Affiliation:
1. 1 Mathematics, College of Engineering & Science , Victoria University , PO Box 14428, Melbourne City, MC 800 , Australia and DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science & Applied Mathematics , University of the Witwatersrand , Johannesburg , South Africa
Abstract
Abstract
For a continuous and positive function w (λ) , λ > 0 and µ a positive measure on (0, ∞) we consider the following integral transform
𝒟
(
w
,
μ
)
(
T
)
:
=
∫
0
∞
w
(
λ
)
(
λ
+
T
)
-
1
d
μ
(
λ
)
,
\mathcal{D}\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right){{\left( {\lambda + T} \right)}^{ - 1}}d\mu \left( \lambda \right),}
where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.
Assume that A ≥ α > 0, δ ≥ B > 0 and 0 < m ≤ B − A ≤ M for some constants α, δ, m, M. Then
0
≤
-
m
𝒟
′
(
w
,
μ
)
(
δ
)
≤
𝒟
(
w
,
μ
)
(
A
)
-
𝒟
(
w
,
μ
)
(
B
)
≤
-
M
𝒟
′
(
w
,
μ
)
(
α
)
,
0 \le - m\mathcal{D}'\left( {w,\mu } \right)\left( \delta \right) \le \mathcal{D}\left( {w,\mu } \right)\left( A \right) - \mathcal{D}\left( {w,\mu } \right)\left( B \right) \le - M\mathcal{D}'\left( {w,\mu } \right)\left( \alpha \right),
where D
′(w, µ) (t) is the derivative of D(w, µ) (t) as a function of t > 0.
If f : [0, ∞) → ℝ is operator monotone on [0, ∞) with f (0) = 0, then
0
≤
m
δ
2
[
f
(
δ
)
-
f
′
(
δ
)
δ
≤
f
(
A
)
A
-
1
-
f
(
B
)
B
-
1
]
≤
M
α
2
[
f
(
α
)
-
f
′
(
α
)
α
]
.
\matrix{ {0 \le {m \over {{\delta ^2}}}\left[ {f\left( \delta \right) - f'\left( \delta \right)\delta \le f\left( A \right){A^{ - 1}} - f{{\left( B \right)}^{B - 1}}} \right]} \cr { \le {M \over {{\alpha ^2}}}\left[ {f\left( \alpha \right) - f'\left( \alpha \right)\alpha } \right].} \cr }
Some examples for operator convex functions as well as for integral transforms D (·, ·) related to the exponential and logarithmic functions are also provided.