Note on the norm involved in the definition of the \(\epsilon\)-spectrum (Q5938932)

There are several definitions of the \(\varepsilon\)-spectrum of matrices from \(M_n({\mathbb C})\). Unlike the spectrum, the \(\varepsilon\)-spectrum definitions depend on the choice of the norm. The authors study connections on norms that make all these definitions equivalent. For example, it is proved that the definitions are equivalent if a matrix norm is an induced one. The Frobenius norm, which is not a subordinated norm, constitutes an example of a norm, for which different definitions of the \(\varepsilon\)-spectrum lead to different sets.