Mathematics

CLAUDIA PINZARI

Educational Goals

General objectives: To provide students with the basics related to the study of functional spaces that intervene in various fields. In particular, linear operators will be studied between Banach or Hilbert spaces and their spectrum will be analyzed. Finally, some non-linear Functional Analysis techniques will be presented, suitable for the study of differential problems.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to the Functional Analysis and to its different applications to differential problems.

A - Knowledge and understanding of Banach spaces, Hilbert spaces, locally convex spaces, Banach algebras, linear and limited operators on Banach and Hilbert spaces, spectral theory for normal operators on Hilbert spaces, continuous functional calculus and Borelian, Gelfand transform, unbounded operators on Hilbert space, closed, closable, self-adjoint.

B – Application skills on knowing how to apply analytical concepts to problems of operator algebras, functional analytical approach to quantum field theory, quantum mechanics.

C - Independence of judgement
The assigned problems will stimulate the ability to solve independently

D – Communication skills
Interaction and collaboration between students and with the teacher will be ensured during the lessons
lessons and receptions

E - Ability to learn
Notes/connections to concepts in future courses

General knowledge of topology, elementary theory of Banach spaces and their applications, of Hilbert, elements of complex analysis. The necessary preliminary results will be recalled in class.

1) Ordered sets: binary relations, order, majorant, minority, ascending or descending filtering orders, lattice, total order, good ordering, examples; axiom of choice, principle of good ordering, Zorn's lemma; equivalence between AC, Zorn, principle of good order (sd); topology, recalls on the axioms of numerability and relations. Weakest and strongest topology among a family of topologies given on the same set; separability; net and ordinary successions.

2) net, subnet, accumulation point for a net; universal nets and convergence at the points of accumulation; existence of universal nets;

3) Filter, universal net, existence of universal subnets. Basic properties of continuous functions on compacts. Compactness and equivalent definitions; elementary properties; weak topology; product topology. Tychonoff's theorem. Comments on cases in which AC is not necessary, finite or countable set of indices. Property T2. Product of spaces T2 is and T2. Compact T2 is T4. Countable product of compact metrics is compact metric in the product topology. Tychonoff's cube.

4) The subsets of Tychonoff's cube are normal metric spaces with a numerable basis. Theorem (sd) Any topological space normal with a countable basis and 'homeomorphic to a subset of the Tychonoff cobo, and therefore metrizable. It is compact if and only and the subset is closed. Relations between topology and net. Normed spaces; Banach spaces, basic properties; linear operator between normed spaces, equivalent conditions of continuity. Space of linear and bounded operators. Sub-multiplication of the standard. Quotient normed space. Injective operator associated with a bounded and linear operator. Banach space quotient. If X is normed, M and X / M are Banach's, then X is Banach's.

5) A subspace of finite dim in a normed space is Banach. Extension of a densely defined operator, recalls on the completion of a metric space., Realization of the completion for a normed space as a canonical immersion in the bidual. Various examples of normed spaces and completion of infinite dimension, continuous functions on a locally compact Hausdorff space, examples with infinite products. Baire's category theorem.

6) Open application theorem, continuity of the inverse of a limited and bi-univocal operator between Banach spaces; closed graph theorem; principle of uniform limitation.

7) Application to the space of continuous and periodic functions in an interval; Fourier algebra on the circle; exercise on the existence of non-derivable functions at any point of an interval. Minkowski functional, extensibility of functionals; theorem of
Hahn-Banach, existence of linear and bounded functionals on a normed space that separate the points. Extension to the quotient space. Subspace annihilator. Canonical isometry of immersion in the bidual and relationship with that of completion. Relation to reflexivity.

8) Classical examples of reflexive Banach spaces. Non-degenerate duality. Added operator. Algebraic properties. Isometry of the adjunct. Automatic boundedness of a linear operator with addition between Banach spaces. Exercises on classical examples of duality between spaces of sequences. Weak topology defined by a family of seminorms; topological vector space; locally convex vector space. Continuous functionals with respect to the topology defined by a duality. Construction of Minkowski functionals in a topological vector space. Equivalence between the two definitions of SVTLC.

9) Existence of continuous functionals that separate the convex ones in an SVT (Geometric form of the Hahn-Banach theorem). Weak topology of a normed space, equality of the dual with respect to the original topology; the two topologies have the same closed convex. ∗ -weak topology of the dual, computation of the topological dual. Alaoglu's theorem.

