Studies on the Deficiency Indices of Product Differential Operators in Direct Sum Spaces

Abstract

In this chapter, we explained the deficiency index problem for the product differential operators which are generated by a general ordinary quasi-differential expressions \(\tau_1\), \(\tau_2\), ..., \(\tau_n\) each of order n with complex coefficients in the direct sum \(\oplus^N_{P=1}\)\(L^2_w (I_p)\) of spaces of functions defined on each of the separate intervals in the circumstances of regular and singular end-points. The domains of these operators are described in terms of boundary conditions featuring \(L^2_w\)-solutions of the differential equations. These findings are more extensive of those of formally symmetric expression \(\tau\) studied in [1 - 8], and those of general quasi-differential expressions \(\tau\) in [9, 10, 11].