A Remark on a Paper by EDMUNDS and TRIEBEL

✍ Scribed by Bernd Carl

• Publisher: John Wiley and Sons
• Year: 1981
• Tongue: English
• Weight: 205 KB
• Volume: 103
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19811030105

✦ Synopsis

The results given in this note cornplemend those of EDMUNDS/TRIEBEL in [4].

Moreover we characterize the "degree of noncompactness" as well as the radius of the essential spectrum for operators acting in RAXACII spaces l y quantities defined hy KOLMOGOROV and GELFAND numbers.

Throughout this paper all BANACH spaces will be complex. UANACH spaces are deiwted by E , F , and G . The symbol 9 ( E , F ) stands for the BANACH space of all (hounded linear) operators from E into F . I n the sequel e(")(S) means the n-th entropy number of an operator S € 9 ( E , F ) denoted by E, in [4]. According to A. PIETSCII [5] the n-th dyadic entropy number is defined hy e,,(S) : = e(2n-1)(S). I n [4] R "measure for non-compactness" for an operator is given by P(S) : = lim P ) ( S ) . n--Coiicerning propertiefi of B we also refer to [4]. I n particular. the formula v,(S)= lim /I''~'@~), lSlt ?(E, E ) , N --

where re(S) is the radius of the essential spectrum of 8, was observed by Nuss-BAI-RI (cf. [-l] for references and comments; there is also a siniplified proof for the 1 1 i i m m 1 ~ space case).