Ondřej Kalenda, KMA MFF UK

Novemeber 4: Bochner integral - computation using duality; computing integral of the function x↦χ_[0,x] with values in L^p[0,1] for p<∞; nonmeasurability and weak measurability for p=∞; computing an integral with values in C[0,1]; Pettis integral, Pettis-integrable functions which are not Bochner integrable, relationship to absolute and unconditional convergence.

November 12: Relationship of Bochner and Pettis integral to absolute and unconditional convergence (continuation); Banach algebras - renorming under which the unit has norm 1; algebra Cⁿ with l_p norm and its renorming; algebra l_p(Γ); adding a unit to an algebra without unit and to an algebra with a unit; the unit of a subalgebra need not be the unit of the surrounding algebra (e.g. for matrices); alternative norms on the algebra with added unit (e.g. on the algebra of compact operators with added ientity); cartesian product of algebras; algebra of continuous functions with values in a Banach algebra; algebra of matrices with entries from a given Banach algebra

November 18: Locally compact abelial groups, Haar measure, convolution algebra L¹(G), algebra M(G); algebras with many left (or right) units, algebra with trivial product and its representation in an algebra of operators, spectrum in an algebra without unit, spectrum w.r.to a subalgebra may be larger.

November 25: Spectrum w.r.to a subalgebra may be larger (finishing the example), holomorphic calculus in the algebra of continuous functions, in the matrix algebra (for a diagonal matrix, for a Jordan cell, for a general matrix, dependence not only on the values on the spectrum, aplication the the exponential of a matrix), spectrum in the algebra l¹(Z_m), holomorphic calculus in the algebra l¹(Z₂).

Decemeber 2: In the matrix algebra there are no nontrivial two-sided ideals, there are one-sided ideals, description of complex homomorphisms on the algebra C(K) (evaluation functionals - a proof using Riesz theorem and a proof without use of Riesz theorem), complex homomorphisms on the algebra l¹(G) as group homomorphisms G→T, concrete description for the groups Z and Z_m.

Decemeber 8: Description of dual groups to Q, Zⁿ, Z^(N) etc., Gelfand transform on the algebra l¹(G) and especially on l¹(Z), relationship to functions with absolutely convergent Fourier series; complex homomorphisms on algebras without unit, complex homomorphisms on the algebra L¹(G) as continuous group homomorphisms G→T.

Decemeber 16: Description of the dual group of R, Gelfand and Fourier transforms; description of the dual group of T, Gelfand transform and Fourier series; l¹(Z) is not a C*-algebra, neither after renorming; continuous calculus on the algebra C(K).

January 6: Characterization of normal matrices and function calculus on them, absolute value of an element of a C*-algebra, polar decomposition of an operator on a Hilbert space, application of polar decomposition to find Schmidt representation of a compact operator, properties of a compact operator and Schmidt representation, spectrum of the operator T(f)(x)=f(-x).

January 13: Spectral decomposition of the operator T(f)(x)=f(-x) and more generally of a symmetry, spectral decomposition of the Plancherel transform, unitary equivalence with a multiplication operator - for the two previous cases, method of the proof in case of existence of a cyclic vector, application to a shift operator on l²(Z).

Ondrej Kalenda

Department of Mathematical Analysis

Faculty of Mathematics and Physics

Charles University in Prague

Obecné informace a odkazy

Zimní semestr 2015/2016

Classes in Functional Analysis 1