### Inequalities on the singular values of an off-diagonal block of a Hermitian matrix.

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Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of $A\u2081,...,{A}_{k}\in V$ by $A\u2081*\cdots *{A}_{k}={A}_{i\u2081}\cdots {A}_{i\u2098}$. This includes the usual product $A\u2081*\cdots *{A}_{k}=A\u2081\cdots {A}_{k}$ and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying ${\mu}^{m}=1$ such that ϕ: V → V has the form A...

Let W(A) and ${W}_{e}\left(A\right)$ be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that ${W}_{e}\left(A\right)$ is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, ${W}_{e}\left(A\right)$ can be obtained as the intersection of all sets of the form $cl\left(W(A\u2081,...,{A}_{i+1},{A}_{i}+F,{A}_{i+1},...,A\u2098)\right)$, where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in ${W}_{e}\left(A\right)$ as star centers....

In this paper, we develop some stochastic dominance theorems for the location and scale family and linear combinations of random variables and for risk lovers as well as risk averters that extend results in Hadar and Russell (1971) and Tesfatsion (1976). The results are discussed and applied to decision-making.

Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T\u2081\ast \cdots \ast {T}_{k}$ on elements in ${}_{i}$, which covers the usual product $T\u2081\ast \cdots \ast {T}_{k}=T\u2081\cdots {T}_{k}$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying $\sigma \left(\Phi \left(A\u2081\right)\ast \cdots \ast \Phi \left({A}_{k}\right)\right)=\sigma \left(A\u2081\ast \cdots \ast {A}_{k}\right)$ whenever any one of ${A}_{i}$’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied by a root...

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