Interpolator Symmetries and New Kalton-Peck Spaces

Abstract

AbstractWe study the six diagrams generated by the first three Schechter interpolators $$\Delta _2(f)= f''(1/2)/2!, \Delta _1(f)= f'(1/2), \Delta _0(f)=f(1/2)$$ Δ 2 ( f ) = f ′ ′ ( 1 / 2 ) / 2 ! , Δ 1 ( f ) = f ′ ( 1 / 2 ) , Δ 0 ( f ) = f ( 1 / 2 ) acting on the Calderón space associated to the pair $$(\ell _\infty , \ell _1)$$ ( ℓ ∞ , ℓ 1 ) . We will study the remarkable and somehow unexpected properties of all the spaces appearing in those diagrams: two new spaces (and their duals), two Orlicz spaces (and their duals) in addition to the third order Rochberg space, the standard Kalton-Peck space $$Z_2$$ Z 2 and, of course, the Hilbert space $$\ell _2$$ ℓ 2 . We will also deal with a nice test case: that of weighted $$\ell _2$$ ℓ 2 spaces, in which case all involved spaces are Hilbert spaces.