Tensor product of left $n$-invertible operators
Volume 215 / 2013
Abstract
A Banach space operator $T\in\mathcal X$ has a left $m$-inverse (resp., an essential left $m$-inverse) for some integer $m\geq 1$ if there exists an operator $S\in\mathcal X$ (resp., an operator $S\in\mathcal X$ and a compact operator $K\in\mathcal X$) such that $\sum_{i=0}^m{(-1)^i\binom mi {S}^{m-i} T^{m-i}}=0$ (resp., $\sum_{i=0}^m{(-1)^i\binom mi {T}^{m-i} S^{m-i}}=K$). If $T_i$ is left $m_i$-invertible (resp., essentially left $m_i$-invertible), then the tensor product $T_1\otimes T_2$ is left $(m_1+m_2-1)$-invertible (resp., essentially left $(m_1+m_2-1)$-invertible). Furthermore, if $T_1$ is strictly left $m$-invertible (resp., strictly essentially left $m$-invertible), then $T_1\otimes T_2$ is: (i) left $(m+n-1)$-invertible (resp., essentially left $(m+n-1)$-invertible) if and only if $T_2$ is left $n$-invertible (resp., essentially left $n$-invertible); (ii) strictly left $(m+n-1)$-invertible (resp., strictly essentially left $(m+n-1)$-invertible) if and only if $T_2$ is strictly left $n$-invertible (resp., strictly essentially left $n$-invertible).
