Search results for "Hilbert spaces"

Frame-related Sequences in Chains and Scales of Hilbert Spaces

2022

Frames for Hilbert spaces are interesting for mathematicians but also important for applications in, e.g., signal analysis and physics. In both mathematics and physics, it is natural to consider a full scale of spaces, and not only a single one. In this paper, we study how certain frame-related properties of a certain sequence in one of the spaces, such as completeness or the property of being a (semi-) frame, propagate to the other ones in a scale of Hilbert spaces. We link that to the properties of the respective frame-related operators, such as analysis or synthesis. We start with a detailed survey of the theory of Hilbert chains. Using a canonical isomorphism, the properties of frame se…

Tensor product characterizations of mixed intersections of non quasianalytic classes and kernel theorems

2009

Mixed intersections of non quasi-analytic classes have been studied in [12]. Here we obtain tensor product representations of these spaces that lead to kernel theorems as well as to tensor product representations of intersections of non quasi-analytic classes on product of open or of compact sets (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Invertibility in tensor products of Q-algebras

2002

MR2986428 Lebedev, Leonid P.(CL-UNC); Vorovich, Iosif I.; Cloud, Michael J. Functional analysis in mechanics. Second edition. Springer Monographs in …

2014

Pietsch's factorization theorem for dominated polynomials

2007

Abstract We prove that, like in the linear case, there is a canonical prototype of a p -dominated homogeneous polynomial through which every p -dominated polynomial between Banach spaces factors.

Operators in Rigged Hilbert spaces: some spectral properties

2014

A notion of resolvent set for an operator acting in a rigged Hilbert space $\D \subset \H\subset \D^\times$ is proposed. This set depends on a family of intermediate locally convex spaces living between $\D$ and $\D^\times$, called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.

Rolle's Theorem for Polynomials of Degree Four in a Hilbert Space

2002

AbstractIn an infinite-dimensional real Hilbert space, we introduce a class of fourth-degree polynomials which do not satisfy Rolle's Theorem in the unit ball. Extending what happens in the finite-dimensional case, we show that every fourth-degree polynomial defined by a compact operator satisfies Rolle's Theorem.

Operators on Partial Inner Product Spaces: Towards a Spectral Analysis

2014

Given a LHS (Lattice of Hilbert spaces) $V_J$ and a symmetric operator $A$ in $V_J$, in the sense of partial inner product spaces, we define a generalized resolvent for $A$ and study the corresponding spectral properties. In particular, we examine, with help of the KLMN theorem, the question of generalized eigenvalues associated to points of the continuous (Hilbertian) spectrum. We give some examples, including so-called frame multipliers.

Generation of Frames

2004

It is well known that, given a generic frame, there exists a unique frame operator which satisfies, together with its adjoint, a double operator inequality. In this paper we start considering the inverse problem, that is how to associate a frame to certain operators satisfying the same kind of inequality. The main motivation of our analysis is the possibility of using frame theory in the discussion of some aspects of the quantum time evolution, both for open and for closed physical systems.

Electronic Processes in Solid State: Dirac Framework

2019

The present paper proposes canonical Dirac framework adapted for application to the electronic processes in solid state. The concern is a spatially periodic structure of atoms distinguished by birth and annihilation of particle states excited due to interaction with the electromagnetic field. This implies replacing the conventional energy-momentum relation specific of the canonical Dirac framework and permissible for particle physics by a case specific relation available for the solid state. The advancement is a unified and consistent mathematical framework incorporating the Hilbert space, the quantum field, and the special relativity. Essential details of the birth and annihilation of the …