Workshop Numerical Ranges and Spectral Enclosures

Block numerical range for linear relations

Abstract: We extend the concept of the block numerical range for $n\times n$ block operator matrices to linear relations in Hilbert spaces, which are given in range representations. Moreover, some fundamental properties of closed linear relations related to their operator parts are used to derive further spectral inclusion results. In particular, under certain assumptions, we show that the block numerical range of a closed linear relation $\mathcal{L}$ contains its eigenvalues and that the approximate point spectrum of $\mathcal{L}$ is contained in the closure of its block numerical range.

On the hyperbolicity of the Krein space numerical range

Abstract: We investigate the Krein space numerical range of two-by-two block matrices, with diagonal blocks as scalar multiples of the identity. For these matrices, we specifically investigate the cases when the respective boundary generating curves consist of hyperbolas. This provides a unified approach to derive established and new results concerning the numerical range hyperbolic shape. 

Product of nonnegative selfadjoint operators in unbounded settings

Abstract: In this talk, necessary and sufficient conditions are established for the factorization of a closed, in general, unbounded operator T = AB into a product of two nonnegative selfadjoint operators A and B. In particular, it is proved that this class of operators can be characterized not only by means of quasi-affinity to an operator S = S∗ ≥ 0, but also via Sebestyén inequality, a result known in the setting of bounded operators T. Furthermore, it is proved that the above T has a positive spectrum and shares several local spectral properties such as the SVEP and the Dunford’s property. The talk is based on joint work with Seppo Hassi (Vaasa).

Essential numerical ranges and multiplier tricks 

Abstract: The essential numerical range of a linear operator is used to describe the set of spectral pollution when approximating the operator by projection or domain truncation methods. We propose to multiply both sides of the eigenvalue problem by an operator and study the essential numerical range of the resulting linear pencil. By taking the intersection over various multipliers, we get sharp enclosures for the set of spectral pollution. We apply the results to indefinite differential operators. This talk is based on joint work with Marco Marletta (Cardiff).

The Essential Numerical Spectrum of a Linear Operator in a Banach Space

Abstract: The purpose of this presentation is to define and develop a new notion of the essential numerical spectrum of an operator on a Banach space  and to study its properties. Our definition is closely related to the essential numerical range. 

On the factorization of monic quadratic operator pencils withs accretive coefficients

Abstract: In this work, a canonical factorization is given for a quadratic pencil of accretive operators in a Hilbert space. We establish a criterion in order that the linear factors, into which the pencil splits, generates an holomorphic semi-group of contraction operators. As an application, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation in the non-homogeneous case. An illustrative example is also given.

Spectral properties of Hamiltonian operator pencils

Abstract: The spectral properties of operator pencils of the form \mathcal{A}(\lambda) = \lambda E-A are studied. This is achieved through a corresponding linear relation in a Krein space, which satisfies certain properties, such as nonnegativity and finitely many negative squares in some Krein space, under suitable assumptions on E and A. One can then use the well-known spectral properties of such linear relations in Krein spaces to derive the spectral properties of the operator pencil \mathcal{A}(\lambda).

Lumer-Phillips version of linear relations that generate holomorphic semigroups

Abstract: We present results on m-dissipative relations, prove generation results and the Lumer-Phillips version of linear relations that generates holomorphic semigroups.

Numerical ranges of non-negative operators in Krein spaces

Abstract: The numerical range of an operator on a Krein space is defined analogously to the classical numerical range, with the indefinite inner product taking the place of the Hilbert space inner product. As a consequence, it is always the union of at most two convex sets. For non-negative operators, it is known that their (necessarily real) spectrum lies in the closure of this numerical range, giving a basic spectral enclosure. In this talk, we provide a complete characterization of the numerical range for non-negative operators, showing that it coincides with the entire real line, except possibly for a bounded interval whose closure contains zero. This demonstrates that the aforementioned spectral enclosure yields information only about small spectral values. To overcome this limitation, we introduce the co-numerical range, analyze its structure, and prove that it delivers a spectral enclosure that captures the large spectral values. Taken together, the two enclosures provide a tight bound on the spectrum. This is joint work with Carsten Trunk (TU Ilmenau).

