Stable and Norm-stable Invariant Subspaces

2010/01/07 by Alexander Borichev, Borichev, Alexander, Don Hadwin +3
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Primary 47A15 #Secondary 47B37 #math.FA #msc:47A15 #msc:47B37

paper · pdf · doi:10.48550/arxiv.1001.1018

arxiv created 2010/08/18 · arxiv updated 2010/08/20

Abstract

We prove that if T is an operator on an infinite-dimensional Hilbert space whose spectrum and essential spectrum are both connected and whose Fredholm index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces of T are the finite-dimensional ones. We also characterize norm-stable invariant subspaces of any weighted unilateral shift operator. We show that quasianalytic shift operators are points of norm continuity of the lattice of the invariant subspaces. We also provide a necessary condition for strongly stable invariant subspaces for certain operators.

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