If $H$, is a seperable Hilberspace, $E$ seperable Banach space. $(h_n)$ orthonormal basis of $H$. Let $T\in B(H,E)$.
If we have the condition on $T$ that $$\sum_k \left\|Th_k\right\|^p <\infty,$$ then what do we call $T$? If we had $T:H\to H$ then it would be some sort of a trace class operator. I thought this operator would be called a p-summing operator. But I found that the definition of $p$-summing is actually different. I am mainly interested in $p=1$.
Also can anyone provide me examples of ( non-finite dimensional ) spaces $H,E$ and an operator $T$ where such condition holds? I know for $p=1$ an example $T:l^2\to l^q$ where $T((x(n)_n) = (\alpha(n)x(n))_n $ is a multiplication operator with an element $(\alpha(n))_n \in l^q$, and $(h_n)$ the standard basis of $l^2$.
I would be very grateful for examples.