Given a Banach space X and a bounded linear operator T on X, a subspace Y of X is almost invariant under T if TY ⊆ Y + F for some finite-dimensional “error” F . In this paper, we study subspaces that are almost invariant under every operator in an algebra A of operators acting on X. We show that if A is norm closed then the dimensions of “errors” corresponding to operators in A must be uniforml...

The objective of this contribution is to characterize the so-called finite fixed poles of the Almost Disturbance Decoupling Problem by state feedback (ADDP) ′ . The most important step towards this result relies on the extension to almost invariant subspaces of the key notion of self-boundedness, as initially introduced by Basile and Marro for perfect controlled-invariants, namely, we introduce...

A complete classification of linear differential operators possessing finite-dimensional invariant subspace with a basis of monomials is presented.

In [1], J. Dyer, A. Pedersen and P. Porcelli announced that an affirmative answer to the invariant subspace problem would imply that every reductive operator is normal. Their argument, outlined in [1], provides a striking application of direct integral theory. Moreover, this method leads to a general decomposition theory for reductive algebras which in turn illuminates the close relationship be...

An important open problem in operator theory is the invariant subspace problem. Since the problem is solved for all finite dimensional complex vector spaces of dimension at least 2, H denotes a separable Hilbert space whose dimension is infinite. It is enough to think for a contraction T , that is, ‖T‖ ≤ 1 on H . Thus, T ( 6= 0) denotes a contraction in L(H). Let λ always denote an element of t...