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In this paper we give and prove a criterion for the normality of unboundedclosed operators, which is a sort of a maximality result which will be called"double maximality". As applications, we show, under some assumptions, that thesum of two symmetric operators is essentially self-adjoint; and that the sum oftwo unbounded normal operators is essentially normal. Some other importantresults are also established.

A Criterion for the Normality of Unbounded Operators and Applications to Self-adjointness