Spectral disjointness and invariant subspaces
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CitationHarte, R. (2019). Spectral disjointness and invariant subspaces. International Conference of Mathematical Sciences (ICMS 2019). s. 7.
Spectral disjointness confers a certain ”independence” upon linear operators. If G is a ring with identity I then an idempotent Q = Q2 ∈ G gives the ring G a block structure G ∼= A M N B where for example A = QGQ; then T = a m n b ∈ G commutes with Q iff it is a ”block diagonal”: T Q = QT ⇐⇒ T = a 0 0 b . Specialising to complex Banach algebras, for block diagonals there is two way implication σA(a) ∩ σB(b) = ∅ ⇐⇒ Q ∈ Holo(T) : Q = f(T) with f : U → G holomorphic on an open neighbourhood of σG(T). Weaker spectral disjointness gives a little less: σ lef t A (a) ∩ σ right B (b) = ∅ =⇒ Q ∈ comm2 (T) : the block structure idempotent Q “double commutes” with T ∈ G. Specializing to G = B(X), the bounded operators on a Banach space, closed complemented subspaces Y ⊆ X give us again the block structure, and operators T ∈ G for which Y is “invariant” become “block triangles”: T(Y ) ⊆ Y ⇐⇒ T = a m 0 b . When Y ⊆ X is not complemented then the block structure is missing and we must resort to the restriction and the quotient: a = TY ∈ A = B(Y ) ; b = T/Y ∈ B(X/Y ) . Now spectral disjointness σA(a) ∩ σB(b) = ∅ ensures that the subspace Y ⊆ X is both hyperinvariant and reducing, in particular complemented.
SourceInternational Conference of Mathematical Sciences (ICMS 2019)
- Makale Koleksiyonu 
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