On $(A,m)$-expansive operators

Sungeun Jung¹ ; Yoenha Kim¹ ; Eungil Ko² ; Ji Eun Lee¹

¹ Institute of Mathematical Sciences Ewha Womans University 120-750 Seoul, Korea
² Department of Mathematics Ewha Womans University 120-750 Seoul, Kore

Studia Mathematica, Tome 213 (2012) no. 1, pp. 3-23

Résumé

We give several conditions for $(A,m)$-expansive operators to have the single-valued extension property. We also provide some spectral properties of such operators. Moreover, we prove that the $A$-covariance of any $(A,2)$-expansive operator $T\in\mathcal{L(H)} $ is positive, showing that there exists a reducing subspace $\cal M$ on which $T$ is $(A,2)$-isometric. In addition, we verify that Weyl's theorem holds for an operator $T\in\mathcal{L(H)} $ provided that $T$ is $(T^{\ast}T,2)$-expansive. We next study $(A,m)$-isometric operators as a special case of $(A,m)$-expansive operators. Finally, we prove that every operator $T\in\mathcal{L(H)} $ which is $(T^{\ast}T,2)$-isometric has a scalar extension.

Détail

DOI : 10.4064/sm213-1-2

Keywords: several conditions expansive operators have single valued extension property provide spectral properties operators moreover prove a covariance expansive operator mathcal positive showing there exists reducing subspace cal which isometric addition verify weyls theorem holds operator mathcal provided ast expansive study isometric operators special expansive operators finally prove every operator mathcal which ast isometric has scalar extension