The Singular Continuous Spectrum and Non-closed Invariant Subspaces
نویسندگان
چکیده
Let A be a bounded self-adjoint operator on a separable Hilbert space H and H0 ⊂ H a closed invariant subspace of A. Assuming that H0 is of codimension 1, we study the variation of the invariant subspace H0 under bounded self-adjoint perturbations V of A that are off-diagonal with respect to the decomposition H = H0 ⊕ H1. In particular, we prove the existence of a one-parameter family of dense non-closed invariant subspaces of the operator A+V provided that this operator has a nonempty singular continuous spectrum. We show that such subspaces are related to non-closable densely defined solutions of the operator Riccati equation associated with generalized eigenfunctions corresponding to the singular continuous spectrum of B.
منابع مشابه
The Singularly Continuous Spectrum and Non-closed Invariant Subspaces
Let A be a bounded self-adjoint operator on a separable Hilbert space H and H0 ⊂ H a closed invariant subspace of A. Assuming that H0 is of codimension 1, we study the variation of the invariant subspace H0 under bounded self-adjoint perturbations V of A that are off-diagonal with respect to the decomposition H = H0 ⊕H1. In particular, we prove the existence of a oneparameter family of dense no...
متن کاملWeak*-closed invariant subspaces and ideals of semigroup algebras on foundation semigroups
Let S be a locally compact foundation semigroup with identity and be its semigroup algebra. Let X be a weak*-closed left translation invariant subspace of In this paper, we prove that X is invariantly complemented in if and only if the left ideal of has a bounded approximate identity. We also prove that a foundation semigroup with identity S is left amenab...
متن کاملSimplicity of singular spectrum in Anderson type Hamiltonians
We study self adjoint operators of the form Hω = H0 + ∑ ω(n)(δn| · )δn, where the δn’s are a family of orthonormal vectors and the ω(n)’s are independent random variables with absolutely continuous probability distributions. We prove a general structural theorem which provides in this setting a natural decomposition of the Hilbert space as a direct sum of mutually orthogonal closed subspaces th...
متن کاملOn the relations between the point spectrum of A and invertibility of I + f(A)B
Let A be a bounded linear operator on a Banach space X. We investigate the conditions of existing rank-one operator B such that I+f(A)B is invertible for every analytic function f on sigma(A). Also we compare the invariant subspaces of f(A)B and B. This work is motivated by an operator method on the Banach space ell^2 for solving some PDEs which is extended to general operator space under some ...
متن کاملA continuous path of singular masas in the hyperfinite II1 factor
Using methods of R.J.Tauer [13] we exhibit an uncountable family of singular masas in the hyperfinite II1 factor R all with Pukánszky invariant {1}, no pair of which are conjugate by an automorphism of R. This is done by introducing an invariant Γ(A) for a masa A in a II1 factor N as the maximal size of a projection e ∈ A for which Ae contains non-trivial centralising sequences for eNe. The mas...
متن کامل