A strongly open set which is not measurable in the weak operator topology

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

How can a sequence of functions be dense in L^2

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

Sufficient criteria for X⊂HX \subset \mathcal{H} to be a Lipschitz (or unif. cont.) retract of H\mathcal{H}

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

Reference Request: De Rham isomorphism with Hilbert space coefficients

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

Eigenvalues and spectrum of the adjoint

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

Is this subspace of B(H)B(\mathcal{H}) known?

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

Drinfeld center of a tensor category

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

Infinite dimensional topological quantum field theories?

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

Weak topology on a pre-Hilbert Space

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