Contents 
This is a joint work with Prof. Ond\v{r}ej Do\v{s}l\'y.
We study the socalled $p$critical difference operators, i.e., the nonnegative operators that can be turned
to negative ones by small (in a certain sense) negative perturbations.
This research is motivated by [2], where the concept of critical operators is introduced for difference
operators of the second order. We have generalized this concept in [1]
for $2n$order SturmLiouville difference operators, and we have suggested a criterium of $p$criticality
for one term $2n$order difference operators $l(y)_k = \Delta^n (r_k \Delta^n y_k)$.
This criterium turned out to be valid and we have proved it
using a structure of the solution space of the equation $l(y)_k = 0$, see [3].
[1] O.~Do\v{s}l\'{y}, P.~Hasil,
\emph{Critical higher order SturmLiouville difference operators},
J. Differ. Equ. Appl., to appear.
[2] F.~Gesztesy, Z.~Zhao,
\emph{Critical and subcritical Jacobi operators defined as Friedrichs extensions},
J. Differential Equations \textbf{103} (1993), 6893.
[3] P.~Hasil,
\emph{Criterium of pcriticality for one term 2norder difference operators},
submitted. 
