arXiv subjects

Josué I. Rios-Cangas

Publications and source records attributed to Josué I. Rios-Cangas.

13 recordsLinked to original sources

arXiv2026

$C_0$-Semigroups on d.m.o. linear relations

This article develops a theory of one-parameter semigroups for linear relations, introducing the notion of domain-multivalued orthogonal (d.m.o.) relations. We establish complete characterizations for generators of uniformly and strongly continuous semigroups, showing them to be maximal and densely d.m.o. relations, respectively. Furthermore, the Hille-Yosida and Lumer-Phillips theorems are extended to this general setting, providing necessary and sufficient conditions for generating $C_0$-semigroups of contractions (identified as maximal anti-accumulative relations) and unitary operators (anti-selfadjoint relations). This rigorous framework sheds new light on the peculiarities of one-parameter semigroups for linear relations.

math-ph↗

arXiv2025

Quantum error-correcting codes via inner products and error bases

In this paper, we address the problem of state communication in finite-level quantum systems through noise-affected channels. Our approach is based on a self-consistent theory of decoding inner products associated with the code and error (or noise) bases defined on corrupting subspaces. This viewpoint yields new necessary and sufficient conditions for the existence of quantum error-correcting codes in terms of these inner products. The obtained results extend the foundations of quantum error correction beyond classical analogies, highlighting the structural insights offered by operator theory and the underlying product space.

quant-ph↗

arXiv2025

On the analytical approach to infinite-mode Boson-Gaussian states

We develop an analytical approach to quantum Gaussian states in infinite-mode representation of the Canonical Commutation Relations (CCR's), using Yosida approximations to define integrability of possibly unbounded observables with respect to a state $ρ$ ($ρ$-integrability). It turns out that all elements of the commutative $*$-algebra generated by a possibly unbounded $ρ$-integrable observable $A$, denoted by $\langle A\rangle$, are normal and $ρ\, $-integrable. Besides, $\langle A\rangle$ can be endowed with the well-defined norm $\|\cdot\|_ρ:= {\rm tr}\,(ρ|\cdot| )$. Our approach allows us to rigorously establish fundamental properties and derive key formulae for the mean value vector and the covariance operator. We additionally show that the covariance operator $S$ of any Gaussian state is real, bounded, positive, and invertible, with the property that $S-iJ\geq 0$, being $J$ the multiplication operator by $-i$ on $\ell_2({\mathbb N})$.

math-ph↗

arXiv2025

Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators

This work presents a rigorous characterization of inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators. We first deal with a general setting of continuous sesquilinear forms on a Hilbert space $\mathcal H$, and provide a characterization of all inner products by means of positive operators in $\mathcal {B(H)}$. Next, we establish necessary and sufficient conditions for an operator in $\mathcal B(S_2)$ to be positive. Identifying an inner product with a positive operator enables us to rigorously describe inner products on $S_2$.

math.FA↗

arXiv2024

The $k$-adjacency operators and adjacency Jacobi matrix on distance-regular graphs

We deal in this work with a class of graphs, namely, the class of distance-regular graphs, in which on the basis of $k$-adjacency operators, the adjacency operator $A$ of a distance-regular graph is identified as a Jacobi matrix. To get so, the set of the $k$-adjacency operators is recognized as a canonical basis in a certain Hilbert space, where the spectrum of the Jacobi matrix coincides with the support of the measure of $A$. The obtained identification permits a deeper spectral analysis of the graph. The finite-dimensional case is addressed by means of the extension theory of nondensely defined, symmetric linear operators.

math-ph↗

arXiv2023

The Krein transform and semi-bounded extensions of semi-bounded linear relations

The Krein transform is the real counterpart of the Cayley transform and gives a one-to-one correspondence between the positive relations and symmetric contractions. It is treated with a slight variation of the usual one, resulting in an involution for linear relations. On the other hand, a semi-bounded linear relation has closed semi-bounded symmetric extensions with semi-bounded selfadjoint extensions. A self-consistent theory of semi-bounded symmetric extensions of semi-bounded linear relations is presented. By using The Krein transform, a formula of positive extensions of quasi-null relations is provided.

math-ph↗

arXiv2023

The fast recurrent subspace on an $N$-level quantum energy transport model

The fast recurrent subspace (the biggest support of all invariant states) of a Weak Coupling Limit Type Quantum Markov Semigroup modeling a quantum transport open system of $N$-energy levels is determined. This is achieved by characterizing the structure of all the invariant state and their spectra in terms of a natural generalization of the Discrete Fourier Transform operator. Finally, the attraction domains and long-time behavior of the evolution are studied on hereditary subalgebras where faithful invariant states exist.

math-ph↗

arXiv2021

A Characterization of Quantum Gaussian States in Terms of Annihilation Moments

We give a rigorous definition of moments of an unbounded observable with respect to a quantum state in terms of Yosida's approximations of unbounded generators of contractions semigroups. We use this notion to characterize Gaussian states in terms of annihilation moments. As a by-product, rigorous formulae for the mean value vector and the covariance matrix of a Gaussian state are obtained.

math-ph↗

arXiv2020

Transition maps between Hilbert subspaces and quantum energy transport

We use a natural generalization of the discrete Fourier transform to define transition maps between Hilbert subspaces and the global transport operator $Z$. By using these transition maps as Kraus (or noise) operators, an extension of the quantum energy transport model of 10.1142/S0219025718500182 describing the dynamics of an open quantum system of $N$-levels is presented. We deduce the structure of the invariant states which can be recovered by transporting states supported on the first level.

math-ph↗

arXiv2019

Canonical decomposition of dissipative linear relations

On the basis of Sz. Nagy-Foiaş-Langer and von Neumann-Wold decompositions, two decompositions for dissipative linear relations are given and they are realized by transforming invariant subspaces for contractions, by means of the Z transform. These decompositions permit the separation of the selfadjoint and completely nonselfadjoint parts of a dissipative relation.

math.FA↗

arXiv2019

Perturbation theory for selfadjoint relations

We study Weyl-type perturbation theorems in the context of linear closed relations. We establish general results on perturbations for dissipative relations. In the particular case of selfadjoint relations, we study finite-rank perturbations and carry out a detailed analysis of the corresponding changes in the spectrum.

math-ph↗

arXiv2018

Dissipative extension theory for linear relations

This work is devoted to dissipative extension theory for dissipative linear relations. We give a self-consistent theory of extensions by generalizing the theory on symmetric extensions of symmetric operators. Several results on the properties of dissipative relations are proven. Finally, we deal with the spectral properties of dissipative extensions of dissipative relations and provide results concerning particular realizations of this general setting.

math-ph↗

arXiv2013

Inner Products on the Space of Complex Square Matrices

In this paper we study the problem of finding explicit expressions for inner products on the space of complex square matrices $\Mn$. We show that, given an inner product $\lip \cdot, \cdot \rip$ on $\Mn$, with some conditions, there exist positive matrices $A_j$ and $B_j \in \Mn$, for $j=1, 2\dots, m$ such that $$ \lip X, Y \rip = \sum_{j=1}^m \tr\left(Y^* A_j X B_j \right), $$ for all $X, Y \in \Mn$. However, we show that the result does not hold for all inner products. In fact, if the above expression does not hold, we show that there exist positive matrices $A_j$ and $B_j \in \Mn$, for $j=1, 2\dots, m$ such that $$ \lip X, Y \rip = -\tr\left(Y^* A_1 X B_1 \right)+ \sum_{j=2}^m \tr\left(Y^* A_j X B_j \right), $$ for all $X, Y \in \Mn$.

math.RA↗