Related papers: On Generalized Numerical Ranges of Quadratic Opera…

We introduce the concept of essential numerical range $W_{\!e}(T)$ for unbounded Hilbert space operators $T$ and study its fundamental properties including possible equivalent characterizations and perturbation results. Many of the…

The numerical range in the quaternionic setting is, in general, a non convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense,…

We give new necessary and sufficient conditions for the numerical range $W(T)$ of an operator $T \in \mathcal{B}(\mathcal{H})$ to be a subset of the closed elliptical set $K_\delta \subseteq \mathbb{C}$ given by \[ K_\delta {\stackrel{\rm…

We investigate the generalized quadratic operator defined by $$T =\left( \begin{array}{cc} a I_H & A \\ c A^* & bI_K \end{array} \right) ,$$ where $H$ and $K$ are Hilbert spaces, $A:K\to H$ is a bounded linear operator, $I_H$ and $I_K$…

We give a description of the essential spectrum of a large class of operators on metric measure spaces in terms of their localizations at infinity. These operators are analogues of the elliptic operators on Euclidean spaces and our main…

In this article we provide a generalized version of the result of L.H. Son and W. Tutschke \cite{tut} on the solvability of first order systems on the plane whose initial functions are arbitrary holomorphic functions. This is achieved by…

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…

It is shown that the numerical range of a linear operator operator in a Hilbert space is a (complete) $(1{+}\sqrt2)$-spectral set. The proof relies, among other things, in the behavior of the Cauchy transform of the conjugates of…

In this paper, we begin by showing a new generalization of the celebrated Cauchy-Schwarz inequality for the inner product. Then, this generalization is used to present some bounds for the Euclidean operator radius and the Euclidean operator…

We prove global subelliptic estimates for quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. In a previous joint work with M. Hitrik, we…

In infinite dimensions and on the level of trace-class operators $C$ rather than matrices, we show that the closure of the $C$-numerical range $W_C(T)$ is always star-shaped with respect to the set $\operatorname{tr}(C)W_e(T)$, where…

The numerical range of a bounded linear operator on a complex Banach space need not be convex unlike that on a Hilbert space. The aim of this paper is to study operators $T$ on $ \ell^2_p $ for which the numerical range is convex. We also…

In this note we characterize the essential numerical range of a block diagonal o\-pe\-ra\-tor $T=\bigoplus_i T_i$ in terms of the numerical ranges $\{W(T_i)\}_i$ of its components. Specifically, the essential numerical range of $T$ is the…

We prove quantitative Runge type approximation results for spaces of smooth zero solutions of several classes of linear partial differential operators with constant coefficients. Among others, we establish such results for arbitrary…

In this paper, we generalize the notion of the $C$-numerical range of a matrix to operators in arbitrary tracial von Neumann algebras. For each self-adjoint operator $C$, the $C$-numerical range of such an operator is defined; it is a…

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 consider a family of gradient Gaussian vector fields on $\Z^d$, where the covariance operator is not translation invariant. A uniform finite range decomposition of the corresponding covariance operators is proven, i.e., the covariance…

Let $D$ be a bounded convex domain in $\mathbb{C}$ with a regular analytic boundary. Suppose that the numerical range $W(A)$ of a bounded linear operator $A$ is contained in $\overline{D}$. If $\overline{W(A)}$ intersects the boundary…

In their LAMA'2016 paper Gau, Wang and Wu conjectured that a partial isometry $A$ acting on $\mathbb{C}^n$ cannot have a circular numerical range with a non-zero center, and proved this conjecture for $n\leq 4$. We prove it for operators…

We generalise the Elliptical Range Theorem to characterise the numerical range of matrices belonging to a subspace of the space of \(3 \times 3\) matrices. Using Specht's Theorem, which characterizes when two matrices are unitarily…