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Multidimensional weak resolvents and spatial equivalence of normal operators

Michał Jasiczak (2006)

Studia Mathematica

The aim of this paper is to answer some questions concerning weak resolvents. Firstly, we investigate the domain of extension of weak resolvents Ω and find a formula linking Ω with the Taylor spectrum. We also show that equality of weak resolvents of operator tuples A and B results in isomorphism of the algebras generated by these operators. Although this isomorphism need not be of the form (1) X ↦ U*XU, where U is an isometry, for normal operators it is always possible...

Nonhermitian systems and pseudospectra

Lloyd N. Trefethen (2005/2006)

Séminaire Équations aux dérivées partielles

Four applications are outlined of pseudospectra of highly nonnormal linear operators.

Norm continuity of c 0 -semigroups

V. Goersmeyer, L. Weis (1999)

Studia Mathematica

We show that a positive semigroup T t on L p ( Ω , ν ) with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of “smoothing properties” of certain convolution operators on general Banach spaces and an extrapolation result for the L p -scale, which may be of independent interest.

Notes on q-deformed operators

Schôichi Ôta, Franciszek Hugon Szafraniec (2004)

Studia Mathematica

The paper concerns operators of deformed structure like q-normal and q-hyponormal operators with the deformation parameter q being a positive number different from 1. In particular, an example of a q-hyponormal operator with empty spectrum is given, and q-hyponormality is characterized in terms of some operator inequalities.

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