A refined polar decomposition for $\boldsymbol{J}$-unitary operators

Abstract

In this paper, we will characterize the components of the polar decomposition for an arbitrary $J$-unitary operator in a Hilbert space (where $J$ is a conjugation). This characterization has a quite different structure from that of symmetric and complex skew-symmetric operators. It is also shown that for a $J$-imaginary closed symmetric operator in a Hilbert space there exists a $J$-imaginary self-adjoint extension in a possibly larger Hilbert space (a linear operator $A$ in a Hilbert space $H$ is said to be $J$-imaginary if $f\in D(A)$ implies $Jf\in D(A)$ and $AJf = -JAf$, where $J$ is a conjugation on $H.$) All Hilbert spaces in this paper are assumed to be separable.