Let $H$ be a Hilbert space and $B(H)$ denotes the space of all continuous linear operators on $H$. I am looking for a class/example of bounded linear functionals $B(H)\to \mathbb C$ which cannot be reduced to the type $$T\mapsto\sum\_{i=1}^\infty k\_i\left<Tx\_i,y\_i\right>$$ for some fixed $x\_i$'s and $y\_i$'s in $H$ and $k\_i$'s in $\mathbb C$. Do we have any explicit examples?



Example of linear functionals on $B(H)$
So, you are asking about non-normal functionals on $B(H)$. This is very similar to the question of what are the functionals on $\ell\_\infty$ that are not in $\ell\_1$?


Fix an ultrafilter $U$ on $\mathbb N$ and let $(e\_n)$ be an ONB for $H$. Define


$$\langle \phi, T\rangle = \lim\_{n,U} \langle Te\_n, e\_n\rangle\quad (T\in B(H)).$$


Then this functional is not normal.


