<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper we establish some vector inequalities for two operators related to Schwarz and Buzano results. We show amongst others that in a Hilbert space <italic>H</italic> we have the inequality
<disp-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/ausm-2016-0005_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>[</mo><mrow><msup><mrow><mrow><mo>〈</mo><mrow><mfrac><mrow><msup><mrow><mrow><mo>|</mo><mtext>A</mtext><mo>|</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>|</mo><mtext>B</mtext><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mtext>x</mtext><mo>,</mo><mtext>x</mtext></mrow><mo>〉</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mrow><mo>〈</mo><mrow><mfrac><mrow><msup><mrow><mrow><mo>|</mo><mtext>A</mtext><mo>|</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>|</mo><mtext>B</mtext><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mtext>y</mtext><mo>,</mo><mtext>y</mtext></mrow><mo>〉</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mrow><mo>|</mo><mrow><mrow><mo>〈</mo><mrow><mfrac><mrow><msup><mrow><mrow><mo>|</mo><mtext>A</mtext><mo>|</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>|</mo><mtext>B</mtext><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow><mtext>2</mtext></mfrac><mtext>x</mtext><mo>,</mo><mtext>y</mtext></mrow></mrow><mo>〉</mo></mrow></mrow><mo>|</mo></mrow><mo>]</mo></mrow><mo>≥</mo><mrow><mo>|</mo><mrow><mrow><mo>〈</mo><mrow><mi>Re</mi><mo> </mo><mo stretchy="false">(</mo><mtext>B</mtext><mo>*</mo><mtext>A</mtext><mo stretchy="false">)</mo><mo> </mo><mtext>x</mtext><mo>,</mo><mtext>y</mtext></mrow><mo>〉</mo></mrow></mrow><mo>|</mo></mrow></mrow></math><tex-math>$${1 \over 2}\left[ {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{x}},{\rm{x}}} \right\rangle ^{1/2} \left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{y}},{\rm{y}}} \right\rangle ^{1/2} + \left| {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over {\rm{2}}}} {\rm{x}},{\rm{y}}\right\rangle } \right|} \right] \ge \left| {\left\langle {{\mathop{\rm Re}\nolimits} ({\rm{B}}*{\rm{A}})\,{\rm{x}},{\rm{y}}} \right\rangle } \right|$$</tex-math></alternatives></disp-formula>
for <italic>A</italic>, <italic>B</italic> two bounded linear operators on <italic>H</italic> such that R<italic>e</italic> (<italic>B</italic>*<italic>A</italic>) is a nonnegative operator and any vectors <italic>x</italic>, <italic>y</italic> ∈ <italic>H</italic>.</p><p>Applications for norm and numerical radius inequalities are given as well.</p></abstract>

In this paper we establish some vector inequalities for two operators related to Schwarz and Buzano results. We show amongst others that in a Hilbert space H we have the inequality
12[〈|A|2+|B|22x,x〉1/2〈|A|2+|B|22y,y〉1/2+|〈|A|2+|B|22x,y〉|]≥|〈Re (B*A) x,y〉|$${1 \over 2}\left[ {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{x}},{\rm{x}}} \right\rangle ^{1/2} \left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{y}},{\rm{y}}} \right\rangle ^{1/2} + \left| {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over {\rm{2}}}} {\rm{x}},{\rm{y}}\right\rangle } \right|} \right] \ge \left| {\left\langle {{\mathop{\rm Re}\nolimits} ({\rm{B}}*{\rm{A}})\,{\rm{x}},{\rm{y}}} \right\rangle } \right|$$
for A, B two bounded linear operators on H such that Re (B*A) is a nonnegative operator and any vectors x, y ∈ H.Applications for norm and numerical radius inequalities are given as well.

Some vector inequalities for two operators in Hilbert spaces with applications

Mathematics, School of Engineering & Science Victoria University, Australia; School of Computational & Applied Mathematics, University of the Witwatersrand, South Africa

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Some vector inequalities for two operators in Hilbert spaces with applications"}

In this paper we establish some vector inequalities for two operators related to Schwarz and Buzano results. We show amongst others that in a Hilbert space H...

Some vector inequalities for two operators in Hilbert spaces with applications

Published Online: Aug 02, 2016

Page range: 75 - 92

Received: Sep 22, 2015

DOI: https://doi.org/10.1515/ausm-2016-0005

Keywords
inner product spaces, Schwarz’s inequality, Buzano’s inequality, projection, selfadjoint operators, unitary operators, operator norm, numerical radius

© 2016 Sever S. Dragomir, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Some vector inequalities for two operators in Hilbert spaces with applications

Published Online: Aug 02, 2016

Page range: 75 - 92

Received: Sep 22, 2015

DOI: https://doi.org/10.1515/ausm-2016-0005

Keywordsinner product spaces, Schwarz’s inequality, Buzano’s inequality, projection, selfadjoint operators, unitary operators, operator norm, numerical radius

© 2016 Sever S. Dragomir, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
inner product spaces, Schwarz’s inequality, Buzano’s inequality, projection, selfadjoint operators, unitary operators, operator norm, numerical radius