On 𝐿¹ isomorphisms

Cambern, Michael

doi:10.1090/S0002-9939-1980-0550500-6

On $L^{1}$ isomorphisms
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by Michael Cambern

Proc. Amer. Math. Soc. 78 (1980), 227-228

DOI: https://doi.org/10.1090/S0002-9939-1980-0550500-6

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Abstract:

Let $({X_1},{\Sigma _1},{\mu _1})$ and $({X_2},{\Sigma _2},{\mu _2})$ be two $\sigma$-finite measure spaces. We show that any isomorphism T of the Banach space ${L^1}({X_1},{\Sigma _1},{\mu _1})$ onto the Banach space ${L^1}({X_2},{\Sigma _2},{\mu _2})$ which satisfies $\left \| T \right \|\;\left \| {{T^{ - 1}}} \right \| < 2$ induces a transformation of the underlying measure spaces.

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