I have been trying to prove this for a while but failed so far. Let $A$ and $B$ are two positive, self-adjoint operators with compact resolvent on a Hilbert space $H$ defined on the same dense domain $D \subset H$ and suppose that $A^{-1}B$ extends from $D$ to a bounded operator on all of $H$. … Read more
Let K be a given smooth, origin-symmetric, strictly convex body in n dimensional euclidean space. At every point x on the boundary of K there exists an origin-symmetric ellipsoid Ex that touches x of second-order, the osculating ellipsoid at x. Denote the family of osculating ellipsoids by F:={Ex:x∈boundary of K}. Moreover, set G:={TE:E∈F & T∈SL(n)}. Is it true … Read more
One may have an equation (with boundary conditions omitted below) ut−Au=f u(0)=u0 which has a weak solution u∈L2(0,T;V)∩C([0,T];H) in the sense that −∫T0∫Ωu(t)ϕt(t)+∫T0∫ΩA12u(t)A12ϕ(t)=∫T0∫Ωf(t)ϕ(t)+∫Ωu0ϕ(0) for all test functions ϕ vanishing at t=T. Since we do not know if ut exists or is a function, the concept of Steklov averages vh(t)=1h∫t+htv(s)ds is useful, since the weak formulation … Read more
How meaningful is it to try to integrate around the branch-cut of a function? For example lets say I have the function log(z2+a2) for a>0 and I choose my branch-cuts to be starting at ±ia and moving up and down the y−axis respectively. Now I am trying to integrate around a small circle around such … Read more
Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem has more complicated operator other then just $\Delta$ below. And it works in $BV$ space. But here I give … Read more
Let $X$ be a Banach space, and let $F(X)$ be the finite-rank operators on $X$. Then $X$ has the $\lambda$-BAP if there exists a net $(S_i)_i$ in $F(X)$ with $\sup_i\|S_i\|\leq\lambda$ such that $S_i\to\mathrm{id}_X$ pointwise, that is, $\|S_i(x)-x\|\to 0$ for every $x\in X$. What is the connection to the following variants of this property: (Explanation of … Read more
I asked this question at math.stackexchange and received no comment. Let A be a C*-algebra and ϕ be a positive linear functional on A. Let ˜ϕ be its unique w∗-continuous extension on A∗∗. Let us put Nϕ:={a∈A:ϕ(a∗a)=0} , N˜ϕ:={x∈A∗∗:ϕ(x∗x)=0} Nϕ forms a closed left ideal in A and N˜ϕ forms a w∗-closed left ideal in A∗∗. It … Read more
For a Lie group G and a locally convex space V let E(G,V) be the locally convex space of smooth functions from G to V, and accordingly E′c(G,V) the space of compactly supported distributions. A G-module V is called differentiable if v→(g→gv) defines a continuous map from V to E(G,V) for all v∈V. In particular … Read more
Conway’s box function is the inverse of Minkowski’s question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). The question is are there functional equations known about the function, which would allow recursive computation? Answer AttributionSource : Link , Question Author : Dimiter P … Read more
Let H be a non-separable Hilbert space and {ei}i∈I be an orthonormal basis for H. Let J be a uncountable proper subset in I. Let us put E={x∈B(H):‖ One may check that E is an open set in the strong operator topology but not in the weak operator topology. Question1: I feel E is not … Read more
