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Properties Of K-Isotropic Functions, Tianpei Jiang

Electronic Thesis and Dissertation Repository

The focus of this work is a family of maps from the space of $n \times n$ symmetric matrices, $S^n$, into the space $S^{{n \choose k}}$ for any $k=1,\ldots, n$, invariant under the conjugate action of the orthogonal group $O^n$. This family, called generated $k$-isotropic functions, generalizes known types of maps with similar invariance property, such as the spectral, primary matrix, isotropic functions, multiplicative compound, and additive compound matrices on $S^n$. The notion of operator monotonicity dates back to a work by L\"owner in 1934. A map $F :S^n \to S^m$ is called {\it operator monotone}, if $A \succeq B …