ACTA issues

On operators which are adjoint to each other

Dan Popovici, Zoltán Sebestyén

Acta Sci. Math. (Szeged) 80:1-2(2014), 175-194

Abstract. Given two linear operators $S$ and $T$ acting between Hilbert spaces $\mathscr{H}$ and $\mathscr{K}$, respectively $\mathscr{K}$ and $\mathscr{H}$, which satisfy the relation $ \langle Sh, k\rangle =\langle h, Tk\rangle, \quad h\in\dom S, k\in\dom T, $ i.e., according to the classical terminology of M.$ $H. Stone, which are adjoint to each other, we provide necessary and sufficient conditions in order to ensure the equality between the closure of $S$ and the adjoint of $T$. A central role in our approach is played by the range of the operator matrix $M_{S, T}=\left({1_{\dom S} -T\atop S 1_{\dom T}}\right )$. We obtain, as consequences, several results characterizing skewadjointness, selfadjointness and essential selfadjointness. We improve, in particular, the celebrated selfadjointness criterion of J. von Neumann.

DOI: 10.14232/actasm-012-857-7

AMS Subject Classification (1991): 47A05, 47A20, 47B25

Keyword(s): unbounded operator, operators which are adjoint to each other, symmetric, skewadjoint, selfadjoint, essentially selfadjoint, closable

Received November 30, 2012, and in final form June 3, 2013. (Registered under 107/2012.)