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Register a new userOn functions of bounded variation on convex domains in Hilbert spaces
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We study functions of bounded variation (and sets of finite perimeter) on a convex open set $Omegasubseteq X$, $X$ being an infinite dimensional real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein--Uhlenbeck operator.
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In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.
Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if $f$ is a nonlinear operator convex function on a bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result concerning a nonconstant operator monotone function on $[0,infty)$ is presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the Lowner--Heinz inequality.
We characterize the trace of magnetic Sobolev spaces defined in a half-space or in a smooth bounded domain in which the magnetic field $A$ is differentiable and its exterior derivative corresponding to the magnetic field $dA$ is bounded. In particular, we prove that, for $d ge 1$ and $p>1$, the trace of the magnetic Sobolev space $W^{1, p}_A(mathbb{R}^{d+1}_+)$ is exactly $W^{1-1/p, p}_{A^{shortparallel}}(mathbb{R}^d)$ where $A^{shortparallel}(x) =( A_1, dotsc, A_d)(x, 0)$ for $x in mathbb{R}^d$ with the convention $A = (A_1, dotsc, A_{d+1})$ when $A in C^1(overline{mathbb{R}^{d+1}_+}, mathbb{R}^{d+1})$. We also characterize fractional magnetic Sobolev spaces as interpolation spaces and give extension theorems from a half-space to the entire space.
In this article we study standard subspaces of Hilbert spaces of vector-valued holomorphic functions on tube domains E + i C^0, where C subeq E is a pointed generating cone invariant under e^{R h} for some endomorphism h in End(E), diagonalizable with the eigenvalues 1,0,-1 (generalizing a Lorentz boost). This data specifies a wedge domain W(E,C,h) subeq E and one of our main results exhibits corresponding standard subspaces as being generated using test functions on these domains. We also investigate aspects of reflection positivity for the triple (E,C,e^{pi i h}) and the support properties of distributions on E, arising as Fourier transforms of operator-valued measures defining the Hilbert spaces H. For the imaginary part of these distributions, we find similarities to the well known Huygens principle, relating to wedge duality in the Minkowski context. Interesting examples are the Riesz distributions associated to euclidean Jordan algebras.
In this work we are concerned with maximality of monotone operators representable by certain convex functions in non-reflexive Banach spaces. We also prove that these maximal monotone operators satisfy a Bronsted-Rockafellar type property. We show that if a function in XxX^* and its conjugate are above the duality product in their respective domains, then this function represents a maximal monotone operator.
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