Narrow Results

Let X₁,…,X_k be quasinormed spaces with quasinorms | ⋅ |_j, j = 1,…,k, respectively. For any f = (f₁,⋯,f_k) ∈ X₁ ×⋯× X_k let ρ(f) be the unique non-negative root of the Cauchy polynomial . We prove that ρ(⋅) (which in general cannot be expressed by radicals when k ≥ 5) is a quasinorm on X₁ ×⋯× X_k, which we call root quasinorm, and we find a characterization of this quasinorm as limit of ratios of consecutive terms of a linear recurrence relation. If X₁,…,X_k are normed, Banach or Banach function spaces, then the same construction gives respectively a normed, Banach or a Banach function space. Norms obtained as roots of polynomials are already known in the framework of the variable Lebesgue spaces, in the case of the exponent simple function with values 1,…,k. We investigate the properties of the root quasinorm and we establish a number of inequalities, which come from a rich literature of the past century.

Let $E$ and $F$ be two symmetric quasi-Banach spaces and let ${ℳ}$ be a semifinite von Neumann algebra. The purpose of this paper is to study the product space $E({ℳ})\odot F({ℳ})$ and the space of multipliers from $E({ℳ})$ to $F({ℳ})$, i.e. $M(E({ℳ}),F({ℳ}))$. These spaces share many properties with their classical counterparts. Let $0<{\alpha }_0,{\alpha }_1<\infty .$ It is shown that if $F$ is ${\alpha }_1$-convex fully symmetric and $E$ is ${\alpha }_0$-convex, then $M(E({ℳ}),F({ℳ}))=M(E,F)({ℳ})$, where $M(E,F)({ℳ})=\{x\in L_0({ℳ}):\mu (x)\in M(E,F)\}$ and $M(E,F)$ is the space of multipliers from $E$ to $F.$ As an application, we give conditions on when $M(E({ℳ}),F({ℳ}))\odot E({ℳ})=F({ℳ}).$ Moreover, we show that the product space can be described with the help of complex interpolation method.

A well known vector measure version of the Radon-Nikodým theorem—comparing two vector measures— can also be interpreted in terms of factorization of an operator—the one associated to the first measure— through the one associated to the second one. Using several results published in recent years, we show the factorization version of this Radon-Nikodým theorem and its applications in the setting of the Operator Theory and the Harmonic Analysis.