### On Tensor Product Characterization of Nuclear Spaces.

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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.

We characterize the reflexivity of the completed projective tensor products $X{\tilde{\otimes}}_{\pi}Y$ of Banach spaces in terms of certain approximative biorthogonal systems.

We observe that a separable Banach space $X$ is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if $\mathcal{L}(X,Y)$ is not reflexive for reflexive $X$ and $Y$ then $\mathcal{L}({X}_{1},Y)$ is is not reflexive for some ${X}_{1}\subset X$, ${X}_{1}$ having a basis.

Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let $X$ and $Y$ be Banach spaces such that $X$ is weakly compactly generated Asplund space and ${X}^{*}$ has the approximation property (respectively $Y$ is weakly compactly generated Asplund space and ${Y}^{*}$ has the approximation property). Suppose that $L(X,Y)\ne K(X,Y)$ and let $1<\lambda <2$. Then $X$ (respectively $Y$) can be equivalently renormed so that any projection $P$ of $L(X,Y)$ onto $K(X,Y)$ has the sup-norm greater or equal to $\lambda $.

In the first part of the paper we prove some new result improving all those already known about the equivalence of the nonexistence of a projection (of any norm) onto the space of compact operators and the containment of ${c}_{0}$ in the same space of compact operators. Then we show several results implying that the space of compact operators is uncomplemented by norm one projections in larger spaces of operators. The paper ends with a list of questions naturally rising from old results and the results...

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