K理论在一类算子逼近中的应用.pdf

Dissertation Submitted to
Hebei University of Technology
for
The Master Degree of
Science in Applied Mathematics
K-Theory be Applied in Approximation of A Class of
Operators
by
Wen Shilin
Supervisor: Prof. He Hua
December 2011
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K-Theory be Applied in Approximation of A Class of Operators
ABSTRACT
K-theory as the ponent of the subject of/mutative topol-
ogy0, has profound and unexpectedly effect on operator algebras. One can learn
about the structure of a given operator by knowing its K-theory, and one can char-
acterize similarity of operators in terms of the K-group of mutant alge-
bras.
Set L(H) be the collection of the bounded linear operators plex sepa-
rable Hilbert space H. In this paper, we show that: for any operator T ∈ L(H),
there exist direct sum of finitely having good property strongly irreducible oper-
n
n 0 0
ators {Ti} = , such that kT −⊕ Tik < ε, and for each i ∈ N, A (Ti)/radA (Ti)
i 1 i=1
0 0
mutative, where radA (Ti) is the Jacobson radical of A (Ti). Moreover,
0 (2) 0 (2) 0
∨A (Ti)  N or N . K0(A (Ti))  Z or Z , where ∨A (Ti) is the semigroup of
0
mutant algebras, and K0(A (Ti)) is its K0-group.
KEY WORDS: strong irreducible operator; direct sum approximation; K0-
group; Cowen − Douglas operator
ii
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