On Coarse Isometries and Linear Isometries between Banach Spaces

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></semantics></math></inline-formula> be two Banach spaces and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow></semantics></math></inline-formula> be a standard coarse isometry. In this paper, we first show a sufficient and necessary condition for the coarse left-inverse operator of general Banach spaces to admit a linearly isometric right inverse. Furthermore, by using the well-known simultaneous extension operator, we obtain an asymptotical stability result when <i>Y</i> is a space of continuous functions. In addition, we also prove that every coarse left-inverse operator does admit a linear isometric right inverse without other assumptions when <i>Y</i> is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mi>p</mi></msub><mrow><mo>(</mo><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> space, or both <i>X</i> and <i>Y</i> are finite dimensional spaces of the same dimension. Making use of the results mentioned above, we generalize several results of isometric embeddings and give a stability result of coarse isometries between Banach spaces.