Positivity 2: 115–122, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
On an Inequality of A. Grothendieck Concerning
Operators on L
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA
(Received: 24 August 1997; accepted in revised form: 12 January 1998)
Abstract. In 1955, A. Grothendieck proved a basic inequality which shows that any bounded linear
operator between L
(µ)-spaces maps (Lebesgue-) dominated sequences to dominated sequences. An
elementary proof of this inequality is obtained via a new decomposition principle for the lattice of
Mathematics Subject Classiﬁcations (1991): 46E30
Key words: lattice of measurable functions, dominated sequences, decomposition principle
Let µ, ν be measures on measurable spaces, and let T : L
(µ) → L
(ν) be a
bounded linear operator (here L
(µ) denotes the real or complex Banach space of
(equivalence classes of) µ-integrable functions). In , (see Corollaire, page 67)
Grothendieck establishes the following fundamental inequality:
| dµ .
We ﬁrst give some motivation for the inequality, then give a proof involving an
apparently new principle concerning the lattice of measurable functions.
It follows easily from (1) that every such operator maps dominated (or order
bounded) sequences into dominated sequences. In fact, it follows that
if F is a family in L
(µ) for which there exists a µ-integrable ϕ with
|f |≤ϕa.e. for all f in F , then there exists a non-negative ν-integrable
ϕdµso that |Tf |≤ψa.e. for all f in F .
This consequence of (1) (which is of course equivalent to (1)) is drawn explicitly
by Grothendieck in  (see Proposition 10, page 66).
In the summer of 1979, during her research visit to the University of Texas at
Austin, I suggested to Mireille Lévy that the inequality (1) might actually char-
acterize those operators from a subspace of L
(µ) to L
(ν), which extend to an
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