All Questions

Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$. Question. Does there exist a Lebesgue measurable ...

I try to understand a particular step on page two from the research paper "A multidimensional analog of a limit theorem of G. Szegö". https://iopscience.iop.org/article/10.1070/...

Consider two positive-semi definite matrices $T_1, T_2$ of unit trace. Let $T(\lambda):=T_1 + \lambda(T_2-T_1)$ be the convex combination of the two. We then study $f(\lambda) := \operatorname{tr}(T(\...

Let $V$ be a finite-dimensional vector space and consider the space $X=V\times V\times V\times V.$ Consider the block matrix $$A = \begin{pmatrix} A_1 & A_2 \\ A_2^* & -A_1\end{pmatrix}$$ ...

We investigate the Hilbert space $\ell^2(\mathbb{N}_0)$ with standard orthonormal basis vectors $e_n:=(0,...,0,1,0,...).$ Consider the family of self-adjoint rank $1$ projections $P_n\bullet:= \...

Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$ consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ...