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Measure of non-compactness of operators interpolated by the real method

Radosław Szwedek (2006)

Studia Mathematica

We study the measure of non-compactness of operators between abstract real interpolation spaces. We prove an estimate of this measure, depending on the fundamental function of the space. An application to the spectral theory of linear operators is presented.

Measure of weak noncompactness under complex interpolation

Andrzej Kryczka, Stanisław Prus (2001)

Studia Mathematica

Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón’s complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if T: A₀ → B₀ or T: A₁ → B₁ is weakly compact, then so is T : A [ θ ] B [ θ ] for all 0 < θ < 1, where A [ θ ] and B [ θ ] are interpolation spaces with respect to the pairs (A₀,A₁) and (B₀,B₁). Some formulae for this measure and relations to other quantities measuring weak noncompactness are established.

Montel and reflexive preduals of spaces of holomorphic functions on Fréchet spaces

Christopher Boyd (1993)

Studia Mathematica

For U open in a locally convex space E it is shown in [31] that there is a complete locally convex space G(U) such that G ( U ) i ' = ( ( U ) , τ δ ) . Here, we assume U is balanced open in a Fréchet space and give necessary and sufficient conditions for G(U) to be Montel and reflexive. These results give an insight into the relationship between the τ 0 and τ ω topologies on ℋ (U).

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