A Banach space on which there are few operators

Abstract

It is shown that there is a nonseparable Banach space on which every operator has the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S plus rho upper I"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>+</mml:mo> <mml:mi> ρ </mml:mi> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">S + \rho I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi> ρ </mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a scalar and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an operator with separable range.