non-closed weak graph limit of symmetric operators

Hi Everyone,

I was recently reading Reed & Simon's functional analysis textbook (the first volume), and it mentions casually on page 294 that weak graph limits of a sequence of symmetric operators are not necessarily closed. However no example (or a reference to one) is given. I've tried to construct one but nothing simple seems to work.

Does anyone know of some such example or at least of a reference where one is discussed?

Thanks

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In my copy (the 1972 edition) it says "It is an open question whether weak graph limits are necessarily closed if each $A_n$ is symmetric". I see in Google Books on-line preview (which is of the 1980 edition) "It is not true that weak graph limits are necessarily closed if each $A_n$ is symmetric". So apparently an example was found (or was brought to the authors' attention) between 1972 and 1980. –  Robert Israel Aug 29 '12 at 8:19