Local behaviour of operators
نویسنده
چکیده
Examples: 1) Local spectral radius r(T, x) of an operator T at a point x ∈ X can be defined by r(T, x) = lim supk→∞ ‖T kx‖1/k, i.e. it is a quantity defined in terms of B. The local spectral radius plays an important role in the local spectral theory. 2) As an analogy to the local spectral radius for the set of all polynomials can be considered the local capacity (see later). 3) The invariant subspace problem can be also easily reformulated by using the sets {p(T )x : p ∈ P}: An operator T in X has no non-trivial invariant subspace if and only if {p(T )x : p ∈ P} is dense for all x ∈ X. Many of the positive results (e.g. results based on the Scott Brown technique) consist in finding x ∈ X such that ‖p(T )x‖ ≥ 1 for all polynomials p with p(0) = 1.
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