII.3 (Compact convex sets) - to Example 20.
Lecture 22 and classes 11 - 12.5.2022
Completion of Section VII.3 - from Theorem 21 to the end of the section.
Combining Krein-Milman and Mazur theorems for weakly compact convex sets in Banach spaces and for closed convex bounded
sets in reflexive spaces. Extreme points of closed unit balls in spaces c0, C([0,1]),
more generally C(K), ℓ1, hint for L1.
Lecture 23 - 17.5.2022
Section VII.4 (Weakly compact sets and operators) - to Lemma 27.
Lecture 24 - 19.5.2022
Completing Section VII.4 - Theorem 28, Proposition 29, then Theorem 26 and the rest of the section.
