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2 votes

"Completeness" for weak convergence of unbounded closed operators on a separable Hilbert space $H$

Not always. What you are talking about is also called "convergence in the sense of sesquilinear forms", because you are taking a pointwise limit in $D\times D$ of a sequence of sesquilinear ...
Pedro Lauridsen Ribeiro's user avatar
2 votes

"Completeness" for weak convergence of unbounded closed operators on a separable Hilbert space $H$

Even if you can define $T$ by this limit, this operator need not be closed. Take for example $H=\ell^2$ and $T_n$ as multiplication by $(0,\ldots, 0, n, n+1,\ldots)$ on its natural domain $D$. Note ...
Christian Remling's user avatar

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