Calkin Algebras and Algebras of Operators on Banach Spaces by S. R. Caradus

By S. R. Caradus

Booklet via S. R. Caradus, W. E. Pfaffenberger

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Example text

Note that if is quasi-regular. This sp(x|A), when A 0 plus the set of all does not X ^ O such An advantage accrued by taking 0 in e A (when there is no identity) is that all x e A, A^ is the algebra obtained where by A. be an element in a normed algebra Xx I. is defined to be the set of complex are quasi-regular. 2 we get (X - y) ^[(Xx)* - (yx)M = -X + (Xx)^x + x(yx)* - (Xx)^x(yx)*. BANACH ALGEBRAS Now let D 23 be the set of complex shows basically that interior of follows. (Xx) * D (int(D)). 2, we see that In fact, for such exists.

Are the operators which satisfy Then we have T ^ (XI - T) = -K^ - (I - XT^) (XI - D T ^ = -K^ - (I - XT^). 40 Choose I - AT^ = тах(||т^|| and I - XT^ so that for are invertible in B(X). |x| < e^, we know that Hence the above equations can be written -(I - XTj^ ) (XI - T) = I + (I - -(XI - 1)1^(1 - XT^)"^ = I + K 2 (I - XT^)"^. 6 implies that XI-T is Fredholm. 7) to the first of these equations and recalling that i(I + K) = 0 for any compact K, we get i(T^) + i(XI - T) = 0. Similarly, from the second equation, we get i(T^) + i(T) = 0.

Let A = O be a pole of R , (T) of order p. Write P for the projection associated with A O {A } and T. Then the formulas for the principal part of the Laurent O series for powers of A give T^” ^ P O ^ 0 = T^ P . O Hence R(P ) c N(T^). The reverse inclusion is valid also as can be seen O “* by recalling the well known formula [15, p. 309] : 1/n 0 }. R(P ) = {x e X : I|T x| Thus R(P^) = N(T^). Suppose now that N(T^^^) = N(T^) x e N(T ); then clearly for each positive integer N(T^ ^) = N(T^) = R(P ), O Hence a (T) = p.

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