Range-kernel characterizations of operators which are adjoint of each other
Tóm tắt
We provide necessary and sufficient conditions for a pair S, T of Hilbert space operators in order that they satisfy
$$S^*=T$$
and
$$T^*=S$$
. As a main result we establish an improvement of von Neumann’s classical theorem on the positive self-adjointness of
$$S^*S$$
for two variables. We also give some new characterizations of self-adjointness and skew-adjointness of operators, not requiring their symmetry or skew-symmetry, respectively.
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