A REFINED POLAR DECOMPOSITION FOR J-UNITARY OPERATORS

• Published in: Sarajevo journal of mathematics Vol. 11; no. 1; pp. 65 - 72
• Main Author: Zagorodnyuk, Sergey M.
• Format: Journal Article
• Language: English
• Published: 16.05.2025
• ISSN: 1840-0655; 2233-1964
• DOI: 10.5644/SJM.11.1.05

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.