Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2021
Volume
Vol. 41
Number
No. 3
Description
Special Issue - Spectral Theory and Applications
Journal Volume
Opuscula Mathematica
Vol. 41 (2021)
Projects
Pages
Articles
New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry
(Wydawnictwa AGH, 2021) Alpay, Daniel; Jørgensen, Palle E.T.
We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.
Perturbation series for Jacobi matrices and the quantum Rabi model
(Wydawnictwa AGH, 2021) Charif, Mirna; Zieliński, Lech
We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings.
Multi-variable quaternionic spectral analysis
(Wydawnictwa AGH, 2021) Cho, Ilwoo; Jørgensen, Palle E.T.
In this paper, we consider finite dimensional vector spaces $\mathbb{H}^n$ over the ring $\mathbb{H}$ of all quaternions. In particular, we are interested in certain functions acting on $\mathbb{H}^n$, and corresponding functional equations. Our main results show that (i) all quaternions of $\mathbb{H}$ are classified by the spectra of their realizations under representation, (ii) all vectors of $\mathbb{H}^n$ are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring $M_n(\mathbb{C})$ of all $(n \times n)$-matrices over the complex numbers C has close connections with certain »non-linear« functional equations on $\mathbb{H}^n$ up to the classification of (ii).
Extensions of dissipative operators with closable imaginary part
(Wydawnictwa AGH, 2021) Fischbacher, Christoph
Given a dissipative operator $A$ on a complex Hilbert space $\mathcal{H}$ such that the quadratic form $f \mapsto \text{Im}\langle f, Af \rangle$ is closable, we give a necessary and sufficient condition for an extension of $A$ to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.
Spectrum localization of a perturbed operator in a strip and applications
(Wydawnictwa AGH, 2021) Gil', Michael
Let $A$ and $\tilde{A}$ be bounded operators in a Hilbert space. We consider the following problem: let the spectrum of $A$ lie in some strip. In what strip the spectrum of $\tilde{A}$ lies if $A$ and $\tilde{A}$ are »close«? Applications of the obtained results to integral operators and matrices are also discussed. In addition, we apply our perturbation results to approximate the spectral strip of a Hilbert-Schmidt operator by the spectral strips of finite matrices.