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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...

Geometry of Banach spaces and biorthogonal systems

S. Dilworth, Maria Girardi, W. Johnson (2000)

Studia Mathematica

A separable Banach space X contains ℓ 1 isomorphically if and only if X has a bounded fundamental total w c 0 * -stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total w c 0 * -biorthogonal system.

Geometry of quotient spaces and proximinality

Yuan Cui, Henryk Hudzik, Yaowaluck Khongtham (2003)

Annales Polonici Mathematici

It is proved that if X is a rotund Banach space and M is a closed and proximinal subspace of X, then the quotient space X/M is also rotund. It is also shown that if Φ does not satisfy the δ₂-condition, then h ⁰ Φ is not proximinal in l ⁰ Φ and the quotient space l ⁰ Φ / h ⁰ Φ is not rotund (even if l ⁰ Φ is rotund). Weakly nearly uniform convexity and weakly uniform Kadec-Klee property are introduced and it is proved that a Banach space X is weakly nearly uniformly convex if and only if it is reflexive and it has the weakly...

Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K

Dale E. Alspach, Elói Medina Galego (2011)

Studia Mathematica

A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of C ( ω ω ) then X contains a copy of c₀. Moreover,...

G-narrow operators and G-rich subspaces

Tetiana Ivashyna (2013)

Open Mathematics

Let X and Y be Banach spaces. An operator G: X → Y is a Daugavet center if ‖G +T‖ = ‖G‖+‖T‖ for every rank-1 operator T. For every Daugavet center G we consider a certain set of operators acting from X, so-called G-narrow operators. We prove that if J is the natural embedding of Y into a Banach space E, then E can be equivalently renormed so that an operator T is (J ○ G)-narrow if and only if T is G-narrow. We study G-rich subspaces of X: Z ⊂ X is called G-rich if the quotient map q: X → X/Z is...

Greedy approximation and the multivariate Haar system

A. Kamont, V. N. Temlyakov (2004)

Studia Mathematica

We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis in L p ( [ 0 , 1 ] ) , 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis d = × . . . × in L p ( [ 0 , 1 ] d ) ,...

Growth of coefficients of universal Dirichlet series

A. Mouze (2007)

Annales Polonici Mathematici

We study universal Dirichlet series with respect to overconvergence, which are absolutely convergent in the right half of the complex plane. In particular we obtain estimates on the growth of their coefficients. We can then compare several classes of universal Dirichlet series.

H ∞ functional calculus in real interpolation spaces

Giovanni Dore (1999)

Studia Mathematica

Let A be a linear closed densely defined operator in a complex Banach space X. If A is of type ω (i.e. the spectrum of A is contained in a sector of angle 2ω, symmetric around the real positive axis, and ∥ λ ( λ I - A ) - 1 ∥ is bounded outside every larger sector) and has a bounded inverse, then A has a bounded H ∞ functional calculus in the real interpolation spaces between X and the domain of the operator itself.

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