Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInequalities for trace on $tau$-measurable operators
192
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert space equipped with a faithful normal semifinite trace $tau$. A closed densely defined operator $x$ affiliated with $mathfrak{M}$ is called $tau$-measurable if there exists a number $lambda geq 0$ such that $tau left(e^{|x|}(lambda,infty)right)<infty$. A number of useful inequalities, which are known for the trace on Hilbert space operators, are extended to trace on $tau$-measurable operators. In particular, these inequalities imply Clarkson inequalities for $n$-tuples of $tau$-measurable operators. A general parallelogram law for $tau$-measurable operators are given as well.
rate research
Read More
Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $tau$ on $M$ let $S(M, tau)$ (resp. $S_0(M, tau)$) be the algebra of all $tau$-measurable (resp. $tau$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,tau)$) is inner, and each derivation on $S_0(M, tau)$ is spatial and implemented by an element from $S(M, tau).$
Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $tau,$ let $S_0(M, tau)$ be the algebra of all $tau$-compact operators affiliated with $M.$ We give a complete description of all derivations on the algebra $S_0(M, tau).$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $S_0(M, tau)$ is spatial.
We establish that trace inequalities $$|D^{k-1}u|_{L^{frac{n-s}{n-1}}(mathbb{R}^{n},dmu)} leq c |mu|_{L^{1,n-s}(mathbb{R}^{n})}^{frac{n-1}{n-s}}|mathbb{A}[D]u|_{L^{1}(mathbb{R}^{n},dmathscr{L}^{n})}$$ hold for vector fields $uin C^{infty}(mathbb{R}^{n};mathbb{R}^{N})$ if and only if the $k$-th order homogeneous linear differential operator $mathbb{A}[D]$ on $mathbb{R}^{n}$ is elliptic and cancelling, provided that $s<1$, and give partial results for $s=1$, where stronger conditions on $mathbb{A}[D]$ are necessary. Here, $|mu|_{L^{1,lambda}}$ denotes the $(1,lambda)$-Morrey norm of the measure $mu$, so that such traces can be taken, for example, with respect to the Hausdorff measure $mathscr{H}^{n-s}$ restricted to fractals of codimension $0<s<1$. The above class of inequalities give a systematic generalisation of Adams trace inequalities to the limit case $p=1$ and can be used to prove trace embeddings for functions of bounded $mathbb{A}$-variation, thereby comprising Sobolev functions and functions of bounded variation or deformation. We moreover establish a multiplicative version of the above inequality, which implies ($mathbb{A}$-)strict continuity of the associated trace operators on $text{BV}^{mathbb{A}}$.
The parallel sum for adjoinable operators on Hilbert $C^*$-modules is introduced and studied. Some results known for matrices and bounded linear operators on Hilbert spaces are generalized to the case of adjointable operators on Hilbert $C^*$-modules. It is shown that there exist a Hilbert $C^*$-module $H$ and two positive operators $A, Binmathcal{L}(H)$ such that the operator equation $A^{1/2}=(A+B)^{1/2}X, Xin cal{L}(H)$ has no solution, where $mathcal{L}(H)$ denotes the set of all adjointable operators on $H$.
We investigate the orthogonality preserving property for pairs of mappings on inner product $C^*$-modules extending existing results for a single orthogonality-preserving mapping. Guided by the point of view that the $C^*$-valued inner product structure of a Hilbert $C^*$-module is determined essentially by the module structure and by the orthogonality structure, pairs of linear and local orthogonality-preserving mappings are investigated, not a priori bounded. The intuition is that most often $C^*$-linearity and boundedness can be derived from the settings under consideration. In particular, we obtain that if $mathscr{A}$ is a $C^{*}$-algebra and $T, S:mathscr{E}longrightarrow mathscr{F}$ are two bounded ${mathscr A}$-linear mappings between full Hilbert $mathscr{A}$-modules, then $langle x, yrangle = 0$ implies $langle T(x), S(y)rangle = 0$ for all $x, yin mathscr{E}$ if and only if there exists an element $gamma$ of the center $Z(M({mathscr A}))$ of the multiplier algebra $M({mathscr A})$ of ${mathscr A}$ such that $langle T(x), S(y)rangle = gamma langle x, yrangle$ for all $x, yin mathscr{E}$. In particular, for adjointable operators $S$ we have $T=(S^*)^{-1}$, and any bounded invertible module operator $T$ may appear. Varying the conditions on the mappings $T$ and $S$ we obtain further affirmative results for local operators and for pairs of a bounded and of an unbounded module operator with bounded inverse, among others. Also, unbounded operators with disjoint ranges are considered. The proving techniques give new insights.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
