Related papers: Spectral Sets: Numerical Range and Beyond
In [{\em The Numerical Range is a $(1 + \sqrt{2})$-Spectral Set}, SIAM J. Matrix Anal. Appl. 38 (2017), pp.~649-655], Crouzeix and Palencia show that the numerical range of a square matrix or linear operator $A$ is a $(1 +…
Crouzeix and Palencia recently showed that the numerical range of a Hilbert-space operator is a $(1+\sqrt2)$-spectral set for the operator. One of the principal ingredients of their proof can be formulated as an abstract functional-analysis…
We study spectral constants for convex domains $\Omega$ containing the spectrum of an operator. We extend the Crouzeix--Palencia framework by obtaining bounds depending on a parameter $\gamma$ and relating these bounds to geometric…
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…
It is known that, if $\Omega$ $\subset$ C is a convex set containing the numerical range of an operator A, then $\Omega$ is a C $\Omega$ -spectral set for A with C $\Omega$ $\le$ 1+ $\sqrt$ 2. We improve this estimate in unbounded cases.
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…
The quadratic numerical range $W^2(A)$ is a subset of the standard numerical range of a linear operator which still contains its spectrum. It arises naturally in operators which have a $2 \times 2$ block structure, and it consists of at…
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…
This is a survey about spectral sets, to appear in the second edition of Handbook of Linear Algebra (L. Hogben, ed.). Spectral sets and K-spectral sets, introduced by John von Neumann, offer a possibility to estimate the norm of functions…
Which convex subsets of the complex plane are the numerical range W(A of some matrix A? This paper gives a precise characterization of these sets. In addition to this we show that for any A there exists a symmetric matrix B of the same size…
We study the relation between the intrinsic and the spatial numerical ranges with the recently introduced "approximated" spatial numerical range. As main result, we show that the intrinsic numerical range always coincides with the convex…
We use results in [M. Crouzeix and A. Greenbaum,Spectral sets: numerical range and beyond, SIAM Jour. Matrix Anal. Appl., 40 (2019), pp. 1087-1101] to derive a variety of K-spectral sets and show how they can be used in some applications.…
GMRES is a popular Krylov subspace method for solving linear systems of equations involving a general non-Hermitian coefficient matrix. The conventional bounds on GMRES convergence involve polynomial approximation problems in the complex…
We define and study a numerical-range analogue of the notion of spectral set. Among the results obtained are a positivity criterion and a dilation theorem, analogous to those already known for spectral sets. An important difference from the…
We verify a conjecture on the structure of higher-rank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higher-rank numerical ranges for a generic unitary…
Suppose that c is a linear operator acting on an n-dimensional complex Hilbert Space H, and let tau denote the normalized trace on B(H). Set b_1 = (c+c*)/2 and b_2 = (c-c*)/2i, and write B for the the spectral scale of {b_1, b_2} with…
We define the complete numerical radius norm for homomorphisms from any operator algebra into ${\mathcal B}({\mathcal H})$, and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show…
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…
Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of…
We derive, similar to Lau and Riha, a matrix formulation of a general best approximation theorem of Singer for the special case of spectral approximations of a given matrix from a given subspace. Using our matrix formulation we describe the…