An inequality is proved in abstract separable Hilbert space <mml:math alttext="$H$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math> where <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> and <mml:math alttext="$B$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math> are bounded self-adjoint positive operators defined in <mml:math alttext="$H$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math> such that <mml:math alttext="$R\left( A \right) = R\left( B \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>B</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> and <mml:math alttext="$R\left( A \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is closed.

International Journal of Mathematics and Mathematical Sciences

An operator inequality

Abstract

Copyright