<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

Publié en ligne: 26 mai 2021

Pages: 158 - 171

Reçu: 24 oct. 2020

Accepté: 07 avr. 2021

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

Mots clés
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

Publié en ligne: 26 mai 2021

Pages: 158 - 171

Reçu: 24 oct. 2020

Accepté: 07 avr. 2021

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

Mots clésoperator 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.

Mots clés
operator monotone functions, operator inequalities, logarithmic operator inequalities, power inequalities