# Representations of the weak Calkin algebra

Gonzàlez, Manuel (1994) Representations of the weak Calkin algebra. Accademia Peloritana dei Pericolanti, Classe di Scienze FF. MM. NN., LXXII. pp. 153-169.

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## Abstract

Let $L(E)$ denote the space of all continuous linear operators in a Banch space $E$. For every $T \in \ L(E)$ the operator $R(T) \in \ L(E^*^*/E)$ is defined by $R(T)(X^*^*+E) = T^*^*x^*^*+E (x^*^* \in \ E^*^*)$. The map $R:T \to \ R(T)$ induces a representation of the weak Calkin algebra $L(E)/W(E)$, rhe quotient of $L(E)$ by the ideal $W(E)$ of all weakly compact operators on $E$, in the algebra $L(E^*^*/(E)$. Here we give a survey of the properties of the map R: if it has dense range or closed range, if it is surjective, etc., and describe some applications. We present examples showing that the properties of R can be very different on different spaces E. In some cases the only compact operator in the image of R is the null operator, in the other cases R is surjective, and in the case of $E = \ell \ ^2 (J)$, where J is James' space, we have that $E^*^*/E \cong \ \ell \ ^2$ and the image of R is the class of lattice regular operators on $\ell \ ^2$. Among the applications, we show how to obtain examples of tauberian operators T so that $T^*^*$ is not tauberian, and operators $T \in \ L(E)$ such that R(T) is invertible in $L(E^*^*/E)$ but T fails to be invertible modulo W(E).

Item Type: Article M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Scienze Fisiche, Matematiche e Naturali > 1994M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Scienze Medico-Biologiche > 1994M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Scienze Giuridiche, Economiche e Politiche > 1994M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Lettere, Filosofia e belle Arti > 1994 Dr A F 19 Sep 2012 11:42 20 Sep 2012 08:14 http://cab.unime.it/mus/id/eprint/680