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In this paper we find a formula for the adjoint of bounded pseudo-differential operators on compact Lie groups, which yields a necessary and sufficient condition for self-adjointness of these operators. By a similar method we find necessary and sufficient conditions for bounded pseudo-differential operators on compact Lie groups to be normal or unitary and we give some examples of our results on . Then we show that similar results are true for some bounded pseudo-differential operators on . [ABSTRACT FROM AUTHOR]

The matrix operator defined by the left-hand side of the Bevis–Hall–Hartwig equation(briefly, the BHH operator) is not linear in the space of complex matrices. However, it becomes a linear operator if every matrixis replaced by the pair of real matrices of the same size, namely, by its real and imaginary parts. Introducing an appropriate scalar product in the resulting real space, we find necessary and sufficient conditions for the BHH operator to be normal. [ABSTRACT FROM PUBLISHER]

In this article we prove an existence result, necessary and sufficient conditions for quasiconvex programming problem with a locally starshaped constraint region. Our optimality conditions are different from the usual optimality conditions, in that the subdifferential of the objective function is replaced by a normal cone operator. Such an optimality condition has advantage over the usual one, i.e. it becomes sufficient even when the objective function is only quasiconvex. As a special case we derive the corresponding results for the class of ‘Quasiconvex-quasiaffine’ MPEC which is a class of mathematical programs with complementarity constraints where the objective function is quasiconvex, the inequality constraint is quasiconvex and the rest of constraints are quasiaffine. [ABSTRACT FROM AUTHOR]