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We can think about a bounded operator $T\colon c_0\to c_0$ as a double-infinite matrix $[T_{mn}]_{m,n\geq 1}$ which acts on a sequence $a=[a_1, a_2, a_3, \ldots ]\in c_0$ in the same way as usual (finite) matrices act on vectors ($n$-tuples of scalars), i.e.

$$ Ta= [T_{mn}][a_n] = \left[ \sum_{n=1}^\infty T_{mn}a_n\right] $$

Suppose $a=[a_1, a_2, a_3, \ldots ]\in \ell^\infty = (c_0)^{**}$. Does the following formula still hold:

$$ T^{**}a= [T_{mn}][a_n] = \left[ \sum_{n=1}^\infty T_{mn}a_n\right] $$

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    Now answered at MO: http://mathoverflow.net/questions/83977/second-conjugate-operators-to-operators-on-c-02011-12-21
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    B. Salkas: could you post this as an answer and accept it so that the question can be considered closed, please?2011-12-21
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    @t.b. I think there may be a system-imposed time limit on how soon one can post/accept one's own answer to a question.2011-12-21

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