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

By S. R. Caradus

Publication through S. R. Caradus, W. E. Pfaffenberger

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Observe that we can always find such decompositions and finite dimensional. 8) 39 THEOREM An operator T in B(X) is Fredholm if and only if invertible in the Calkin algebra Proof. If T тг(Т) is C(X). 6, we have 7T(T^)tt(T) = 7t(I + K^) = TT(I) TT(T)TTd^) = 7Г(1 + K^) = TT(I). Thus tt(T) invertible. tt(TS has a left and right inverse in Conversely, if - I) = 7t(ST - I) = 0. tt(T) C(X). ■ is an infinite dimensional Banach space and T e B(X), then cannot be the entire complex plane. Proof. 10) If tt(T) for all X.

I But X- ^ pole of E -C O n V (i = 1 ,2). ) since in a finite dimensional space, the spectrum of 50 an operator consists entirely of poles of the resolvent. ) ^ we see that X O -I ^ be a pole of (XI - T) . 5 The West decomposition If C + Q C is a compact operator and is a Riesz operator. e. can every Riesz operator be written as a sum, C + Q. In the case where the operators are defined on a Hilbert space, T. T. West [66] was able to obtain an affirmative result; the general problem is still unsettled and constitutes an important open question in this area.

In X, V n' If T is not one-to-one with closed range there is [22, p. 57] a sequence ^ X*n e choose X* U (x) = x*(x)x . n n n each ^ ^^n^ ^ with Ilx*|I = ' ' n '' Clearly x| I 4 I. X, x U Therefore such that I. We define Also = I. TU^ 0 in each U n e B(X) |t U^( x )|| < | | B(X), so n by the rule T t ( x^ ) | | for is a left topological divisor of zero (b) of zero in U T n 0. I|u*|I = Suppose that B(X) But then we have I and R(T) = X . there is a sequence T*U* -> 0 . T* If T is a right topological divisor where each as one-to-one on X I I“ ^ with closed range, This is impossible by part (a) of this theorem.

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