Every operator has almost-invariant subspaces

Almost Self-Bounded Controlled-Invariant Subspaces and Almost Disturbance Decoupling

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...

Invariant Subspaces of Voiculescu's Circular Operator

Almost every graph is divergent under the biclique operator

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. The biclique graph of G denoted by KB(G), is the intersection graph of all the bicliques of G. The biclique graph can be thought as an operator between the class of all graphs. The iterated biclique graph of G denoted by KBk(G), is the graph obtained by applying the biclique operator k successive times to G. The asso...

Prevalence: a Translation-invariant “almost Every” on Infinite-dimensional Spaces

We present a measure-theoretic condition for a property to hold “almost everywhere” on an infinite-dimensional vector space, with particular emphasis on function spaces such as C and L. Like the concept of “Lebesgue almost every” on finite-dimensional spaces, our notion of “prevalence” is translation invariant. Instead of using a specific measure on the entire space, we define prevalence in ter...

Almost Every Number Has a Continuum of β-Expansions

where b·c denotes the integral part of a number and {·} stands for its fractional part. If x belongs to [1, 1/(β − 1)), then we put ` = min {k ≥ 1 : x− β−1 − · · · − β−k ∈ (0, 1)} and apply the greedy algorithm to x − β−1 − · · · − β to obtain the digits ε`+1, ε`+2, etc. Finally, if x = 1/(β − 1), then εn = 1 for all n. The question then arises: Are there any other β-expansions of a given x bes...