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

nominal

(Square) Roots of Bounded OperatorsLet . ∈ .(.). We say that . ∈ .(.) is a square root of . if .. = ..

Tdd526

blight

SpectrumLet . ∈ .(.) where . is a complex Hilbert space. The set . is called the spectrum of ..

联合

Spectral Radius, Numerical RangeLet . be in .(.). The spectral radius of . is defined as

菊花

Functional CalculiThe functional calculus aims to define .(.) where . is a fixed operator, and . belongs to some classes of functions defined in domains containing .(.), say. We already know that this is possible for any polynomial .. We also know how to define the exponential of . at an undergraduate level (this will be recalled in Chap. .).

EXUDE

calamity

aesthetic

Similarity and Unitary EquivalenceClearly, . 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.

Proponent

The Sylvester EquationConsider the operator equation: . where ., ., . ∈ .(.) are given and . ∈ .(.) is the unknown. This equation is more commonly known as the Sylvester equation.

consolidate