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.

Keywords: Subscalar operators, growth conditions, resolvents.

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