Domination of unbounded operators and commutativity. (English) Zbl 1037.47003

Stochel, Jan; Szafraniec, Franciszek Hugon

J. Math. Soc. Japan 55, No. 2, 405-437 (2003).

The authors prove that pointwise commuting formally normal operators which are dominated by a single essentially normal operator are essentially normal, and their closures commute spectrally. It is also shown that for a formally normal operator \(A\), the normality of a polynomial \(p(A)\) implies the essential normality of \(A\) and of all polynomials of \(A\) whose degrees do not exceed the degree of \(p\). For commuting families of formally normal operators, some Nelson type criteria of essential spectral commutation are obtained; these results are in fact new even for symmetric operators, where the condition upon the sum of their squares may now be substituted by conditions upon certain more general polynomials.
In terms of domination, the authors also treat the problem of joint subnormality for systems of operators. Applications to multi-dimensional moment problems are discussed. In particular, determinacy conditions are found for the multi-dimensional Hamburger and complex moment problems.

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

47A13	Several-variable operator theory (spectral, Fredholm, etc.)
47B20	Subnormal operators, hyponormal operators, etc.
47B15	Hermitian and normal operators (spectral measures, functional calculus, etc.)
47A57	Linear operator methods in interpolation, moment and extension problems
47B25	Linear symmetric and selfadjoint operators (unbounded)

Keywords:

formally normal operator; essentially normal operator; multi-dimensional moment problem; spectral commutation