On a factorization of operators on finite dimensional Hilbert spaces

Authors: JIAWEI LUO, JUEXIAN LI, GENG TIAN

Abstract: As is well known, for any operator $T$ on a complex separable Hilbert space, $T$ has the polar decomposition $T=U|T|$, where $U$ is a partial isometry and $|T|$ is the nonnegative operator $(T^*T)^{\frac{1}{2}}$. In 2014, Tian et al. proved that on a complex separable infinite dimensional Hilbert space, any operator admits a polar decomposition in a strongly irreducible sense. More precisely, for any operator $T$ and any $\varepsilon>0$, there exists a decomposition $T=(U+K)S$, where $U$ is a partial isometry, $K$ is a compact operator with $||K||<\varepsilon$, and $S$ is strongly irreducible. In this paper, we will answer the question for operators on two-dimensional Hilbert spaces.

Keywords: Polar decomposition, strongly irreducible operator, Jordan block

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