# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Quasi-constricted linear operators on Banach spaces

### Tom 144 / 2001

Studia Mathematica 144 (2001), 169-179 MSC: Primary 47A65; Secondary 47A35, 47A10. DOI: 10.4064/sm144-2-5

#### Streszczenie

Let $X$ be a Banach space over $\mathbb C$. The bounded linear operator $T$ on $X$ is called quasi-constricted if the subspace $X_0:=\{ x\in X: \lim_{n\to \infty }\| T^nx\| =0\}$ is closed and has finite codimension. We show that a power bounded linear operator $T\in L(X)$ is quasi-constricted iff it has an attractor $A$ with Hausdorff measure of noncompactness $\chi _{\| \cdot \| _1}(A)< 1$ for some equivalent norm $\| \cdot \| _1$ 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 $\overline {\lambda }T$ is mean ergodic for all $\lambda$ in the peripheral spectrum $\sigma _\pi (T)$ of $T$ is constricted and power bounded, and hence has a compact attractor.

#### Autorzy

• Eduard Yu. Emel'yanovSobolev Institute of Mathematics at Novosibirsk
630090 Novosibirsk, Russia
Mathematisches Institut der
Universität Tübingen
A. D. Morgenstelle 2
D-72076 Tübingen, Germany
e-mail
• Manfred P. H. WolffMathematisches Institut der
Universität Tübingen
A. D. Morgenstelle 2
D-72076 Tübingen, Germany
e-mail

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