Some alternative definitions for the "plus-minus" interpolation spaces $\left\langle A_{0},A_{1}\right\rangle _{\theta}$ of Jaak Peetre

TL;DR

This paper proves the equivalence of different definitions of Peetre's plus-minus interpolation spaces using a new continuous integral approach, clarifying their relation to Calderon complex interpolation spaces.

Contribution

It provides an explicit proof that various discrete definitions of the plus-minus spaces are equivalent through a new continuous integral formulation.

Findings

Different discrete definitions are equivalent via a continuous integral approach.

The continuous definition may clarify connections to Calderon complex interpolation.

Results could extend to Gustavsson-Peetre variants.

Abstract

The Peetre "plus-minus" interpolation spaces $\left\langle A_{0},A_{1}\right\rangle _{\theta}$ are defined variously via conditions about the unconditional convergence of certain Banach space valued series whose terms have coefficients which are powers of 2 or, alternatively, powers of $e$. It may seem intuitively obvious that using powers of 2, or of $e$, or powers of some other constant number greater than 1 in such definitions should produce the same space to within equivalence of norms. To allay any doubts, we here offer an explicit proof of this fact, via a "continuous" definition of the same spaces where integrals replace the above mentioned series. This apparently new definition, which is also in some sense a "limiting case" of the above mentioned "discrete" definitions, may be relevant in the study of the connection between the Peetre "plus-minus" interpolation spaces and Calderon complex interpolation spaces when both the spaces of the underlying couple are are Banach lattices on the same measure space. Related results can probably be obtained for the Gustavsson-Peetre variant of the "plus-minus" spaces.