10) Sequential Alaoglu's theorem, independence from the axiom of choice; consequences of Alaoglu's theorem: the unitary disk closed in norm of a Banach space is weakly compacted if and only if the space is reflexive (only half proof). Existence of weakly convergent ordinary subsequences of normally bounded ordinary sequences in a reflexive Banach space (Eberlein-Smulian, s.d.), example with Lp (X) with 1

11) Radon integral; extension to space L1 (sd). Radon charges. Positive Radon measurements. Probability measurements on a compact Hausdorff space. Riesz representation theorem (sd). Banach space of measures. Characterization of positive radon measurements oncompact Hausdorff. Characterization of extremal measures (s.d.). For thesis: proof of the characterization theorem of positive measures and extremal points on a Hausdorff compact (Pedersen), Spaces of distributions (Yosida). Hilbert space. Sesquilinear forms. Polarization. Scalar product. Cauchy-Schwarz. Parallelogram. and characterization with the dot product norm

12) Prehilbertian space. Completion. Buildings. Projection theorem on a closed convex. Case of the closed subspace. Riesz representation theorem of continuous functionals. Orthonormal basis. Existence. Hilbert dimension. Relation between sesquilinear forms and bounded operators. Added operator. Algebraic properties of the adjunct. Property C ∗ of the norm.

13) Self-adjoint operators. Core of the adjunct. delimitation from zero. Characterization of the invertibility of a bounded operator on Hilbert space. Normal, positive operators. Property. Construction of the square root of a positive operator. Orthogonal projections, unit operators, isometries.

14) partial isometries, polar decomposition, finite rank operators. Closed disk of a Hilbert space. Weak compactness. Compact operators and various characterizations. Diagonalizable operators. Characterization of the compactness of diagonal operators.

15) Existence of eigenvalues ​​of maximum absolute value for normal compact operators. Calculation of the norm of a normal compact operator (dim only for positive operators.). Diagonalizability of compact normal operators on complex Hilbert space. Calkin algebra. Compact perturbation of a limited operator. Atkinson's theorem. Fredholm operators. Index. Product of Fredholm and Fredholm operators. Basic properties. Examples of Fredholm operators for each index value. Fredholm alternative.

16) Analogy between B (H) and L∞ (X). Trace of a positive operator. Relationship between trace and compactness. Trace class B1 (H) operators. Hilbert-Schmidt B2 (H) operators. Position in B (H). Spectrum of an operator.

17) Hilbert space structure in B2 (H). Estimating the trace of a product. Trace properties. Banach algebra structure in B1 (H). Duality between compact operators, trace class operators and bounded operators.

18) Duality between compact operators, trace class operators and limited operators. Characterization of the Hilbert-Schmidt operators on L2 (X). Fredholm integral equations. Banach algebras. Closed ideals. Quotient algebra. Adding the identity. Examples: C0 (X), B0 (H), L1 (Rn), B (H). Invertibility of the elements in the open unit disk of I. Complex case. Spectrum of an element. Solving function. Spectrum of the analytical functional calculation of an element.

19) Spectrum of the analytical functional calculation of an element. Non-triviality of the spectrum of an element of a complex non-zero Banach algebra. Analyticity of the resolving function. calculation of the spectral radius. A unital Banach algebra in which every non-null element is invertible, coincides with C. Spectrum of an element in C (X), with Hausdorff compact X. Exercise on the spectrum of AB and BA. Point, continuous and residual spectrum of an element of B (X), with X Banach space. Continuous functional calculus for a self-adjoint operator on Hilbert space. Introduction to the Gelfand transform.

20) Complex commutative Banach algebra with I. Maximal ideals. characters. Spectrum of algebra. Relationship between the spectrum and the space of maximal ideals. Calculation
of the spectrum of each element. Automatic continuity of characters. Core of the Gelfand transformation. Semi-simple algebra. Example with C (X), with Hausdorff compact X. Gelfand-Naimark theorem for commutative Banach algebras with I. Stone-Weierstrass theorem.

21) Example: l1 (Z). Computation of the spectrum and identification of the Gelfand transform with the Fourier transform. Fourier algebra (image of the Gelfand transform). C ∗ -algebra, abstract definition. ∗ -Banach algebra (involutive Banach algebra for which the property C ∗ of the norm is replaced by ∥A ∗ ∥ = ∥A∥ for each element A of the algebra. Computation of the spectral radius of a normal element of a C ∗ - alegbra. Unitary (self-adjoint) elements have spectrum in the (real) circle. Gelfand-Naimark theorem for C ∗ -comutative algebras with I. Exercise on the uniform density of trigonometric polynomials in the algebra of continuous and periodic functions. regular of a Banach algebra C ∗ -algebra with the added identity.
22) Spectrum of an element of a C ∗ -algebra without identity. Stone-Weierstrass theorem for locally compact topological spaces. Continuous functional calculus for a normal operator on a C ∗ -algebra. Invariance of the spectrum of a normal element with respect to the C ∗ -algebra that contains it. Gelfand-Naimark theorem for C ∗ -computative algebras without identity. Compacting of Stone-Check. Concrete continuous functional calculation: Spectrum of shift operators and their additions. Calculation of the residual and continuous point spectrum. Self-adjoint operators have real spectrum. Triviality of the residual spectrum for self-adjoint operators on Hilbert space.

