
rashidkhon
, April 16, 2011
These wellwritten notes treat, via Banach algebras, important classes of bounded linear operators on a Banach space, e.g., semiFredholm operators and Riesz operators.
Chapter one includes some basic properties of bounded semiFredholm operators on a Banach space. A Fredholm operator which has finite ascent and descent is called a RieszSchauder operator. The authors' characterization of such an operator appears to be new. They show that an operator is RieszSchauder if and only if it is the sum of a bijection and an operator of finite rank which commutes with the bijection.
Chapter two gives some basic facts about Banach algebras which are needed in the remaining chapters.
Chapter three is concerned with Riesz operators and semiFredholm operators. Roughly speaking, an operator is a Riesz operator if it has the same spectral properties as a compact operator. Several characterizations of Riesz operators are given. For example, it is shown that $T$T is a Riesz operator if and only if for every $\lambda\neq 0$λ≠0, $\lambda IT$λI�'T is Fredholm or, equivalently, the canonical image of $T$T in the Calkin algebra of bounded operators modulo the compact operators is quasinilpotent. A proof is given of T. T. West's theorem which states that an operator on a Hilbert space is a Riesz operator if and only if it is the sum of a compact operator and a quasinilpotent operator.
Chapter four is an approach to a theory of semiFredholm operators by means of the Calkin algebra. Basic perturbation theorems are proved. A drawback to this approach is that no information is given as how large the norm of the perturbing operator can be in order to retain stability of the index.
Chapter five investigates various norm closed ideals in the space of bounded linear operators. A proof is given of the GohbergMarkusFelʹdman theorem which asserts that the compact operators form the only (non trivial) norm closed ideal in the space of bounded linear operators on $l_p$lp, $1\leq p<><∞. examples="" are="" given="" to="" show="" how="" complicated="" ideals="" of="" operators="" can="" be.="" chapter="" six,="" the="" final="" chapter,="" discusses="" some="" recent="" generalizations="" of="" fredholm="" theory.="" due="" to="" the="" wide="" range="" of="" topics="" covered,="" proofs="" are,="" understandably,="" sketched="" or="" omitted.="" some="" topics="" are:="" fredholm="" theory="" of="" rings="" which="" have="" no="" nilpotent="" ideals,="" index="" theory="" for="" rings="" with="" unit,="" $c^\ast$c�algebras="" of="" operators="" on="" a="" hilbert="" space="" including="" $c^\ast$c�algebras="" generated="" by="" toeplitz="" operators,="" index="" theory="" of="" von="" neumann="" algebras="" with="" applications="" to="" group="" algebras="" and="" finite="" difference="" equations.="">
