Search results for "Inner product spaces"

showing 5 items of 5 documents

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.

Partial inner product spacesPure mathematicsGeneral MathematicsFOS: Physical sciencesresolventLattice (discrete subgroup)01 natural sciencessymbols.namesakeInner product spaceSettore MAT/05 - Analisi MatematicaPIP-spaceframe multipliers}lattices of Hilbert spacesSpectral analysis0101 mathematicsEigenvalues and eigenvectorsMathematical PhysicsMathematicsResolventframe multipliers010102 general mathematicsSpectrum (functional analysis)Spectral propertiesHilbert spaceMathematical Physics (math-ph)010101 applied mathematicssymbolsspectral properties of symmetric operatorsSpectral theory46Cxx 47A10 47B37
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The Partial Inner Product Space Method: A Quick Overview

2010

Many families of function spaces play a central role in analysis, in particular, in signal processing (e.g., wavelet or Gabor analysis). Typical are spaces, Besov spaces, amalgam spaces, or modulation spaces. In all these cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. It turns out that all these space families are, or contain, scales or lattices of Banach spaces, which are special cases ofpartial inner product spaces(PIP-spaces). In this context, it is often said that such families should be taken as a whole and operators, bases, and frames on them should be defined globally, for the whole family, instead o…

Partial inner product spacesPure mathematicsNuclear operatorPhysicsQC1-999Applied MathematicsTopological tensor productGeneral Physics and AstronomyOperator theorySpace (mathematics)Compact operator on Hilbert spaceSettore MAT/05 - Analisi MatematicaFréchet spaceInterpolation spaceLp spaceMathematicsAdvances in Mathematical Physics
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Banach partial *-algebras: an overview

2019

A Banach partial $*$-algebra is a locally convex partial $*$-algebra whose total space is a Banach space. A Banach partial $*$-algebra is said to be of type (B) if it possesses a generating family of multiplier spaces that are also Banach spaces. We describe the basic properties of these objects and display a number of examples, namely, $L^p$-like function spaces and spaces of operators on Hilbert scales or lattices. Finally we analyze the important cases of Banach quasi $*$-algebras and $CQ^*$-algebras.

Pure mathematicsMathematics::Functional AnalysisAlgebra and Number Theorypartial inner product spacesPartial *-algebra Banach partial *-algebra CQ*-algebra partial inner product space operators on Hilbert scale.Partial algebraPartial *-algebraspartial $*$-algebraCQ*-algebraspartial inner product spaceSettore MAT/05 - Analisi Matematica$CQ^*$-algebraBanach partial *-algebrasoperators on Hilbert scaleBanach partial $*$-algebra46J1008A55Analysis47L60Mathematics
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PIP-Space Valued Reproducing Pairs of Measurable Functions

2019

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space Z : = Z ( X , μ ), where ( X , μ ) is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a pip-space; (iii) Y and Z are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case.

Pure mathematicspartial inner product spacesMeasurable functionLogicGeneralizationreproducing pairs; continuous frames; upper and lower semi-frames; partial inner product spacesStructure (category theory)upper and lower semi-framecontinuous frameAbstract spaceSpace (mathematics)01 natural sciencesMeasure (mathematics)symbols.namesakeSettore MAT/05 - Analisi Matematica0103 physical sciences0101 mathematics010306 general physicsreproducing pairMathematical PhysicsMathematicscontinuous framesAlgebra and Number Theorylcsh:Mathematics010102 general mathematicsHilbert spaceupper and lower semi-frameslcsh:QA1-939reproducing pairssymbolsGeometry and TopologyAnalysis
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Partial inner product spaces: Theory and Applications

2010

Partial Inner Product (PIP) Spaces are ubiquitous, e.g. Rigged Hilbert spaces, chains of Hilbert or Banach spaces (such as the Lebesgue spaces Lp over the real line), etc. In fact, most functional spaces used in (quantum) physics and in signal processing are of this type. The book contains a systematic analysis of PIP spaces and operators defined on them. Numerous examples are described in detail and a large bibliography is provided. Finally, the last chapters cover the many applications of PIP spaces in physics and in signal/image processing, respectively. As such, the book will be useful both for researchers in mathematics and practitioners of these disciplines.

Settore MAT/05 - Analisi MatematicaInner productInner product spaces
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