Linearizing an operator

This question is more about a curious identity I have come across, than to do with explicit research. The question is somewhat advanced so I’m posting it here rather than on math stackexchange. It comes across as a sort of juvenile identity, but the work put in to show the result is rather tenuous. For … Read more

Matrices with almost constant coefficient have a simple eigenvalue

As a by-product of a general result for bounded operators of a Banach space, I have the following: A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$, on all row $i$ it holds \begin{equation} \frac{1}{n}\sum_k \lvert \ell_{ik}-c\rvert\le \frac{|c|}{6}, \label{eq:Perron} \end{equation} must have a simple eigenvalue. Under a slightly stronger … Read more

Operator topologies

Let L(H) be the space of bounded operators on some Hilbert space. We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT). It is immediate that the norm topology is stronger than the SOT which is again stronger than the WOT. My question is … Read more

On decidability of an infinite dimensional dynamic system

Consider two infinite dimensional Toeplitz integer matrices $A,B$ where $B$ is just a shift operator and an infinite dimensional vector $V_0$. Given a prime $p$ and an infinite dimensional integer vector $U_0$ with $\|U_0\|_1\in\Bbb Z$ and $\|U_0\|_1<\infty$ is it easy to decide whether there is a $k$ such that the sequence of vectors $$U_k=AU_{k-1}+V_{k}=AU_{k-1}+BV_{k-1}$$ satisfies … Read more

Minkowski determinant inequality for the Fuglede-Kadison determinant

For positive-semidefinite matrices $A, B$ in $M_n(\mathbb{C})$, the Minkowski determinant theorem tells us that $\det(A+B)^{\frac{1}{n}} \ge \det(A)^{\frac{1}{n}} + \det(B)^{\frac{1}{n}}$. For a matrix $A$ in $M_n(\mathbb{C})$, the Fuglede-Kadison determinant is given by $\Delta(A) = |\det(A)|^{\frac{1}{n}}$. A natural guess is that an inequality of the form $\Delta(A+B) \ge \Delta(A) + \Delta(B)$ should hold for positive operators $A, … Read more

Best Approximation in Operator/non-Frobenius Norm

Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator X with a scalar multiple of another operator Y minimize:‖ has a closed form solution \alpha = \frac{\langle X, Y \rangle}{\langle Y,Y \rangle} via an orthogonal projection. What if I want to consider a non-Frobenius … Read more

A bounded operator associated with a principal bundle

Assume that (X,B,G) is a G– principal bundle where G is a compact topological or Lie group. The normalized Haar measure of (each fiber of ) X is denoted by μ. The space of continuos complex valued function on X is denoted by C(X) which is equiped with the sup norm and the standard operation. … Read more

Extension of Coburn’s theorem on isometry and Toeplitz algebra

\newcommand{\id}{\mathrm{id}}Let H be a Hilbert space, and X \in B(H) a proper isometry (i.e. X^{\star}X = \id and XX^{\star} \neq \id). Coburn’s theorem states that {\rm C}^{\star}(X), the {\rm C}^{\star}-algebra generated by X, is the Toeplitz algebra. We are interested in an extension of Coburn’s theorem. Consider Y \in B(H) such that Y^{\star}Y = \id+p, … Read more

Reductive Operator Problem

In the 1972 paper ”An equivalent Formulation of the Invariant Subspace Conjecture” Dyer, Pedersen, and Porcelli announce the following result: The Invariant Subspace Problem has a positive solution if and only if every reductive operator is normal (the Reductive Operator Problem). This result is cited in a few subsequent papers, as well as in Radjavi … Read more

One-parameter unitary group preserving invariant domain of infinitesimal generator

Let H be a separable Hilbert space (e.g. L2(Rd)), and let D1⊂H be a dense subspace (e.g. S(Rd)). Suppose that an operator A:D1→D1 is essentially self-adjoint. By Stone’s theorem, we know that there exists a strongly continuous one-parameter group eitA with infinitesimal generator A, i.e. limt→0−it(eitA−Id)f=Af,f∈Dom(A). My question is the following. Question (attempt 1). When … Read more