Using the recent theory of Krein--von Neumann extensions for positive functionals we present several simple criteria to decide whether a given positive functional on the full operator algebra is normal. We also characterize those functionals defined on the left ideal of finite rank operators that have a normal extension.
We introduce the so-called Bishop-Phelps-Bollobás property for positive functionals, a particular case of the Bishop-Phelps-Bollobás property for positive operators. First we show a version of the Bishop-Phelps-Bollobás theorem for positive elements and positive functionals in the dual of any Banach lattice. We also characterize the strong Bishop-Phelps-Bollobás property for positive functionals in a Banach lattice. We prove that any finite-dimensional Banach lattice has the the Bishop-Phelps-Bollobás property for positive functionals. A sufficient and a necessary condition to have the Bishop-…
Let $\operatorname{Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space $M$ that vanish at a base point. We show that every normal functional in $\operatorname{Lip}_0(M)^\ast$ is weak$^*$ continuous, answering a question by N. Weaver.
Let $F$ be an ordered topological vector space (over $\mathbb{R}$) whose positive cone $F_+$ is weakly closed, and let $E \subseteq F$ be a subspace. We prove that the set of positive continuous linear functionals on $E$ that can be extended (positively and continuously) to $F$ is weak-$*$ dense in the topological dual wedge $E_+'$. Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces.
For a positive linear map F and a normal matrix N, we show that |F(N)| is bounded by some simple linear combinations in the unitary orbit of F(|N|). Several elegant sharp inequalities are derived, especially for the Schur product.
We prove Lieb type convexity and concavity results for trace functionals associated with positive operator monotone (decreasing) functions and certain monotone concave functions. This gives a partial generalization of Hiai's recent work on trace functionals associated with power functions.
We analyze the role played by $n$-convexity for the fulfillment of a series of linear functional inequalities that extend the Hornich-Hlawka functional inequality, $f\left( x\right) +f\left( y\right) +f\left( z\right) +f\left( x+y+z\right) \geq f\left( x+y\right) +f\left( y+z\right)+f\left( z+x\right) +f(0),$ including extensions to the case of positive operators.
Let $M$ be a discrete-time normal martingale that has the chaotic representation property. Then, from the space of square integrable functionals of $M$, one can construct generalized functionals of $M$. In this paper, by using a type of weights, we introduce a class of continuous linear operators acting on generalized functionals of $M$, which we call generalized weighted number (GWN) operators. We prove that GWN operators can be represented in terms of generalized annihilation and creation operators (acting on generalized functionals of $M$). We also examine commutation relations between a GW…
In this paper, using the recently discovered notion of the $S$-spectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be $-1$). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spec…
Given a closed Riemann surface $(Σ,g)$ and any positive smooth weight, we use a minmax scheme together with compactness, quantization results and with sharp energy estimates to prove the existence of positive critical points of the functional $$J_{p,β}(u)=\frac{2-p}{2}\left(\frac{p\|u\|_{H^1}^2}{2β} \right)^{\frac{p}{2-p}}-\ln \int_Σ(e^{u_+^p}-1) f dv_g,$$ for every $p\in (1,2)$ and $β>0$, {or} for $p=1$ and $β\in (0,\infty)\setminus 4π\mathbb{N}$. Letting $p\uparrow 2$ we obtain positive critical points of the Moser-Trudinger functional $$F(u):=\int_Σ(e^{u^2}-1)f dv_g$$ constrained to $\mathc…
