The self-adjointness conditions for a higher order differential operator with an operator coefficient

M. G. Gimadislamov

Matematičeskie zametki, Tome 5 (1969) no. 6, pp. 697-707

Résumé

Certain sufficient conditions are found for self-adjointness of the differential operator generated by the expressionl $$ l(y)=(-1)^ny^{2n}+Q(x)y, \quad -\infty<x<\infty, $$ where $Q(x)$ is for each fixed value of $x$ a bounded self-adjoint operator acting from the Hilbert space $H$ into $H$, and $y(x)$ is a vector function of $H_1$ for which $$ \int_{-\infty}^\infty\|y\|_H^2\,dx<\infty. $$