scholarly journals Kriging for Hilbert-space valued random fields: The operatorial point of view

2016 ◽  
Vol 146 ◽  
pp. 84-94 ◽  
Author(s):  
Alessandra Menafoglio ◽  
Giovanni Petris
Keyword(s):  
Hilbert Space ◽  
Random Fields ◽  
Point Of View

scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

scholarly journals Paths of unitary access to exceptional points

2021 ◽  
Vol 2038 (1) ◽  
pp. 012026
Author(s):  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Phase Transitions ◽  
Quantum Phase Transitions ◽  
Quantum Systems ◽  
Point Of View ◽  
Explicit Description ◽  
Direct Access ◽  
Exceptional Point ◽  
Exceptional Points ◽  
Innovative Idea

Abstract With an innovative idea of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson’s papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato’s exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states ℒ into a triplet (viz., in our notation, spaces K and ℋ besides the conventional ℒ ). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.

MAGNETIC GROUP ON THE PSEUDOSPHERE

1992 ◽  
Vol 06 (21) ◽  
pp. 3525-3537 ◽  
Author(s):  
V. BARONE ◽  
V. PENNA ◽  
P. SODANO
Keyword(s):  
Magnetic Field ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Constant Magnetic Field ◽  
Coherent States ◽  
Compact Operators ◽  
Point Of View ◽  
Algebraic Point ◽  
Planar Limit ◽  
Discrete Part

The quantum mechanics of a particle moving on a pseudosphere under the action of a constant magnetic field is studied from an algebraic point of view. The magnetic group on the pseudosphere is SU(1, 1). The Hilbert space for the discrete part of the spectrum is investigated. The eigenstates of the non-compact operators (the hyperbolic magnetic translators) are constructed and shown to be expressible as continuous superpositions of coherent states. The planar limit of both the algebra and the eigenstates is analyzed. Some possible applications are briefly outlined.

scholarly journals Convergence theorems for fixed point iterative methods defined as admissible perturbations of a nonlinear operator

2013 ◽  
Vol 29 (1) ◽  
pp. 9-18
Author(s):  
VASILE BERINDE ◽  
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Nonlinear Operator ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Of View ◽  
Iterative Approximation ◽  
Convergence Theorems ◽  
Fixed Point Iterative Method

The aim of this paper is to prove some convergence theorems for a general fixed point iterative method defined by means of the new concept of admissible perturbation of a nonlinear operator, introduced in [Rus, I. A., An abstract point of view on iterative approximation of fixed points, Fixed Point Theory 13 (2012), No. 1, 179–192]. The obtained convergence theorems extend and unify some fundamental results in the iterative approximation of fixed points due to Petryshyn [Petryshyn, W. V., Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl. 14 (1966), 276–284] and Browder and Petryshyn [Browder, F. E. and Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), No. 2, 197–228].

The cardinal series in Hilbert space

1949 ◽  
Vol 45 (3) ◽  
pp. 335-341 ◽  
Author(s):  
J. D. Weston
Keyword(s):  
Hilbert Space ◽  
Point Of View ◽  
Mean Square ◽  
Interpolation Theory ◽  
Smooth Approximation ◽  
Cardinal Function ◽  
Image Position ◽  
Orthogonal Set

The name ‘cardinal function’ was given toby E. T. Whittaker (1), who considered it as a ‘smooth’ approximation to a function f(x), having the same values as f(x) at the points a + rw (r = 0, ± 1, ± 2, …). It has since been extensively studied (2), mainly from the point of view of interpolation theory. Hardy (3), however, observed that the functions νr(t) defined byform a normal orthogonal set on the interval (−∞, ∞), for r = 0, ± 1, ± 2, …. This fact suggests a discussion of the cardinal series from the point of view of mean-square approximation.

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