Titlebook: Counterexamples in Operator Theory; Mohammed Hichem Mortad Textbook 2022 The Editor(s) (if applicable) and The Author(s), under exclusive

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,Glaubwürdigkeit: ein Forschungsüberblick,One of the most powerful tools in the theory of normal operators is the following Fuglede theorem.

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Norbert Konegen,Klaus SondergeldClearly, . and . have the same eigenvalues which, in this setting, means that . and . have equal spectra. To see why . and . are not unitarily equivalent, remember that two unitarily equivalent operators are simultaneously (e.g.) self-adjoint. Since . is self-adjoint and . is not, it follows that they cannot be unitarily equivalent.

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Norbert Konegen,Klaus SondergeldConsider the operator equation: . where ., ., . ∈ .(.) are given and . ∈ .(.) is the unknown. This equation is more commonly known as the Sylvester equation.

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Norbert Konegen,Klaus SondergeldShow that the mapping .↦.. defined from .(.) into .(.) is not weakly continuous, that is, find a sequence (..) in .(.) that converges weakly to . ∈ .(.) yet . does not converge weakly to ...

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Some Basic PropertiesThroughout this chapter, . and . denote two Hilbert spaces over . unless otherwise stated.

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Basic Classes of Bounded Linear OperatorsLet . be a Hilbert space, and let . ∈ .(.). Let . be the identity operator on ..

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Operator TopologiesLet . be a Hilbert space, and let (..) be a sequence in .(.).

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