Related papers: A note on the essential numerical range of block d…

In this paper we completely describe the numerical range of Toeplitz operators on weighted Bergman spaces with harmonic symbol. We also characterize the numerical range of weighted composition operators on weighted Bergman spaces and…

We investigate the relationship between ideal membership of an operator and its pieces relative to several canonical types of partitions of the entries of its matrix representation with respect to a given orthonormal basis. Our main…

Employing the Orlicz functions we extend the Buzano's inequality which is a refinement of the Cauchy-Schwarz inequality. Also using the Orlicz functions we obtain several numerical radius inequalities for a bounded linear operator as well…

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\left\{xy = n\right\}$ has between $\Omega(n^{1/3})$ and $O(n^{1/3} \log n)$ vertices. The same bounds apply to any hyperbola with rational…

We consider the nonconvex set $\mathcal S_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \geq 0,\; z \in \{0,1\}^n\}$, which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best…

Let $\A$ be a $C^*-$algebra with unit element $1$ and unitary group $U.$ Let $a=(a_1, ..., a_k)$ and $b=(b_1, ..., b_k)$ two $k-$tuples of elements in $\A.$ The elementary operator associated to $a$ and $b$ is defined by $ R_{a, b} (x)=…

Suppose that c is an operator on a Hilbert Space H such that the von Neumann algebra N generated by c is finite. Suppose that tau is a faithful normal tracial state on N. Let B denote the spectal scale of c with respect to tau. We show that…

Let $\mathbb{A}=\left( \begin{array}{cc} A & 0 \\ 0 & A \\ \end{array} \right)$ be the $2\times2$ diagonal operator matrix determined by a positive bounded operator $A$. For semi-Hilbertian operators $X$ and $Y$, we first show that…

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by deleting fewer than $k$ vertices. The block number $\beta(G)$ of $G$ is the maximum integer $k$ for which $G$ contains a…

We establish new metric characterizations for the norm (respectively, ultraweak) closure of the convex hull of a bounded set in an arbitrary $C^*$-algebra (respectively, von Neumann algebra), and provide applications of these results to the…

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix $A \in M_n$ has eigenvalues $a_1, \..., a_n$, then its higher rank…

We show that one can enumerate the vertices of the convex hull of integer points in polytopes whose constraint matrices have bounded and nonzero subdeterminants, in time polynomial in the dimension and encoding size of the polytope. This…

A commuting triple of operators $(A,B,P)$ on a Hilbert space $\mathcal{H}$ is called a tetrablock contraction if the closure of the set $$ E = \{\underline{x}=(x_1,x_2,x_3)\in \mathbb{C}^3: 1-x_1z-x_2w+x_3zw \neq 0 \text{whenever}|z| \leq…

The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correction. We will discuss some of the real algebraic and convex…

$\def\DD{{{\bf \delta}}}\def\CH{{\mathop{\mathrm{ConvexHull}}}}\newcommand{\LL}{{\cal {L}}} \newcommand{\ZZ}{\mathbb{Z}} $ Given a convex body $C$ in the plane, its discrete hull is $C^0 = \CH( C \cap \LL )$, where $\LL = \ZZ \times \ZZ$ is…

The Betke-Henk-Wills conjecture provides an upper bound for the lattice point enumerator $G(K, \Lambda)$ of a convex body in terms of its successive minima. While the conjecture is established for orthogonal parallelotopes, its validity for…

The goal of this note is to study the spectrum of a self-adjoint convolution operator in $L^2(\mathbb R^d)$ with an integrable kernel that is perturbed by an essentially bounded real-valued potential tending to zero at infinity. We show…

A complete analysis of the essential spectrum of matrix-differential operators $\mathcal A$ of the form \begin{align} \begin{pmatrix} -\displaystyle{\frac{\rm d}{\rm d t}} p \displaystyle{\frac{\rm d}{\rm d t}} + q &…

We study properties of convex hulls of (co)adjoint orbits of compact groups, with applications to invariant theory and tensor product decompositions. The notion of partial convex hulls is introduced and applied to define two numerical…

A bounded linear operator $T$ on a Banach space $X$ is said to be generalized Drazin-meromorphic invertible if there exists a bounded linear operator $S$ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is meromorphic. We shall say that…