Integral representations of rpow
#
This file contains an integral representation of the rpow
function between 0 and 1: we show
that there exists a measure on ℝ such that x ^ p = ∫ t, rpowIntegrand₀₁ p t x ∂μ
for
the integrand rpowIntegrand₀₁ p t x := t ^ p * (t⁻¹ - (t + x)⁻¹)
.
This representation is useful for showing that rpow
is operator monotone and operator concave
in this range; that is, cfc rpow
is monotone/concave. The integrand can be shown to be
operator monotone and concave through direct means, and this integral lifts these properties
to rpow
.
Notes #
Here we only compute the integral up to a constant, even though the actual constant can be computed via contour integration. We chose to avoid this, as the constant is seldom if ever relevant in applications, and would needlessly complicate the proof.
Main declarations #
rpowIntegrand₀₁ p t x := t ^ p * (t⁻¹ - (t + x)⁻¹)
exists_measure_rpow_eq_integral
: there exists a measure onℝ
such thatx ^ p = ∫ t, rpowIntegrand₀₁ p t x ∂μ
CFC.exists_measure_nnrpow_eq_integral_cfcₙ_rpowIntegrand₀₁
: the corresponding statement wherex ^ p
is defined via the CFC.CFC.monotone_nnrpow
,CFC.monotone_rpow
:a ↦ a ^ p
is operator monotone forp ∈ [0,1]
CFC.monotone_sqrt
:CFC.sqrt
is operator monotone
TODO #
- Show operator concavity of
rpow
overIcc 0 1
- Give analogous representations for the ranges
Ioo (-1) 0
andIoo 1 2
.
References #
- [Car10] Eric A. Carlen, "Trace inequalities and quantum entropies: An introductory course" (see Lemma 2.8)
The integral representation of the function x ↦ x ^ p
(where p ∈ (0, 1)
) .
The integral representation of the function x ↦ x ^ p
(where p ∈ (0, 1)
).
rpowIntegrand₀₁ p t
is operator monotone for all p ∈ Ioo 0 1
and all t ∈ Ioi 0
.
a ↦ a ^ p
is operator monotone for p ∈ [0,1]
.
CFC.sqrt
is operator monotone.
a ↦ a ^ p
is operator monotone for p ∈ [0,1]
.