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Consider the following (somewhat simplified) situation. Let $\mathcal{H}$ be separable Hilbert space and $\mathcal{B}(\mathcal{H})$ the Banach algebra of bounded linear operators acting on $\mathcal{H …

It is well known that the spectrum is continuous as function of operator. More precisely, let $\mathcal{H}$ be separable Hilbert space and $\mathcal{B}(\mathcal{H})$ the Banach algebra of linear opera …

I will formulate this question in the language of Jacobi operators and spectral measures although it could be entirely rewritten in terms of orthogonal polynomials and measures of orthogonality. Obj …

Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with …

It is well-known that the Hilbert matrix $H$, i.e., the symmetric Hankel matrix with entries $$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$ determines a bounded operator on $\ell^{2}(\mathbb{N …

I have several questions concerning history of Jacobi matrices. Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? What was his contribution …

You can find a hint in this article, where a tridiagonal operator with similar properties is diagonalized.

Let $A=(a_{i,j})_{i,j=1}^{\infty}$ be a semi-infinite matrix with real entries. Suppose further that $A$ and $A^{T}$ (matrix transpose) represent bounded operators on $\ell^{p}$ for $p\geq1$. Denote f …

Let $T_{n}(b)$ be the $n\times n$ Toeplitz matrix determined by the symbol $$ b(z)=\frac{1}{z}+\sum_{j=0}^{k}a_{j}z^{j} $$ where $k\in\mathbb{N}$ and $a_{0},\dots,a_{k}\in\mathbb{R}$, $a_{k}\neq0$. T …

This is not a complete answer to the question. However, the following example indicates that the curve $\Lambda(b)$ actually can separate the plane $\mathbb{C}$. However, this is just a numerically co …