Entropy and Widths of Multiplier Operators on Compact Globally Symmetric Spaces of Rank 1

The main aim of the present article is to give a unified treatment for a wide range of multiplier operators $\Lambda$ on symmetric manifolds. Namely, we investigate entropy numbers and $n$-widths of decaying multipliers sequences of real numbers $\Lambda = \{\lambda_{n}\}_{k=1}^{\infty}$, $|\lambda_{1}| \geq |\lambda_{2}| \geq \ldots$, $\Lambda:L_{p}(M^{d}) \rightarrow L_{q}(M^{d})$ on $M^{d}$, on the compact globally symmetric spaces of rank 1 or on two-point homogeneous spaces $M^{d}$: $S^{d}$, $P^{d}(\RR)$, $P^{d}(\CC)$, $P^{d}(\HH)$, $P^{16}({\rm Cay})$. In particular, wegive sharp orders of entropy and $n$-widths of standard Sobolev's classes $W^{\gamma}_{p}(M^{d})$ in $L_{q}(M^{d})$ for all $1 < p, q < \infty$.