23) Continuous functional calculation. Spectral mapping theorem. Spectral measurements. Cyclic vector for a self-adjoint operator. Operators without multiplicity and spectral theorem (realization by means of multiplication operators). Construction of normal operators with an arbitrary spectrum and a dense succession of eigenvalues; of operators without eigenvalues ​​with spectrum the closure of a bounded open in C. The isolated points in the spectrum of a normal operator are eigenvalues. In this case, the characteristic function is continuous. Normal operators with real spectrum are self-adjoint; with positive spectrum they are positive. Decomposition of a self-adjoint operator as the difference of orthogonal positive operators.

24) Introduction to the Borelian functional calculus. Strong and weak topology of B (H). A limited growing net of self-adjoint operators on Hilbert space strongly converges at the upper bound. Bounded Borelian functions on a locally compact Hausdorff space. Characterization of the real part. Bicommutant of a normal operator. Commutant of a self-adjoint subset. Von Neumann bicommutant theorem (s.d.). Definition of von Neumann algebra in B (H). The commutant of a self-adjoint set of operators is a von Neumann algebra. Theorem on the Borelian functional calculus for a normal operator and discussion that assumes values ​​in the von Neumann algebra of the operator (without dim the homomorphism property). Spectral family with operator values. Example using the Borelian functional calculation of a normal operator.

From a spectral measure to the spectral family and vice versa. Spectral theorem for a normal operator in the form of the integral of the identical function with respect to a spectral family (for the theory of integration starting from a spectral family, only stated).

25) Introduction to densely defined unlimited operators. Domain and graph of a densely defined operator. Closed and closable operators. Characterization of a subspace of a Cartesian product of Banach spaces so that it is the graph of an operator. Closable operators and closing of an operator. Example with the derivation operators id / dx in AC ([0,1]) and multiplication operator in L2 (R). Added an operator between Hilbert spaces. Each densely defined operator between Hilbert spaces has an added closed. It is closable if and only if the addition is densely defined, and the closure addition is the operator's addition. Summary of the proof. Relationship between core and image of the adjunct for a densely defined closable operator. Spectrum and resolvent of a closed operator. Example with the differential operators in L2 ([a, b]) (outline). Symmetric and self-adjoint operators. Essentially self-adjoint operators. Basic criterion for self-adjoint and essential self-adjoint. Example with various domains in AC [0, 1] ⊂ L2 ([0, 1]). Summary of non-essential self-adjoint and self-adjoint extensions for the domain f (0) = f (1) = 0. Outline of Stone's theorem for unitary groups with a strongly continuous parameter and associated self-adjoint operator.

Further possible insights
Theorem on the characterization of the extremal states of the algebra of continuous functions on a compact topological space by Hausdorff (Pedersen)
Details on Fredholm operators and connections with the Atiyah-Singer index theory (Pedersen for the analytical part)
Palais: Seminar on the Atiyah-Singer index theorem
Gilkey: Invariant theory, the heat equation and the Atiyah-Singer index Theorem Unlimited operators, as in the June 1 lesson (Reed-Simon).
Stone's theorem (Reed-Simon)
Spectral theorem for unbounded operators (Reed-Simon)
Further reading, for future developments
A. Connes: Noncommutative geometry

1)J.B. Conway: A course in functional analysis.
2)S. Doplicher: Note del corso di Analisi Funzionale
3) G.K. Pedersen: Analysis Now, Springer
4) S. Doplicher: An invitation to Quantum Mechanics, Lecture notes on some mathematical aspects, March 2021
5) M. Reed, B. Simon: Functional Analysis, I,
6)J. Dixmier: C∗-algebras
7)J. Dixmier: von Neumann algebras
8)R. Haag: Local Quantum Physics, Springer.
9)J.M. Gracia Bondia, J.C. Varilly, H. Figueroa: Elements of noncommutative geometry

The lectures consist in the presentation of the general theory with the most important results and related demonstrations accompanied by targeted examples. Students will be stimulated so that they can develop and acquire skills in solving the proposed problems.

Lesson attendance is strongly attended

Evaluation based on the questions asked in the oral test on the program, any in-depth topic among those proposed, possible resolution of the proposed problems.

The exam questions, which consists of an oral test, will focus on the course program, on optional insights proposed during the lessons or on proposed problems.