In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only pair idempotents with commutator common subspace. We also present geometric characterization subspaces classify are essentially idempotent.

The complemented subspace problem asks, in general, which closed subspaces M of a Banach space X are complemented; i.e. there exists a closed subspace N of X such that X = M ⊕ N? This problem is in the heart of the theory of Banach spaces and plays a key role in the development of the Banach space theory. Our aim is to investigate some new results on complemented subspaces, to present a history...

Recursive subspace model identification (RSMI) has been developed for a decade. Most of RSMIs are only applied for open loop data. In this paper, we propose a new recursive subspace model identification which can be applied under open loop and closed loop data. The key technique of this derivation of the proposed algorithm is to bring the Vector Auto Regressive with eXogenous input (VARX) model...

in this paper, we first discuss about canonical dual of g-frameλp = {λip ∈ b(h, hi) : i ∈ i}, where λ = {λi ∈ b(h, hi) :i ∈ i} is a g-frame for a hilbert space h and p is the orthogonalprojection from h onto a closed subspace m. next, we provethat, if λ = {λi ∈ b(h, hi) : i ∈ i} and θ = {θi ∈ b(k, hi) :i ∈ i} be respective g-frames for non zero hilbert spaces hand k, and λ and θ are unitarily e...

Let H be an integral domain, and let Σ be a collection of integrally closed overrings of H. We show that if A is an overring of H such that H = ( T R∈Σ R)∩A, and if Σ is a Noetherian subspace of the space of all integrally closed overrings of H, then there exists a weakly Noetherian subspace Γ of integrally closed overrings of H such that H = ( T R∈Γ R) ∩ A, and no member of Γ can be omitted fr...