ABSTRACT
Throughout this chapter, let ‖ | · ‖ | $ \Vert |\,\cdot \Vert |\, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math5_1.tif"/> denote a unitarily invariant norm on C n × n $ \mathbb C _{n\times n} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math5_2.tif"/> . A function f : R n → R $ f: \mathbb R ^n \rightarrow \mathbb R $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math5_3.tif"/> is called a symmetric gauge function if f is a vector norm, if f ( S x ) = f ( x ) $ f(Sx) = f(x) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math5_4.tif"/> for all x ∈ R n $ x \in \mathbb R ^n $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math5_5.tif"/> and for all permutation matrices S ∈ C n × n $ S \in \mathbb C _{n\times n} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math5_6.tif"/> , and if f ( x ) = f ( | x | ) $ f(x) = f(|x|) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math5_7.tif"/> for all x ∈ R n $ x \in \mathbb R ^n $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math5_8.tif"/> . An important characterization of unitarily invariant norms given by von Neumann [vN37] is that a function f : C n × n → R $ f: \mathbb C _{n\times n}\rightarrow \mathbb R $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429468940/0047beac-a8ee-41c4-af5b-5b2978d30b96/content/inline-math5_9.tif"/> is a unitarily invariant norm if and only if f(A) is a symmetric gauge function on the singular values of A (see [Bh97, p.91]).
