The generalized Fredholm operators

Abstract

Let <italic>X, Y</italic> be Banach spaces over either the real field or the complex field. A continuous linear operator will be called a <italic>generalized Fredholm operator</italic> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is closed in <italic>Y</italic> , and Ker <italic>T</italic> and Coker <italic>T</italic> are reflexive Banach spaces. A theory similar to the classical Fredholm theory exists for the generalized Fredholm operators; and the similarity brings out the correspondence: Reflexive Banach spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-right-arrow"> <mml:semantics> <mml:mo stretchy="false"> ↔ </mml:mo> <mml:annotation encoding="application/x-tex">\leftrightarrow</mml:annotation> </mml:semantics> </mml:math> </inline-formula> finite-dimensional spaces, weakly compact operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-right-arrow"> <mml:semantics> <mml:mo stretchy="false"> ↔ </mml:mo> <mml:annotation encoding="application/x-tex">\leftrightarrow</mml:annotation> </mml:semantics> </mml:math> </inline-formula> compact operators, generalized Fredholm operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-right-arrow"> <mml:semantics> <mml:mo stretchy="false"> ↔ </mml:mo> <mml:annotation encoding="application/x-tex">\leftrightarrow</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Fredholm operators, Tauberian operators with closed range <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-right-arrow"> <mml:semantics> <mml:mo stretchy="false"> ↔ </mml:mo> <mml:annotation encoding="application/x-tex">\leftrightarrow</mml:annotation> </mml:semantics> </mml:math> </inline-formula> semi-Fredholm operators.