Tensor product of left n-invertible operators

B. P. Duggal ; Vladimir Müller

Studia Mathematica, Tome 215 (2013), p. 113-125 / Harvested from The Polish Digital Mathematics Library

Résumé

A Banach space operator T ∈ has a left m-inverse (resp., an essential left m-inverse) for some integer m ≥ 1 if there exists an operator S ∈ (resp., an operator S ∈ and a compact operator K ∈ ) such that $∑_{i=0}^{m} (-1)^{i} \binom{m}{i} S^{m-i} T^{m-i} = 0$ (resp., $∑_{i=0}^{m} (-1)^{i} \binom{m}{i} T^{m-i} S^{m-i} = K$). If $T_{i}$ is left $m_{i}$-invertible (resp., essentially left $m_{i}$-invertible), then the tensor product T₁ ⊗ T₂ is left (m₁ + m₂-1)-invertible (resp., essentially left (m₁ + m₂-1)-invertible). Furthermore, if T₁ is strictly left m-invertible (resp., strictly essentially left m-invertible), then T₁ ⊗ T₂ is: (i) left (m + n - 1)-invertible (resp., essentially left (m + n - 1)-invertible) if and only if T₂ 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 T₂ is strictly left n-invertible (resp., strictly essentially left n-invertible).

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Publié le : 2013-01-01

Zbl 1273.47037

EUDML-ID : urn:eudml:doc:285583