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Geometric characteristics for convergence and asymptotics of successive approximations of equations with smooth operators

Boris Godunov, Petr Zabreĭko (1995)

Studia Mathematica

We discuss the problem of characterizing the possible asymptotic behaviour of the iterates of a sufficiently smooth nonlinear operator acting in a Banach space in small neighbourhoods of a fixed point. It turns out that under natural conditions, for the most part of initial approximations these iterates tend to "lie down" along a finite-dimensional subspace generated by the leading (peripherical) eigensubspaces of the Fréchet derivative at the fixed point and moreover the asymptotic behaviour of...

Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces

Catalin Badea, Yuri I. Lyubich (2010)

Studia Mathematica

According to the von Neumann-Halperin and Lapidus theorems, in a Hilbert space the iterates of products or, respectively, of convex combinations of orthoprojections are strongly convergent. We extend these results to the iterates of convex combinations of products of some projections in a complex Banach space. The latter is assumed uniformly convex or uniformly smooth for the orthoprojections, or reflexive for more special projections, in particular, for the hermitian ones. In all cases the proof...

Géométrie du spectre dans une algèbre de Banach et domaine numérique

Mohamed Chraibi Kaadoud (2004)

Studia Mathematica

Dans une algèbre de Banach et dans deux cas particuliers, nous montrons la continuité du centre du plus petit disque contenant le spectre. Pour a ∈ , on donne une condition nécessaire et suffisante pour avoir R K = d ( a ) où d(a) est la distance de a aux scalaires et R K le rayon du plus petit disque contenant K qui représente le spectre ou le domaine numérique algébrique de a. Dans un espace de Hilbert complexe, K peut représenter certains types de spectres ou de domaines numériques de a.

Geometry of oblique projections

E. Andruchow, Gustavo Corach, D. Stojanoff (1999)

Studia Mathematica

Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections P a determined by the different involutions a induced by positive invertible elements a ∈ A. The maps φ : P → P a sending p to the unique q ∈ P a with the same range as p and Ω a : P a → P a sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q,r ∈ A with ||q-r|| < 1 such that...

Geometry of the spectral semidistance in Banach algebras

Gareth Braatvedt, Rudi Brits (2014)

Czechoslovak Mathematical Journal

Let A be a unital Banach algebra over ℂ , and suppose that the nonzero spectral values of a and b ∈ A are discrete sets which cluster at 0 ∈ ℂ , if anywhere. We develop a plane geometric formula for the spectral semidistance of a and b which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that a and b are quasinilpotent equivalent if...

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