Скачать или смотреть Marcin Moszyński | Spectral theory for self-adjoint (finitely) cyclic operators | Lecture 4 of 6

Lecture 4 ###
Multiplication operators in L²(M)
Self-adjoint finitely cyclic operators

A B S T R A C T

This mini-course is devoted to а rigorous proof of the result, called here "x-Multiplication Unitary Equivalence" (abbr.: xMUE) Тheorem, for finitely cyclic self-adjoint operators, which is an analog of the "xMUE" Theorem for self-adjoint cyclic operators (according to the opinion of Sergei Naboko, most probably, it is one of the Stone's results for cyclic – i.e.,"simple spectrum" – operators).

A finitely cyclic operator is a generalisation of a cyclic operator: the cyclic ("generating") vector is replaced here by a finite system of vectors (the name an "operator with finite multiplicity of the spectrum" is also used). To do this, we introduce and rigorously develop some foundations of the theory of matrix measure L² spaces, and we study spectral properties of multiplication operators (by scalar functions) in such spaces.

Because of the "mini" character of the course, some proofs will be omitted during the lectures, but all the details can be found in the manuscript which is available here:

https://indico.eimi.ru/e/STMP21Offlin...