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First results on spectrally bounded operators

M. Mathieu, G. J. Schick (2002)

Studia Mathematica

A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is defined to be spectrally bounded if there is a constant M ≥ 0 such that r(Tx) ≤ Mr(x) for all x ∈ E, where r(·) denotes the spectral radius. We study some basic properties of this class of operators, which are sometimes analogous to, sometimes very different from, those of bounded operators between Banach spaces.

Fragmentable mappings and CHART groups

Warren B. Moors (2016)

Fundamenta Mathematicae

The purpose of this note is two-fold: firstly, to give a new and interesting result concerning separate and joint continuity, and secondly, to give a stream-lined (and self-contained) proof of the fact that "tame" CHART groups are topological groups.

Generalized first class selectors for upper semi-continuous set-valued maps in Banach spaces

R. W. Hansell, L. Oncina (2005)

Czechoslovak Mathematical Journal

In this paper we deal with weakly upper semi-continuous set-valued maps, taking arbitrary non-empty values, from a non-metric domain to a Banach space. We obtain selectors having the point of continuity property relative to the norm topology for a large class of compact spaces as a domain. Exact conditions under which the selector is of the first Borel class are also investigated.

Generalized induced norms

S. Hejazian, M. Mirzavaziri, Mohammad Sal Moslehian (2007)

Czechoslovak Mathematical Journal

Let ∥ · ∥ be a norm on the algebra ℳ n of all n × n matrices over ℂ . An interesting problem in matrix theory is that “Are there two norms ∥ · ∥ 1 and ∥ · ∥ 2 on ℂ n such that ∥ A ∥ = max { ∥ A x ∥ 2 ∥ x ∥ 1 = 1 } for all A ∈ ℳ n ?” We will investigate this problem and its various aspects and will discuss some conditions under which ∥ · ∥ 1 = ∥ · ∥ 2 .

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.

Higher-Order Partial Differentiation

Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

In this article, we shall extend the formalization of [10] to discuss higher-order partial differentiation of real valued functions. The linearity of this operator is also proved (refer to [10], [12] and [13] for partial differentiation).

Holomorphic germs on Banach spaces

Chae Soo Bong (1971)

Annales de l'institut Fourier

Let E and F be two complex Banach spaces, U a nonempty subset of E and K a compact subset of E . The concept of holomorphy type θ between E and F , and the natural locally convex topology 𝒯 ω , θ on the vector space ℋ θ ( U , F ) of all holomorphic mappings of a given holomorphy type θ from U to F were considered first by L. Nachbin. Motived by his work, we introduce the locally convex space ℋ θ ( K , F ) of all germs of holomorphic mappings into F around K of a given holomorphy type θ , and study its interplay with ℋ θ ( U , F ) and some...

Hopf Extension Theorem of Measure

Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama (2009)

Formalized Mathematics

The authors have presented some articles about Lebesgue type integration theory. In our previous articles [12, 13, 26], we assumed that some σ-additive measure existed and that a function was measurable on that measure. However the existence of such a measure is not trivial. In general, because the construction of a finite additive measure is comparatively easy, to induce a σ-additive measure a finite additive measure is used. This is known as an E. Hopf's extension theorem of measure [15].

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