New Trends in contemporarty mathematics: Multipliers

These lectures introduce the theory arising from the entrywise product of matrices. We will explore the motivation for studying this product, and prove several important characterisations and properties. Finally, we will investigate other situations where this entrywise product occurs, and understand how this can be unified with our study of matrices.

Mathematics, cycle 2

Discipline: Theoretical Mathematics

Year of study 2; semester 4

Lecture: 15 hours

ECTS credits: 1

Type: elective course

Learning activities: 15 hours (100%) lectures.

Outside class: 2 hours homework for grading

The study of multipliers began with Schur's results on the entrywise product of matrices (with infinite index sets). The theory was further developed and used by Grothendieck, in his investigation of Banach spaces. Parallel to this is the theory of Fourier multipliers -- functions which multiply the terms of a Fourier series to give a new Fourier series. In abstract harmonic analysis this notion is generalised to arbitrary locally compact groups, where Fourier multipliers are known as Herz-Schur multipliers. The connections between these two parallel notions were put in their final form by Bozejko and Fendler. It was also realised that this can be unified using groupoids, and multipliers of groupoids opened more areas of investigation.

The purpose of this course is to give a tour of the theory of multipliers, beginning with the entrywise product of matrices, and generalising to operators on Hilbert space. We will then look at Herz-Schur multipliers, developing the parallel theory of multipliers on groups, and giving the connections between these two notions of multipliers. In the short final section we introduce groupoids and indicate how they can be used to unify and generalise these results.

Topics in matrix analysis; Horn and Johnson; Cambridge University Press; 2009.

Herz-Schur multipliers; Todorov; lectures given at Fields Institute; 2014. Available at https://www.fields.utoronto.ca/programs/scientific/13-14/harmonicanalysis/operatoralg/slides/Todorovfieldslec15.pdf.

KA7_UW09 (Can use mathematical analysis, including Banach and Hilbert spaces)

KA7_UK05 (English)

KA7_UU02 (Can independently search literature)

Attendance.

Grading.

These lectures introduce the theory arising from the entrywise product of matrices. We will explore the motivation for studying this product, and prove several important characterisations and properties. Finally, we will investigate other situations where this entrywise product occurs, and understand how this can be unified with our study of matrices.

Mathematics, cycle 2

Discipline: Theoretical Mathematics

Year of study 2; semester 4

Lecture: 15 hours

ECTS credits: 1

Type: elective course

Learning activities: 15 hours (100%) lectures.

Outside class: 2 hours homework for grading

The study of multipliers began with Schur's results on the entrywise product of matrices (with infinite index sets). The theory was further developed and used by Grothendieck, in his investigation of Banach spaces. Parallel to this is the theory of Fourier multipliers -- functions which multiply the terms of a Fourier series to give a new Fourier series. In abstract harmonic analysis this notion is generalised to arbitrary locally compact groups, where Fourier multipliers are known as Herz-Schur multipliers. The connections between these two parallel notions were put in their final form by Bozejko and Fendler. It was also realised that this can be unified using groupoids, and multipliers of groupoids opened more areas of investigation.

The purpose of this course is to give a tour of the theory of multipliers, beginning with the entrywise product of matrices, and generalising to operators on Hilbert space. We will then look at Herz-Schur multipliers, developing the parallel theory of multipliers on groups, and giving the connections between these two notions of multipliers. In the short final section we introduce groupoids and indicate how they can be used to unify and generalise these results.

Topics in matrix analysis; Horn and Johnson; Cambridge University Press; 2009.

Herz-Schur multipliers; Todorov; lectures given at Fields Institute; 2014. Available at https://www.fields.utoronto.ca/programs/scientific/13-14/harmonicanalysis/operatoralg/slides/Todorovfieldslec15.pdf.