Ukrainian Mathematical Bulletin
Volume 1, Issue 2, 2004 pp. 177-219.
On Cyclic Subspaces and Unicellularity of the Operator $(Vf)(x)=q(x)\int\limits_0^xw(t)f(t)dt$Authors: I.Yu.Domanov
Author institution: Institute of Applied Mathematics and Mechanics of the Ukrainian National Academy of Sciences
Summary: Let $q(x)\in L_p[0,1]$, let $w(x)\in L_{p'}[0,1]$, and let $\overline{q(x)w(x)}=q(x)w(x)\ne 0$ for almost all $x\in [0,1]$. We describe the cyclic sub\-spaces, spectral multiplicity and disc-characteristic of the operator $(V_{q,w}f)(x)=q(x)\int_0^xf(t)w(t)dt$ and its natural powers in the space $L_p[0,1]$. It is also shown that the unicellularity of the operator $V_{q,w}^n$ is equivalent to its quasisimilarity to the operator $cJ^n:=cV^n_{1,1}$ and to the condition sign\,$ q(x)w(x)=const$ almost everywhere on [0,1].
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