On symmetrizable operators of which some iteration satisfies a positive definite condition

Considered are linear (in general, unbounded) operators $A$, defined on a set $R$ which is dense in the Hilbert Space $X$, which are symmetrizable by a symmetric operator $H$ in $R$. Under the condition that there exists an integer $p\ge0$ for which $(HA^px,x)\ge0$ for any $x\in R$, the spectral properties of the operator $A$ and the solutions of the equation $x-\lambda Ax=y,~x,y\in R$ are investigated. The results obtained are applied to investigating some boundary-value problems for differential equations.