Tensor product of left -invertible operators
Tom 215 / 2013
Streszczenie
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).