- ホーム
- > 洋書
- > 英文書
- > Science / Mathematics
Full Description
An operator $C$ on a Hilbert space $\mathcal H$ dilates to an operator $T$ on a Hilbert space $\mathcal K$ if there is an isometry $V:\mathcal H\to \mathcal K$ such that $C= V^* TV$. A main result of this paper is, for a positive integer $d$, the simultaneous dilation, up to a sharp factor $\vartheta (d)$, expressed as a ratio of $\Gamma $ functions for $d$ even, of all $d\times d$ symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space.
Contents
Introduction
Dilations and Free Spectrahedral Inclusions
Lifting and Averaging
A Simplified Form for $\vartheta $
$\vartheta$ is the Optimal Bound
The Optimality Condition $\alpha =\beta $ in Terms of Beta Functions
Rank versus Size for the Matrix Cube
Free Spectrahedral Inclusion Generalities
Reformulation of the Optimization Problem
Simmons' Theorem for Half Integers
Bounds on the Median and the Equipoint of the Beta Distribution
Proof of Theorem 2.1
Estimating $\vartheta (d)$ for Odd $d$.
Dilations and Inclusions of Balls
Probabilistic Theorems and Interpretations continued
Bibliography
Index.
