Related papers: Enclosure of the Numerical Range and Resolvent Est…
In this paper we introduce an enclosure of the numerical range of a class of rational operator functions. In contrast to the numerical range the presented enclosure can be computed exactly in the infinite dimensional case as well as in the…
We identify subsets of the joint numerical range of an operator tuple in terms of its joint spectrum. This result helps us to transfer weak convergence of operator orbits into certain approximation and interpolation properties for powers in…
We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations $\ddot{z}(t) + D \dot{z} (t) + A_0 z(t) = 0$ in a Hilbert space. Our main tool is the quadratic…
In this paper we compute the closure of the numerical range of certain periodic tridiagonal operators. This is achieved by showing that the closure of the numerical range of such operators can be expressed as the closure of the convex hull…
In this note we introduce the concept of the numerical range of a bounded linear operator with respect to a family of projections. We give a precise definition and elaborate on its connection to the classical numerical range as well as to…
We study containment regions of the numerical range of the product of operators $A$ and $B$ such that $W(A)$ and $W(B)$ are line segments. It is shown that the containment region is equal to the convex hull of elliptical disks determined by…
We study resolvent estimates for non-selfadjoint semiclassical pseudodifferential operators with double characteristics. Assuming that the quadratic approximation along the double characteristics is elliptic, we obtain polynomial upper…
We study the relation between the spectrum of a self-adjoint operator and its multidimensional numerical range. It turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the…
We study the spectrum of a periodic non-self-adjoint Dirac operator, and its dependence on a semiclassical parameter is also considered. Several bounds on the spectrum are obtained which provide sharp spectral enclosure estimates.…
Sharp resolvent bounds for non-selfadjoint semiclassical elliptic quadratic differential operators are established, in the interior of the range of the associated quadratic symbol.
We prove a semiclassical resolvent estimate for a broad class of non-self-adjoint, non-elliptic pseudodifferential operators in the low-lying spectral regime. The proof relies on improved ellipticity properties for the symbol of the…
We study the convergence of a family of numerical integration methods where the numerical integral is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, the convergence has already been…
We present some upper and lower bounds for the numerical radius of a bounded linear operator defined on complex Hilbert space, which improves on the existing upper and lower bounds. We also present an upper bound for the spectral radius of…
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…
The convergence of the so-called quadratic method for computing eigenvalue enclosures of general self-adjoint operators is examined. Explicit asymptotic bounds for convergence to isolated eigenvalues are found. These bounds turn out to…
Numerical range of a Hermitian operator X is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken…
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…
We investigate the relation between the spectrum of a non-normal matrix and the norm of its resolvent. We provide spectral estimates for the resolvent of matrices whose largest singular value is bounded by $1$ (so-called Hilbert space…
We establish new integral inequalities for the numerical radius and the operator norm of bounded linear operators on Hilbert spaces. Our results refine classical triangle-type and operator matrix inequalities by incorporating convex…
This chapter deals with the notion of the resolvent of a self-adjoint operator. We pay special attention to the convergence of unbounded self-adjoint operators in several resolvent senses, and how they are related to the convergence of…