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
Assume Ω is a bounded domain in Rd with sufficiently smooth boundary. Let {λn,φn}∞n=1 be an orthonormal eigensystem of the Laplacian opertor −Δ, that is −Δφn=λnφn,φn∈H2(Ω)∩H10(Ω),n=1,2,… Setting cn(x)=∫T0ψ(x,t)exp(−λn(T−t))dt, where ψ∈C(¯Ω×[0,T]) and ψ>0 on ¯Ω×[0,T], I want to ask whether the span of {cnφn} is dense in L2(Ω), that is ∫Ωf(x)cn(x)φn(x)dx=0,∀n=1,2,… implies f≡0 in Ω. Thank … Read more
I am interested in sufficient criteria which ensure that a subset X of a Hilbert space H is a Lipschitz (or at least uniformly continuous) retract of H. Under which conditions on X⊂H is there a (nonlinear) Lipschitz (or uniformly continuous) map π:H→X with πx=x for all x∈X? Note: It is a crucial requirement here … Read more
Let M be a smooth, closed manifold, equipped with a smooth (finite) triangulation K. Further, let H be a Hilbert space, G:=π1(M) and let ρ:G→GL(H) be a representation (with GL(H) denoting the group of (continuous) automorphisms of H). Via the action of ρ, we can therefore regard H as a (left-) C[G]-module with. The triangulation … Read more
Let (Y,d) be a complete and separable CAT(0) space, fix y∈Y. Then, consider the tanget cone (TyY,dy) at y, i.e. the metric cone over the space of directions, and denote by 0y the ‘tip’ of such cone. It is well known that (TyY,dy) is CAT(0) space and, for any C⊂TyY convex and closed, the CAT(0) … Read more
In a finite-dimensional Hilbert space, the eigenvalues of the adjoint $A^*$ of an operator $A$ are the complex conjugates of the eigenvalues of $A$. But in infinite dimensions this need no longer be the case. For example, an annihilation operator $a$ has all complex numbers as eigenvalues (and coherent states as normalizable eigenstates), while the … Read more
Let H be a Hilbert space. Suppose that I take a fixed ONB of H let us call it {ei}i∈N and then I define ‖ where the supremum is over all pairs of subsequences of \mathbb{N}. Then look at all A \in B(\mathcal{H}) such that \| A \|_{D} < \infty. On this space it holds … Read more
Firstly, apologies for the imprecise language, I’m a physicist trying to understand anyonic excitations from the lens of category theory. If I have a category (say Rep(Z2)) then computing its Drinfeld centre will give you the anyonic excitations in the quantum double models (In this case: 1, e, m, ψ). The category corresponding to these … Read more
A topological quantum field theory is a functor $Z:Cob(n) \to Hilb$ mapping from the $n$-dimensional cobordisms to $Hilb$, the category of Hilbert spaces with morphisms being bounded linear operators. It’s a simple theorem that every such TQFT always takes values of finite-dimensional Hilbert spaces. This set of notes (prop. 2.6) says that it is possible … Read more
Since there was essentially no answers on my previous question, I will ask a partial case of it, which is very easy to state. Let (X,⟨⋅,⋅⟩) be an inner product (pre-Hilbert) space. Is it possible to describe the weak topology on X explicetely in terms of ⟨⋅,⋅⟩ as a function on X×X? That means without … Read more
