#### Title

Properties of k-isotropic functions

#### Degree

Doctor of Philosophy

#### Program

Statistics and Actuarial Sciences

Hristo S. Sendov

#### Abstract

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 $A \succeq B \mbox{ implies } F(A) \succeq F(B)$. (Here, $\succeq$' denotes the semidefinite partial order in $S^n$.) Often, the function $F$ is defined in terms of an underlying simpler function $f$. Of main interest is to find the properties of $f$ that characterize operator monotonicity of $F$. In that case, it is said that $f$ is also operator monotone. Classical examples are the L\"owner's operators and the spectral (scalar-valued isotropic) functions. Operator monotonicity for these two classes of functions is characterized in seemingly very different ways. The work in Chapter 1 extends the notion of operator monotonicity to symmetric functions $f$ on $k$ arguments. The latter is used to define {\it (generated) $k$-isotropic functions} $F : S^n \to S^{n \choose k}$ for any $n \ge k$. Necessary and sufficient conditions are given for $f$ to characterize an operator monotone $k$-isotropic map $F$. Then, in Chapter 2, we give necessary and sufficient conditions for the analyticity of (generated) $k$-isotropic functions. When $k=1$, the $k$-isotropic map becomes a L\"owner's operator and when $k=n$ it becomes a spectral functions. This allows us to reconcile and explain the differences between the conditions for monotonicity and analyticity for the L\"owner's operators and the spectral functions. We say that a function $F:S^n \rightarrow S^{\cramped{n^k}}$ is {\it $k$-tensor isotropic}, if it satisfies $$F(UA\trans{U}) = (U^{\otimes k}) F(A) \trans{(U^{\otimes k})}$$ for all $U \in O^n$ and all $A$ in the domain of $F$. Here, ${\otimes k}$' denotes the $k$-th tensor power. The goal of Chapter 3 is to investigate the internal structure of the $k$-tensor isotropic functions and formulate a canonical representation of $F$ in terms of simpler functions on $\mathbb{R}^n$. We achieve this goal in the case when $k=2$ for any natural $n$. In the process, we characterize the structure of the matrices in the centralizer of certain orthogonal subgroups of $O^{\cramped{n^k}}$. An orthogonally invariant class of operator functions, $F^H$ is studied in Chapter 4. Then, we connect $F^H$ to (generated) $k$-isotropic functions, when it is restricted to block diagonal matrices. This connection allows us to establish various smoothness properties of $F^H$.

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