Dimension of the space of unitary equivariant translation invariant tensor valuations

凸函数上的等变赋值

Equivariant Valuations on Convex Functions

本研究对满足对偶上图平移不变性、关于保体积线性映射等变或反变的有限凸函数空间连续赋值进行分类，找到差体映射的赋值论泛函类似物，证明投影体映射推广不存在且该结论适用于原点邻域有限的凸函数赋值。

Rotation equivariant Minkowski valuations

The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only non-trivial operator with these properties.

Monge-Ampère operators and valuations

Two classes of measure-valued valuations on convex functions related to Monge-Ampère operators are investigated and classified. It is shown that the space of all valuations with values in the space of complex Radon measures on $\mathbb{R}^n$ that are locally determined, continuous, dually epi-translation invariant as well as translation equivariant, is finite dimensional. Integral representations of these valuations and a description in terms of mixed Monge-Ampère operators are established, as well as a characterization of $\mathrm{SO}(n)$-equivariant valuations in terms of Hessian measures.

Even Minkowski Valuations

A new integral representation of smooth translation invariant and rotation equivariant even Minkowski valuations is established. Explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived. Moreover, the action of Alesker's Hard Lefschetz operators on these Minkowski valuations is explored in detail.

A product formula for homogeneous characteristic functions

A bounded linear operator $T$ on a Hilbert space is said to be homogeneous if $φ(T)$ is unitarily equivalent to $T$ for all $φ$ in the group Möb of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $σ$ of Möb is said to be associated with an operator T if $φ(T)= σ(φ)^\star T σ(φ)$ for all $φ$ in Möb. In this paper, we develop a Möbius equivariant version of the Sz.-Nagy--Foias model theory for completely non-unitary (cnu) contractions. As an application, we prove that if T is a cnu contraction with associated (projective unitary) representation $σ$, then there …