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The Cesàro and related operators, a survey

V. G. Miller — 2007

Banach Center Publications

We provide a survey of properties of the Cesàro operator on Hardy and weighted Bergman spaces, along with its connections to semigroups of weighted composition operators. We also describe recent developments regarding Cesàro-like operators and indicate some open questions and directions of future research.

The Kato-type spectrum and local spectral theory

T. L. MillerV. G. MillerMichael M. Neumann — 2007

Czechoslovak Mathematical Journal

Let T ( X ) be a bounded operator on a complex Banach space X . If V is an open subset of the complex plane such that λ - T is of Kato-type for each λ V , then the induced mapping f ( z ) ( z - T ) f ( z ) has closed range in the Fréchet space of analytic X -valued functions on V . Since semi-Fredholm operators are of Kato-type, this generalizes a result of Eschmeier on Fredholm operators and leads to a sharper estimate of Nagy’s spectral residuum of T . Our proof is elementary; in particular, we avoid the sheaf model of Eschmeier and...

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