Let V be an n-dimensional complex linear space and L(V) the algebra of all linear transformations on V . We prove that every linear map on L(V), which maps every operator into an operator with isomorphic lattice of invariant subspaces, is an inner automorphism or an inner antiautomorphism multiplied by a nonzero constant and additively perturbed by a scalar type operator. The same result holds ...

Given a nilpotent endomorphism, we consider the manifold of invariant subspaces having a fixed Segre characteristic. In [Linear Algebra Appl., 332–334 (2001) 569], the implicit form of a miniversal deformation of an invariant subspace with respect to the usual equivalence relation between subspaces is obtained. Here we obtain the explicit form of this deformation when the invariant subspace is ...

Each normal rational curve Γ in PG(n, F ) admits a group PΓL(Γ) of automorphic collineations. It is well known that for characteristic zero only the empty and the entire subspace are PΓL(Γ)–invariant. In case of characteristic p > 0 there may be further invariant subspaces. For #F ≥ n+ 2, we give a construction of all PΓL(Γ)–invariant subspaces. It turns out that the corresponding lattice is to...

We study the global behaviour of a Newton algorithm on the Grassmann manifold for invariant subspace computation. It is shown that the basins of attraction of the invariant subspaces may collapse in case of small eigenvalue gaps. A Levenberg-Marquardt-like modification of the algorithm with low numerical cost is proposed. A simple strategy for choosing the parameter is shown to dramatically enl...

A hypersurface of order (n+ 1)(ph − 1) in projective space of dimension n of prime characteristic p has an invariant monomial. This implies that a hypersurface of order (n+1)(ph−1)−1 determines an invariant point. A hypersurface of order d < n+ 1 in a projective space of dimension n of characteristic two has an invariant set of subspaces of dimension d− 1 determined by one linear condition on t...

We introduce the concept of generalized S C,A, B -pairs which is related to generalized S A,B -invariant subspaces and generalized S C,A -invariant subspaces for infinite-dimensional systems. As an application the parameter-insensitive disturbance-rejection problem with dynamic compensator is formulated and its solvability conditions are presented. Further, an illustrative example is also exami...

A compactification of the set of conditioned invariant subspaces of fixed dimension for an observable pair (C, A) is proposed. It contains the almost conditioned invariant subspaces of the same dimension. In certain cases the compactification is shown to be smooth and a complete geometric description is given in the case of a single output system.

*Correspondence: [email protected] Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Avenida Reina Mercedes, Seville, 41012, Spain Abstract We survey several applications of fixed point theorems in the theory of invariant subspaces. The general idea is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an operator ...

Let T be a contraction and A the strong limit of {T∗nTn}n≥1. We prove the following theorem: if a hyponormal contraction T does not have a nontrivial invariant subspace, then T is either a proper contraction of class 00 or a nonstrict proper contraction of class 10 for which A is a completely nonprojective nonstrict proper contraction. Moreover, its self-commutator [T∗,T ] is a strict contracti...

In this paper we construct infinitely many selfadjoint solutions of the control algebraic Riccati equation using invariant subspaces of the associated Hamiltonian. We do this under the assumption that the system operator is normal and has compact inverse, and that the Hamiltonian possesses a Riesz basis of invariant subspaces.