Journal of Operator Theory

Volume 56, Issue 2, Fall 2006 pp. 249-258.

Subscalar operators and growth of resolvent

Authors: Catalin Badea (1) and Vladimir Mueller
Author institution: (1) Departement de Mathematiques, UMR CNRS no. 8524, Universite Lille I, F--59655 Villeneuve d'Ascq, France
(2) Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67 Prague 1, Czech Republic

Summary: We construct a Banach space bounded linear operator

$T$ which is not

$\et$ -subscalar but

$\|(T-z)^{-1}\| \leqslant (|z|-1)^{-1}$ for

$|z|>1$ and

$m(T-z) \geqslant \hbox{const}\cdot(1-|z|)^{3}$ for

$|z|<1$ (here

$m$ denotes the minimum modulus). This gives a negative answer to a variant of a problem of K.B.~Laursen and M.M.~Neumann. We also give a sufficient condition (in terms of growth of resolvent and of an analytic left inverse of

$T-z$ ) implying that

$T$ is an

$\et$ -subscalar operator. This condition is also necessary for Hilbert space operators.

Contents Full-Text PDF