Messanae Universitas Studiorum

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
    Subjects: M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Scienze Fisiche, Matematiche e Naturali > 1994
    M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Scienze Medico-Biologiche > 1994
    M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Scienze Giuridiche, Economiche e Politiche > 1994
    M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Lettere, Filosofia e belle Arti > 1994
    Divisions: UNSPECIFIED
    Depositing User: Dr A F
    Date Deposited: 19 Sep 2012 13:42
    Last Modified: 20 Sep 2012 10:14
    URI: http://cab.unime.it/mus/id/eprint/680

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