DIVERSITY OF LORENTZ-ZYGMUND SPACES OF OPERATORS DEFINED BY APPROXIMATION NUMBERS

We prove the following dichotomy for the spaces L-p,q,alpha((a)) (X, Y) of all operators T is an element of L(X, Y) whose approximation numbers belong to the Lorentz-Zygmund sequence spaces l(p,q)(log l)(alpha): If X and Y are infinite-dimensional Banach spaces, then the spaces L-p,q,alpha((a))(X, Y) with 0 < p < infinity, 0 < q <= infinity and alpha is an element of R are all different from each other, but otherwise, if X or Y are finite-dimensional, they are all equal (to L(X, Y)). Moreover we show that the scale {L-infinity,q((a)) (X, Y)}(0

Automated summary

We analyze the properties of the approximation numbers of operators T in the spaces L-p,q,alpha((a))(X, Y). We find that when X and Y are infinite-dimensional Banach spaces, the spaces L-p,q,alpha((a))(X, Y) are all different from each other; but when X or Y is finite-dimensional, they are all equal (to L(X, Y)). Additionally, we discover the scale {L-infinity,q((a)) (X, Y)}(0