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On quasi-compactness of operator nets on Banach spaces

100%

Emel'yanov E.

Studia Mathematica

2011

tom 203

nr 2

163-170

The paper introduces a notion of quasi-compact operator net on a Banach space. It is proved that quasi-compactness of a uniform Lotz-Räbiger net $(T_{λ})_{λ}$ is equivalent to quasi-compactness of some operator $T_{λ}$. We prove that strong convergence of a quasi-compact uniform Lotz-Räbiger net implies uniform convergence to a finite-rank projection. Precompactness of operator nets is also investigated.

Quasi-constricted linear operators on Banach spaces

63%

Emel'yanov E., Wolff M.

Studia Mathematica

2001

tom 144

nr 2

169-179

Let X be a Banach space over ℂ. The bounded linear operator T on X is called quasi-constricted if the subspace $X₀: = {x ∈ X: lim_{n→ ∞} ||Tⁿx|| = 0}$ is closed and has finite codimension. We show that a power bounded linear operator T ∈ L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness $χ_{||·||₁}(A) < 1$ for some equivalent norm ||·||₁ on X. Moreover, we characterize the essential spectral radius of an arbitrary bounded operator T by quasi-constrictedness of scalar multiples of T. Finally, we prove that every quasi-constricted operator T such that λ̅T is mean ergodic for all λ in the peripheral spectrum $σ_{π}(T)$ of T is constricted and power bounded, and hence has a compact attractor.

Ograniczanie wyników

2 Studia Mathematica

2 Emel'yanov E.

1 Wolff M.

1 2011

1 2001

Wyniki wyszukiwania

On quasi-compactness of operator nets on Banach spaces

Quasi-constricted linear operators on Banach spaces