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Backward shifts on Banach spaces \(C(X)\). (English) Zbl 0878.47014

A backward shift on a Banach space \(E\) is an operator \(T\) with one-dimensional kernel such that the corresponding operator from the quotient of \(E\) by the kernel of \(T\) is an isometry and so that \(\bigcup_{n\geq 1}\text{Ker }T^n\) is dense in \(E\). The authors settle a conjecture of J. R. Holub (in a stronger form) by proving that for no infinite compact space \(X\) is there a backward shift on \(C(X)\).
Reviewer: J.B.Cooper (Linz)

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46B25 Classical Banach spaces in the general theory
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