Commutativity, Majorization, and Reduction in Fan–Theobald–von Neumann Systems

Abstract

A Fan–Theobald–von Neumann system (Gowda in Optimizing certain combinations of linear/distance functions over spectral sets, 2019. arXiv:1902.06640v2) is a triple $({{\mathcal {V}}},{{\mathcal {W}}},\lambda )$, where ${{\mathcal {V}}}$ and ${{\mathcal {W}}}$ are real inner product spaces and $\lambda :{{\mathcal {V}}}\rightarrow {{\mathcal {W}}}$ is a norm-preserving map satisfying a Fan–Theobald–von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decompositions systems (Eaton triples). In Gowda (Optimizing certain combinations of linear/distance functions over spectral sets, 2019. arXiv:1902.06640v2), we presented some basic properties of such systems and described results on optimization problems dealing with certain combinations of linear/distance and spectral functions. We also introduced the concept of commutativity via the equality in the Fan–Theobald–von Neumann-type inequality. In the present paper, we elaborate on the concept of commutativity and introduce/study automorphisms, majorization, and reduction in Fan–Theobald–von Neumann systems.

Appendix

1.1 Normal Decomposition Systems

Definition 10.6

[20]. Let ${{\mathcal {V}}}$ be a real inner product space, ${{\mathcal {G}}}$ be a closed subgroup of the orthogonal group of ${{\mathcal {V}}}$, and $\gamma : {{\mathcal {V}}}\rightarrow {{\mathcal {V}}}$ be a map satisfying the following conditions:

• (a) $\gamma$ is ${{\mathcal {G}}}$-invariant, that is, $\gamma (Ax) = \gamma (x)$ for all $x \in {{\mathcal {V}}}$ and $A \in {{\mathcal {G}}}$.

• (b) For each $x\in {{\mathcal {V}}}$, there exists $A\in {{\mathcal {G}}}$ such that $x=A\gamma (x)$.

• (c) For all $x,y\in {{\mathcal {V}}}$, we have $\langle x, y \rangle \le \langle \gamma (x), \gamma (y) \rangle$.

Then, $({{\mathcal {V}}}, {{\mathcal {G}}}, \gamma )$ is called a normal decomposition system.

Items (a) and (b) in the above definition show that $\gamma ^2=\gamma$ and $\left\| \gamma (x) \right\| = \left\| x \right\|$ for all x. We state a few relevant properties.

Proposition 10.7

[20, Proposition 2.3 and Theorem 2.4]. Let $({{\mathcal {V}}},{{\mathcal {G}}},\gamma )$ be a normal decomposition system. Then,

• (i) For any two elements x and y in ${{\mathcal {V}}}$, we have

  $$\begin{aligned} \max _{A \in {{\mathcal {G}}}}\, \langle Ax, y \rangle = \langle \gamma (x), \gamma (y) \rangle . \end{aligned}$$

  Also, $\langle x, y \rangle = \langle \gamma (x), \gamma (y) \rangle$ holds for two elements x and y if and only if there exists an $A \in {{\mathcal {G}}}$ such that $x = A\gamma (x)$ and $y = A\gamma (y)$.

• (ii) The range of $\gamma$, denoted by F, is a closed convex cone in ${{\mathcal {V}}}$.

1.2 Eaton Triples

Eaton triples were introduced and studied in [7,8,9] from the perspective of majorization techniques in probability. They were also extensively studied in the papers of Tam and Niezgoda, see the references.

Definition 10.8

Let ${{\mathcal {V}}}$ be a finite dimensional real inner product space, ${{\mathcal {G}}}$ be a closed subgroup of the orthogonal group of ${{\mathcal {V}}}$, and F be a closed convex cone in ${{\mathcal {V}}}$ satisfying the following conditions:

• (a) $\textrm{Orb}(x)\cap F \ne \emptyset$ for all $x \in {{\mathcal {V}}}$, where $\textrm{Orb}(x) := \{Ax : A \in {{\mathcal {G}}}\}$.

• (b) $\langle x, Ay \rangle \le \langle x, y \rangle$ for all $x, y \in F$ and $A \in {{\mathcal {G}}}$.

Then, $({{\mathcal {V}}}, {{\mathcal {G}}}, F)$ is called an Eaton triple.

It has been shown (see [26], page 14) that in an Eaton triple $(V, {{\mathcal {G}}}, F)$, $\text {Orb}(x)\cap F$ consists of exactly one element for each $x\in {{\mathcal {V}}}$. Defining $\gamma : {{\mathcal {V}}}\rightarrow {{\mathcal {V}}}$ such that $\text {Orb}(x) \cap F = \{\gamma (x)\}$, it has been observed that $({{\mathcal {V}}}, {{\mathcal {G}}}, \gamma )$ is a normal decomposition system. Also, given a finite dimensional normal decomposition system $({{\mathcal {V}}},{{\mathcal {G}}}, \gamma )$ with $F := \gamma ({{\mathcal {V}}})$, $({{\mathcal {V}}}, {{\mathcal {G}}}, F)$ becomes an Eaton triple. Thus, finite dimensional normal decomposition systems are equivalent to Eaton triples [20, 21, 27].

