<abstract xmlns="http://www.w3.org/1999/xhtml">For a continuous and positive function <italic>ω</italic> (<italic>λ</italic>); <italic>λ</italic>> 0 and <italic>μ</italic> a positive measure on [0; ∞) we consider the following 𝒟-<italic>logarithmic integral transform</italic><disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2021-0004_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>𝒟</mi><mi>ℒ</mi><mi>o</mi><mi>g</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><mn>1</mn><mtext>n</mtext><mrow><mo>(</mo><mrow><mfrac><mrow><mi>λ</mi><mo>+</mo><mi>T</mi></mrow><mi>λ</mi></mfrac></mrow><mo>)</mo></mrow><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}\mathcal{L}og\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right)1{\rm{n}}\left( {{{\lambda + T} \over \lambda }} \right)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>.We show among others that, if <italic>A</italic>, <italic>B ></italic> 0 with <italic>BA</italic> + <italic>AB</italic> ≥ 0, then
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2021-0004_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>𝒟</mi><mi>ℒ</mi><mi>o</mi><mi>g</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><mi>ℒ</mi><mi>o</mi><mi>g</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><mi>𝒟</mi><mi>ℒ</mi><mi>o</mi><mi>g</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math><tex-math>\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( A \right) + \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( B \right) \ge \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( {A + B} \right).</tex-math></alternatives></disp-formula>In particular we have
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2021-0004_eq_003.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mrow><mi>π</mi></mrow><mn>2</mn></msup><mo>+</mo><mtext>di</mtext><mo>log</mo><mrow><mo>(</mo><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><mo>)</mo></mrow><mo>≥</mo><mtext>di</mtext><mo>log</mo><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>+</mo><mtext>di</mtext><mo>log</mo><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>,</mo></mrow></math><tex-math>{1 \over 6}{\pi ^2} + {\rm{di}}\log \left( {A + B} \right) \ge {\rm{di}}\log \left( A \right) + {\rm{di}}\log \left( B \right),</tex-math></alternatives></disp-formula>
where the <italic>dilogarithmic function</italic> dilog : [0; ∞) → ℝ is defined by
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2021-0004_eq_004.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtext>di</mtext><mo>log</mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mrow><msubsup><mo>∫</mo><mn>1</mn><mi>t</mi></msubsup><mrow><mfrac><mrow><mn>1</mn><mi>n</mi><mi>s</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>s</mi></mrow></mfrac><mi>d</mi><mi>s</mi><mo>,</mo></mrow></mrow><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>t</mi><mo>≥</mo><mn>0.</mn></mrow></math><tex-math>{\rm{di}}\log \left( t \right): = \int_1^t {{{1ns} \over {1 - s}}ds,} \,\,\,\,t \ge 0.</tex-math></alternatives></disp-formula>Some examples for integral transform 𝒟𝒧<italic>og</italic> (˙;˙) related to the operator monotone functions are also provided.</abstract>

For a continuous and positive function ω (λ); λ> 0 and μ a positive measure on [0; ∞) we consider the following 𝒟-logarithmic integral transform𝒟ℒog(w,μ)(T):=∫0∞w(λ)1n(λ+Tλ)dμ(λ),\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( T \right): = \int_0^\infty  {w\left( \lambda  \right)1{\rm{n}}\left( {{{\lambda  + T} \over \lambda }} \right)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 A, B > 0 with BA + AB ≥ 0, then
𝒟ℒog(w,μ)(A)+𝒟ℒog(w,μ)(B)≥𝒟ℒog(w,μ)(A+B).\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( A \right) + \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( B \right) \ge \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( {A + B} \right).In particular we have
16π2+dilog(A+B)≥dilog(A)+dilog(B),{1 \over 6}{\pi ^2} + {\rm{di}}\log \left( {A + B} \right) \ge {\rm{di}}\log \left( A \right) + {\rm{di}}\log \left( B \right),
where the dilogarithmic function dilog : [0; ∞) → ℝ is defined by
dilog(t):=∫1t1ns1-sds,    t≥0.{\rm{di}}\log \left( t \right): = \int_1^t {{{1ns} \over {1 - s}}ds,} \,\,\,\,t \ge 0.Some examples for integral transform 𝒟𝒧og (˙;˙) related to the operator monotone functions are also provided.

Operator Subadditivity of the 𝒟-Logarithmic Integral Transform for Positive Operators in Hilbert Spaces

Mathematics, College of Engineering & Science

and DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science & Applied Mathematics

Annales Mathematicae Silesianae

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

For a continuous and positive function ω (λ); λ> 0 and μ a positive measure on [0; ∞) we consider the following 𝒟-logarithmic integral...

Operator Subadditivity of the 𝒟-Logarithmic Integral Transform for Positive Operators in Hilbert Spaces

Published Online: May 26, 2021

Page range: 158 - 171

Received: Oct 24, 2020

Accepted: Apr 07, 2021

DOI: https://doi.org/10.2478/amsil-2021-0004

Keywords
operator monotone functions, operator inequalities, logarithmic operator inequalities, power inequalities

© 2021 Silvestru Sever Dragomir, published by Sciendo

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

Operator Subadditivity of the 𝒟-Logarithmic Integral Transform for Positive Operators in Hilbert Spaces

Published Online: May 26, 2021

Page range: 158 - 171

Received: Oct 24, 2020

Accepted: Apr 07, 2021

DOI: https://doi.org/10.2478/amsil-2021-0004

Keywordsoperator monotone functions, operator inequalities, logarithmic operator inequalities, power inequalities

© 2021 Silvestru Sever Dragomir, published by Sciendo

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

Keywords
operator monotone functions, operator inequalities, logarithmic operator inequalities, power inequalities