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

<p>For a continuous and positive function <italic>w </italic>(<italic>λ</italic>), <italic>λ > </italic>0 and <italic>µ </italic>a positive measure on (0, <italic><bold>∞</bold></italic>) we consider the following <italic>integral transform</italic>
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2022-0007_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><mstyle><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup><mrow><mi>w</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>λ</mi><mo>+</mo><mi>T</mi></mrow><mo>)</mo></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></mrow></mrow></mstyle><mo>,</mo></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>.</p>
<p>We show among others that, if <italic>B, A > </italic>0, then
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2022-0007_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable><mtr><mtd><mrow><mrow><mo>[</mo><mrow><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></mrow><mo>]</mo></mrow><mrow><mo>(</mo><mrow><mi>B</mi><mo>−</mo><mi>A</mi></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mstyle><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup><mrow><mi>w</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mstyle><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msup><mrow><mrow><mo>[</mo><mrow><mi>λ</mi><mo>+</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>t</mi></mrow><mo>)</mo></mrow><mi>B</mi><mo>+</mo><mi>t</mi><mi>A</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mi>B</mi><mo>−</mo><mi>A</mi></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup><mi>d</mi><mi>t</mi></mrow></mrow></mstyle></mrow><mo>)</mo></mrow><mi>d</mi><mi>μ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>.</mo></mrow></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math>
<tex-math>\matrix{{\left[{\mathcal{D}\left({w,\mu}\right)\left(A\right)-\mathcal{D}\left({w,\mu}\right)\left(B\right)}\right]\left({B-A}\right)}\cr{=\int_0^\infty{w\left(\lambda\right)\left({\int_0^1{{{\left[{\lambda+\left({1-t}\right)B+tA{)^{-1}}\left({B-A}\right)}\right]}^2}dt}}\right)d\mu\left(\lambda\right).}}\cr}</tex-math>
</alternatives>
</disp-formula>
We also provide some sufficient conditions for the operators <italic>A, B > </italic>0 such that the inequality
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2022-0007_eq_003.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>A</mi><mo>)</mo></mrow><mi>B</mi><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><mi>A</mi><mo>≥</mo><mi>A</mi><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>B</mi><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></mrow></math>
<tex-math>\mathcal{D}\left({w,\mu}\right)\left(A\right)B+\mathcal{D}\left({w,\mu}\right)\left(B\right)A{\ge}A\mathcal{D}\left({w,\mu}\right)\left(A\right)+B\mathcal{D}\left({w,\mu}\right)\left(B\right)</tex-math>
</alternatives>
</disp-formula>
holds. Some examples for power and logarithmic functions are also provided.</p>
</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.
We show among others that, if B, A > 0, then



[𝒟(w,μ)(A)−𝒟(w,μ)(B)](B−A)=∫0∞w(λ)(∫01[λ+(1−t)B+tA)−1(B−A)]2dt)dμ(λ).
\matrix{{\left[{\mathcal{D}\left({w,\mu}\right)\left(A\right)-\mathcal{D}\left({w,\mu}\right)\left(B\right)}\right]\left({B-A}\right)}\cr{=\int_0^\infty{w\left(\lambda\right)\left({\int_0^1{{{\left[{\lambda+\left({1-t}\right)B+tA{)^{-1}}\left({B-A}\right)}\right]}^2}dt}}\right)d\mu\left(\lambda\right).}}\cr}


We also provide some sufficient conditions for the operators A, B > 0 such that the inequality



𝒟(w,μ)(A)B+𝒟(w,μ)(B)A≥A𝒟(w,μ)(A)+B𝒟(w,μ)(B)
\mathcal{D}\left({w,\mu}\right)\left(A\right)B+\mathcal{D}\left({w,\mu}\right)\left(B\right)A{\ge}A\mathcal{D}\left({w,\mu}\right)\left(A\right)+B\mathcal{D}\left({w,\mu}\right)\left(B\right)


holds. Some examples for power and logarithmic functions are also provided.

Simple Operator Asynchronicity of an Integral Transform with Applications

College of Engineering & Science, Department of Mathematics

General Mathematics

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

{"article-title":"Simple Operator Asynchronicity of an Integral Transform with Applications"}

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

Simple Operator Asynchronicity of an Integral Transform with Applications

Published Online: Nov 24, 2023

Page range: 97 - 110

Received: Sep 27, 2022

Accepted: Oct 28, 2022

DOI: https://doi.org/10.2478/gm-2022-0007

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

© 2022 Silvestru Sever Dragomir, published by Sciendo

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

Simple Operator Asynchronicity of an Integral Transform with Applications

Published Online: Nov 24, 2023

Page range: 97 - 110

Received: Sep 27, 2022

Accepted: Oct 28, 2022

DOI: https://doi.org/10.2478/gm-2022-0007

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

© 2022 Silvestru Sever Dragomir, published by Sciendo

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

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