We suggest a method of renorming of spaces of operators which are suitably approximable by sequences of operators from a given class. Further we generalize J. Johnsons’s construction of ideals of compact operators in the space of bounded operators and observe e.g. that under our renormings compact operators are $u$-ideals in the: space of 2-absolutely summing operators or in the space of operators factorable through a Hilbert space.

Let $X$ be a Banach space. We give characterizations of when $ℱ(Y,X)$ is a $u$-ideal in $𝒲(Y,X)$ for every Banach space $Y$ in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when $ℱ(X,Y)$ is a $u$-ideal in $𝒲(X,Y)$ for every Banach space $Y$, when $ℱ(Y,X)$ is a $u$-ideal in $𝒲(Y,X^{**})$ for every Banach space $Y$, and when $ℱ(Y,X)$ is a $u$-ideal in $𝒦(Y,X^{**})$ for every Banach space $Y$.

We investigate sequences and operators via the unconditionally p-summable sequences. We characterize the unconditionally p-null sequences in terms of a certain tensor product and then prove that, for every 1 ≤ p < ∞, a subset of a Banach space is relatively unconditionally p-compact if and only if it is contained in the closed convex hull of an unconditionally p-null sequence.