Control subspaces of minimal dimension. Elementary introduction. Discotheca

V. I. Vasyunin ; N. K. Nikol'skii

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 41-75

Résumé

In this paper there is introduced and studied the following characteristic of a linear operator $A$ acting on a Banach space $X$: $$ \operatorname{disc}A\stackrel{\mathrm{def}}=\sup\{\min(\dim R'\colon R'\subset R,\ R'\in\operatorname{Cyc}A)\colon R\in\operatorname{Cyc}A\}, $$ where $\operatorname{Cyc}A=\{R\colon R\ \text{is a~subspace of}~X,\ \dim R+\infty,\ \operatorname{span}(A^nR\colon n\ge0)=X\}$. Always $\operatorname{disc}A\ge\mu_A=$ (the multiplicity of the spectrum of the operator $A$) $\stackrel{\mathrm{def}}=\min(\dim R\colon R\in\operatorname{Cyc}A)$, where (by definition) in each $A$-cyclic subspace there is contained a cyclic subspace of dimension $\le\operatorname{disc}A$. For a linear dynamical system $x(t)=Ax(t)+Bu(t)$ which is controllable, the characteristic $\operatorname{disc}A$ of the evolution operator $A$ shows how much the control space can be diminished without losing controllability. In this paper there are established some general properties of $\operatorname{disc}$ (for example, conditions are given under which $\operatorname{disc}(A\oplus B)=\max(\operatorname{disc}A,\operatorname{disc}B)$; $\operatorname{disc}$ is computed for the following operators: $S$ ($S$ is the shift in the Hardy space $H^2$); $\operatorname{disc}S=2$ (but $\mu_S=1$); $\operatorname{disc}S^*_n=n$ (but $\mu_{S^*_n}=1$) , where $S_n=S\oplus\dots\oplus S$; $\operatorname{disc}S=2$ (but $\mu_S=1$), where $S$ is the bilateral shift. It is proved that for a normal operator $N$ with simple spectrum, $\operatorname{disc}N=\mu_N=1$ $\Longleftrightarrow$ (the operator $N$ is reductive). There are other results also, and also a list of unsolved problems.