Q-difference operator and q^(-1)-hyponormality

In this thesis, the minimal and maximal operators generated by q-derivative operator expression, the adjoints of the minimal and maximal operators and the spectral problems of these operators are dealed on the Hilbert spaces L_q^2 (0,1),L_q^2 (0,+∞) ve L_q^2 (R). In addition, the q^(-1)-formal normality and q^(-1)-hyponormality of the minimal operator defined by the q-derivative operator expression are examined and the problem apart from the q^(-1)-normal extensions is also discussed in these Hilbert spaces.