<abstract xmlns="http://www.w3.org/1999/xhtml">

For a continuous and positive function <italic>w</italic> (<italic>λ</italic>) , λ <italic>></italic> 0 and <italic>µ</italic> a positive measure on (0, ∞) we consider the following <italic>integral transform</italic> 
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0008_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mrow><msubsup><mo>∫</mo><mn>0</mn><mo>∞</mo></msubsup><mrow><mi>w</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>λ</mi><mo>+</mo><mi>T</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>d</mi><mi>μ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>,</mo></mrow></mrow></mrow></math>
<tex-math>\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),}</tex-math>
</alternatives>
</disp-formula>
where the integral is assumed to exist for <italic>T</italic> a positive operator on a complex Hilbert space <italic>H</italic>.
Assume that <italic>A ≥ α ></italic> 0, <italic>δ ≥ B ></italic> 0 and 0 <italic>&lt; m ≤ B − A ≤ M</italic> for some constants <italic>α, δ, m, M.</italic> Then 
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0008_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mn>0</mn><mo>≤</mo><mo>-</mo><mi>m</mi><msup><mi>𝒟</mi><mo>′</mo></msup><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>δ</mi><mo>)</mo></mrow><mo>≤</mo><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>-</mo><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>≤</mo><mo>-</mo><mi>M</mi><msup><mi>𝒟</mi><mo>′</mo></msup><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>,</mo></mrow></math>
<tex-math>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),</tex-math>
</alternatives>
</disp-formula>
where <italic>D</italic>′(<italic>w, µ</italic>) (<italic>t</italic>) is the derivative of <italic>D</italic>(<italic>w, µ</italic>) (<italic>t</italic>) as a function of <italic>t ></italic> 0.
If <italic>f</italic> : [0, <italic>∞</italic>) → ℝ is operator monotone on [0, <italic>∞</italic>) with <italic>f</italic> (0) = 0, then 
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0008_eq_003.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable><mtr><mtd><mrow><mn>0</mn><mo>≤</mo><mfrac><mi>m</mi><mrow><msup><mrow><mi>δ</mi></mrow><mn>2</mn></msup></mrow></mfrac><mrow><mo>[</mo> <mrow><mi>f</mi><mrow><mo>(</mo><mi>δ</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>δ</mi><mo>)</mo></mrow><mi>δ</mi><mo>≤</mo><mi>f</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><msup><mrow><mi>A</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mi>f</mi><msup><mrow><mrow><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi>B</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow> <mo>]</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>≤</mo><mfrac><mi>M</mi><mrow><msup><mrow><mi>α</mi></mrow><mn>2</mn></msup></mrow></mfrac><mrow><mo>[</mo> <mrow><mi>f</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mi>α</mi></mrow> <mo>]</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math>
<tex-math>\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 }</tex-math>
</alternatives>
</disp-formula>

Some examples for operator convex functions as well as for integral transforms <italic>D</italic> (<italic>·, ·</italic>) related to the exponential and logarithmic functions are also provided.
</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)-1dμ(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≤-m𝒟′(w,μ)(δ)≤𝒟(w,μ)(A)-𝒟(w,μ)(B)≤-M𝒟′(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≤mδ2[ f(δ)-f′(δ)δ≤f(A)A-1-f(B)B-1 ]≤Mα2[ 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.

Gradient Inequalities for an Integral Transform of Positive Operators in Hilbert Spaces

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

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)-1dμ(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  0.
If f : [0, ∞) → ℝ is operator monotone on [0, ∞) with f (0) = 0, then 



0≤mδ2[ f(δ)-f′(δ)δ≤f(A)A-1-f(B)B-1 ]≤Mα2[ 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.

For a continuous and positive function w (λ) , λ > 0 and µ a positive measure on (0, ∞) we consider the following integral transform...

Gradient Inequalities for an Integral Transform of Positive Operators in Hilbert Spaces

Published Online: Jul 26, 2023

Page range: 248 - 265

Received: Sep 27, 2022

Accepted: Jun 05, 2023

DOI: https://doi.org/10.2478/amsil-2023-0008

Keywords
operator monotone functions, operator convex functions, operator inequalities, Löwner–Heinz inequality, logarithmic operator inequalities

© 2023 Silvestru Sever Dragomir, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Gradient Inequalities for an Integral Transform of Positive Operators in Hilbert Spaces

Published Online: Jul 26, 2023

Page range: 248 - 265

Received: Sep 27, 2022

Accepted: Jun 05, 2023

DOI: https://doi.org/10.2478/amsil-2023-0008

Keywordsoperator monotone functions, operator convex functions, operator inequalities, Löwner–Heinz inequality, logarithmic operator inequalities

© 2023 Silvestru Sever Dragomir, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
operator monotone functions, operator convex functions, operator inequalities, Löwner–Heinz inequality, logarithmic operator inequalities