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On the adjoint of Hilbert space operators (1711.08391v1)

Published 22 Nov 2017 in math.FA

Abstract: In general, it is a non trivial task to determine the adjoint $S*$ of an unbounded operator $S$ acting between two Hilbert spaces. We provide necessary and sufficient conditions for a given operator $T$ to be identical with $S*$. In our considerations, a central role is played by the operator matrix $M_{S,T}=\left(\begin{array}{cc} I & -T\ S & I\end{array}\right)$. Our approach has several consequences such as characterizations of closed, normal, skew- and selfadjoint, unitary and orthogonal projection operators in real or complex Hilbert spaces. We also give a self-contained proof of the fact that $T*T$ always has a positive selfadjoint extension.

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