1.3 Rearrangement Inequality for Measurable Functions

The notion of rearrangement of a function, systematically introduced by Hardy and Littlewood, has played a key role in proving inequalities in classical and applied analysis. The definitions and properties in this subsection can be found in [5], Chapter 2.

Let $(\Omega , \Sigma , \mu )$ denote a $\sigma$-finite measure space.

Definition 10.9

Let $f : \Omega \rightarrow {\mathcal {R}}$ be a $\Sigma$-measurable function.

• The function $\mu _f : [0, \infty ) \rightarrow [0, \infty ]$ defined by

  $$\begin{aligned} \mu _f(\alpha ) = \mu \big ( \{x \in \Omega : \left| f(x) \right| > \alpha \} \big ) \end{aligned}$$

  is called the distribution function of f.

• The decreasing rearrangement of f is the function $f^* : [0, \infty ) \rightarrow [0, \infty ]$ defined by

  $$\begin{aligned} f^*(t) := \inf \{ \alpha \ge 0 : \mu _f(\alpha ) \le t \}, \end{aligned}$$

  where we use the convention that $\inf \emptyset = \infty$.

In the next two propositions, we record some basic properties of the distribution function and decreasing rearrangement.

Proposition 10.10

The following properties hold:

• (i) f and $f^*$ are equimeasurable, that is,

  $$\begin{aligned} \mu \big ( \{x \in \Omega : \left| f(x) \right|> \alpha \} \big ) = m \big ( \{t> 0 : f^*(t) > \alpha \} \big ) \end{aligned}$$

  for all $\alpha \ge 0$, where m is the Lebesgue measure.

• (ii) $\left\| f \right\| _{L_{p}(\Omega )} = \left\| f^* \right\| _{L_{p}[0, \infty )}$ for all positive real numbers p.

• (iii) The Hardy–Littlewood–Pólya inequality holds, i.e.,

  $$\begin{aligned} \int _{\Omega } \left| fg \right| \; d\mu \le \int _{0}^{\infty } f^{*}(t)g^{*}(t) \; dt. \end{aligned}$$

Proposition 10.11

There exists only one right-continuous decreasing function $f^*$ equimeasurable with f. Hence, the decreasing rearrangement is unique.

Example 10.12

Consider the measure space $(\Omega , \Sigma , \mu ) = ({\mathbb {N}}, 2^{{\mathbb {N}}}, \mu )$, where $\mu$ is the counting measure on ${\mathbb {N}}$. Then, any $\Sigma$-measurable function $f : \Omega \rightarrow {\mathcal {R}}$ can be realized as a sequence $(x_n)$. The decreasing rearrangement $f^*$ is a function defined on $[0, \infty )$, but can be interpreted as a sequence $(x_n^*)$, where, for any $n\in {\mathbb {N}}$,

$$\begin{aligned} x_n^* := f^*(t) \;\; \text {for} \;\; n-1 \le t < n. \end{aligned}$$

Formally, for any n,

$$\begin{aligned} x_n^* = \inf \Big \{ \alpha \ge 0 : \mu \big ( \{k \in {\mathbb {N}} : \left| x_k \right| > \alpha \} \big ) \le n-1 \Big \}. \end{aligned}$$

The following can easily be observed:

• (a) If $(x_n)$ has finite number of nonzero entries, say, k of them, then $x^*=(\left| x_{n_1} \right| , \left| x_{n_2} \right| , \ldots , \left| x_{n_k} \right| , 0, 0, \ldots )$ with $\left| x_{n_1} \right| \ge \left| x_{n_2} \right| \ge \cdots \ge \left| x_{n_k} \right|$.

• (b) If $(x_n)$ has infinitely many nonzero entries, then $x^*$ consists of absolute values of these entries arranged in the decreasing order; in particular, every entry of $x^*$ is nonzero.

For example, if $x = \Big ( 1, 0, \frac{1}{2}, 0, \frac{1}{3}, 0, 0, \ldots \Big )$, then $x^* = \Big ( 1, \frac{1}{2}, \frac{1}{3}, 0, 0, \ldots \Big )$. On the other hand, if $x = \Big ( 1, 0, \frac{1}{2}, 0, \frac{1}{3}, 0, \frac{1}{4}, 0, \ldots \Big )$, then $x^* = \Big ( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \Big )$.