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Deviation from weak Banach–Saks property for countable direct sums

100%

Kryczka A.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2014

tom 68

nr 2

We introduce a seminorm for bounded linear operators between Banach spaces that shows the deviation from the weak Banach–Saks property. We prove that if (Xv) is a sequence of Banach spaces and a Banach sequence lattice E has the Banach–Saks property, then the deviation from the weak Banach–Saks property of an operator of a certain class between direct sums E(Xv) is equal to the supremum of such deviations attained on the coordinates Xv. This is a quantitative version for operators of the result for the Köthe–Bochner sequence spaces E(X) that if E has the Banach–Saks property, then E(X) has the weak Banach–Saks property if and only if so has X.

Measure of weak noncompactness under complex interpolation

63%

Kryczka A., Prus S.

Studia Mathematica

2001

tom 147

nr 1

89-102

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.

Ograniczanie wyników

1 Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

1 Studia Mathematica

2 Kryczka A.

1 Prus S.

1 2014

1 2001

Wyniki wyszukiwania

Deviation from weak Banach–Saks property for countable direct sums

Measure of weak noncompactness under complex interpolation