Multicyclicity Phenomenon. I. An Introduction and Maxi-Formulas

Abstract

The notion of the spectral multiplicity plays an important role in the theory of self-adjoint and normal operators. It serves a complete unitary invariant for operators of these classes, and in fact forms a background of the whole theory. As the main object it is proved to be the multiplicity function µ _N (·) of a normal operator N on a separable Hubert space H, i.e. the dimension function

$$ {\mu _N}\left( \zeta \right) = \dim \;H\left( \zeta \right) $$

of the spectral decomposition of N

$$ H \cong \int\limits_\mathbb{C} { \oplus \;H\left( \zeta \right)d\lambda \left( \zeta \right)} $$

,

$$ Nf\left( \zeta \right)\;\; = \zeta f\left( \zeta \right),\quad \quad \quad \quad \lambda \;\; - \;\;a.e.\;\;\zeta $$

where ≅ stands for a unitary equivalence and λ for a Borel (scalar) spectral measure of N .In principle, all properties of N can be derived from properties of the multiplicity function µ _N (·). In the case of a pure point spectrum one has simply

$$ {\mu _N}\left( \zeta \right)\;\; = \dim \;Kei\;\left( {T - \zeta I} \right),\quad \quad \quad \quad \lambda \;\; - \;\;a.e.\;\;\zeta $$

.