Invariance of the essential spectra of closed linear operator pencils on a Banach space

Abstract: In this presentation, we detail some results of the essential spectra of closed linear operators pencils defined on a Banach space X. This is done by enacting a relation between the essential spectrum of closed linear operator pencils and that of linear relations. We identify some essential spectra of an unbounded 2* 2 block operator matrices pencils with non diagonal domain acting in Banach spaces.

New spectral bounds for systems with strong damping: numerical range versus quadratic numerical range

In this talk we present two different spectral enclosures for quadratic operator polynomials with unbounded coefficients, one using the quadratic numerical range of the companion operator and the other one using the numerical range of the operator polynomial. The results are applied to abstract second order Cauchy problems with non-selfadjoint damping and  to damped wave equations in $\R^d$. Our results provide much tighter enclosures than standard bounds using  the numerical range of the companion operator, for both uniformly accretive and sectorial damping, and even in the case of selfadjoint damping.

(joint work with B.  Jacob, C. Trunk and H. Vogt as well as with N. Hefti)

Spectral properties of indefinite Laplace operators in star graphs

Abstract: A complete characterization of the spectrum of an indefinite (sign-changing) Laplace operator on a star graph is presented. We use the Krein space framework and boundary triple techniques to investigate this problem.

Selfadjoint pencils and relations in indefinite inner product spaces

Abstract: We investigate the connections between selfadjoint pencils and selfadjoint linear relations (which are subspaces) in $\mathbb{C}^n\times \mathbb{C}^n$. The pencil $sE-A, \,s\in \mathbb{C},$ with $E,A \in \mathbb{C}^{n\times n}$ is called selfadjoint if $E=E^*$ and $A=A^*$. The pencil is regular if $\det(sE-A)$ is not identical zero, otherwise it is singular. Let $J=J^*\in \C^{n\times n}$ be invertible. With $J$ as the Gramian a possibly indefinite $J$-inner product is defined in $\mathbb{C}^n$ via $[x,y]:=y^*Jx$. Let $\mathcal{R}\subseteq\mathbb{C}^n\times \mathbb{C}^n$ be a linear relation in $\mathbb{C}^n$ with a range representation $\mathcal{R}=\left\{(Ez, Az)\in \mathbb{C}^n\times \mathbb{C}^n :~z\in\mathbb{C}^n\right\}.$ Then $\mathcal{R}$ is selfadjoint with respect to the $J$-inner product if and only if $\dim \mathcal{R} = n$ and $A^*JE=E^*JA$. It is shown that any selfadjoint linear relation with nonempty resolvent set corresponds to a selfadjoint regular matrix pencil. Using the Thompson normal form for selfadjoint pencils the structure of the corresponding Gramian $J$ is presented in the case of real, non-real and infinite eigenvalues. In the case of singular selfadjoint pencils, it turns out that $J$ is not invertible anymore, but selfadjoint singular relations correspond to matrix pencils with a structure closely related to the Thompson form. This talk is based on a joint work with Timo Reis and Carsten Trunk (TU Ilmenau).

Maximally dissipative extensions of symmetric operators

Abstract: Using boundary triples, we develop an abstract framework to investigate the complete non-selfadjointness of the maximally dissipative extensions of dissipative operators of the form S+iV, where S is symmetric with equal finite defect indices and V is a bounded non-negative operator. Our key example is the dissipative Schrödinger operator on the interval. This is joint work with C. Fischbacher and A. Patiño.

Computing the Quadratic Numerical Range

Abstract: We present a new algorithm to compute the quadratic numerical range of a matrix. It performs significantly better than random vector sampling, the canonical method used so far. The quadratic numerical range, introduced by Heinz Langer and Christiane Tretter in 1998, is a non-convex subset of the numerical range of a matrix or operator that still contains the spectrum. While non-convexity is an advantage, as it allows for tighter spectral enclosures, it is at the same time one of the reasons that make the quadratic numerical range hard to compute numerically. We describe how these difficulties can be overcome by using a steepest ascent method with a penalty term to locate non-convex parts of the boundary. We also present a theoretical result that explains why random vector sampling becomes infeasible already for moderately sized matrices. The talk is based on joined work with Birgit Jacob and Lukas Vorberg.