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Monotonicity properties of norms. (English) Zbl 0717.15015
Let $${\mathbb{C}}^ d$$ denote complex d space, i.e., the set of $$d\times 1$$ complex matrices. Let n be a norm on $${\mathbb{C}}^ d$$ (n: $${\mathbb{C}}^ d\to {\mathbb{R}}$$ has n(x)$$\geq 0$$ with equality only if $$x=0$$, $$n(\alpha x)=| \alpha | n(x)$$, and $$n(x+y)\leq n(x)+n(y)$$ for all $$x,y\in {\mathbb{C}}^ d$$, $$\alpha\in {\mathbb{C}})$$. The notations $$| \cdot |$$ and $$\leq$$ on $${\mathbb{C}}^ d$$ are taken componentwise. n is absolute if $$n(x)=n(| x|)$$ for all $$x\in {\mathbb{C}}^ d$$; n is monotone if n(x)$$\leq n(y)$$ for all $$x,y\in {\mathbb{C}}^ d$$ with $$| x| \leq | y|$$. Several properties involving these concepts are considered for that of a particular norm n (which n may or may not satisfy). A complete set of implications among the properties is given.
Reviewer: R.Sinkhorn

##### MSC:
 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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##### References:
  Bauer, F.L.; Stoer, J.; Witzgall, C., Absolute and monotonic norms, Numer. math., 3, 257-264, (1961) · Zbl 0111.01602  Deutsch, E., Solution of advanced problem 6249, Amer. math. monthly, 87, 831, (1980)  Funderlic, R.E., Some characterizations of orthant monotonic norms, Linear algebra appl., 28, 77-83, (1979) · Zbl 0419.15001  Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge U. P · Zbl 0576.15001  R. A. Horn and R. Mathias, An analog of the Cauchy-Schwarz inequality for Hadamard products and unitarily invariant norms, SIAM J. Matrix Anal. Appl., to appear. · Zbl 0722.15019  Johnson, C.R., Two submatrix properties of certain induced norms, J. res. nat. bur. standards, 79B, 97-101, (1975) · Zbl 0333.15014  Maítre, J.F., Sur certaines normes et fonctionelles dans LES espaces de matrices et d’operateurs, ()  Malek-Shamirzadi, M., On monotonic and orthant monotonic norms, Linear algebra appl., 56, 169-175, (1984) · Zbl 0525.15018  Merikoski, J.K., On operator norms of submatrices, Linear algebra appl., 36, 173-183, (1981) · Zbl 0455.15024  Merikoski, J.K., Some notes on absolute norms, () · Zbl 0467.15013  Okubo, K., Hölder type norm inequalities for Schur products of matrices, Linear algebra appl., 91, 13-28, (1987) · Zbl 0633.15013  Stoer, J., A characterization of Hölder norms, J. soc. indust. appl. math., 12, 634-648, (1964) · Zbl 0196.29803
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