$q$-Difference Operator on $L_q^{2}( 0, + \infty )$

In this research, the minimal and maximal operators defined by $q$- difference expression are given in the Hilbert space $L_q^{2}( 0, + \infty )$. The existence problem of a $q^{-1}$-normal extension for the minimal operator is mentioned. In addition, the sets of the minimal operator spectrum and the maximal operator spectrum are examined.

Keywords:

$q$- difference operator, minimal operator, maximal operator, $q$-formally normal operator $q$-normal operator, spectrum,

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