We prove asymptotic formulas for the behavior of Gelfand and Kolmogorov numbers of Sobolev embeddings between Besov and Triebel-Lizorkin spaces of radial distributions. Our method works also for Weyl numbers.

For a precompact subset K of a Hilbert space we prove the following inequalities: $n^{1/2}cₙ\left(cov\left(K\right)\right)\le c_K(1+{\sum }_{k=1}^{ⁿ}{k}^{-1/2}{e}_k\left(K\right))$, n ∈ ℕ, and $k^{1/2}{c}_{k+n}\left(cov\left(K\right)\right)\le c[lo{g}^{1/2}(n+1)\epsilon ₙ\left(K\right)+{\sum }_{j=n+1}^{\infty }{\epsilon }_j\left(K\right)/\left(jlo{g}^{1/2}(j+1)\right)]$, k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and ${\epsilon }_k\left(K\right)$ and $e_k\left(K\right)$ denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K) are slowly...

Generalizing A. Grothendieck’s (1955) and V. B. Lidskiĭ’s (1959) trace formulas, we have shown in a recent paper that for p ∈ [1,∞] and s ∈ (0,1] with 1/s = 1 + |1/2-1/p| and for every s-nuclear operator T in every subspace of any $L_p\left(\nu \right)$-space the trace of T is well defined and equals the sum of all eigenvalues of T. Now, we obtain the analogous results for subspaces of quotients (equivalently: for quotients of subspaces) of $L_p$-spaces.