arXiv subjects

David R. Larson

Publications and source records attributed to David R. Larson.

15 recordsLinked to original sources

arXiv2021

Dilations for operator-valued quantum measures

This paper concerns the dilations of Banach space operator-valued quantum measures. While the recently developed general dilation theory can lead to a projection (idempotent) valued dilation for any quantum measure over the projection lattice for a von Neumann algebra that dose not contain type $I_{2}$ direct summand, such a dilation does not necessarily guarantee the preservation of countable additivity of the quantum measure. So it remain an open question whether every countably additive $B(X)$-valued quantum measure can be dilated to a countably additive projection-valued measure.The main purpose of this paper is to prove that such a dilation can be constructed if one of the following two conditions is satisfied: (i) the underling Banach space $X = \ell_{p}$ $(1\leq p < 2$) or it has Schur property, (ii) the quantum measure has bounded $p$-variation for some $ 1\leq p < \infty $. All of these were achieved by establishing a non-commutative version of a minimal dilation theory on the so-called elementary dilation space equipping with an appropriate dilation norm. In particular, the newly introduced $p$-variation norm on the elementary dilation space allows us to prove that every operator-valued quantum measure with bounded $p$-variation has a projection-valued quantum measure dilation that preserves the boundedness of the $p$-variation.

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arXiv2015

Some Structural Properties of Homomorphism Dilation Systems for Linear Maps

Inspired by some recent development on the theory about projection valued dilations for operator valued measures or more generally bounded homomorphism dilations for bounded linear maps on Banach algebras, we explore a pure algebraic version of the dilation theory for linear systems acting on unital algebras and vector spaces. By introducing two natural dilation structures, namely the canonical and the universal dilation systems, we prove that every linearly minimal dilation is equivalent to a reduced homomorphism dilation of the universal dilation, and all the linearly minimal homomorphism dilations can be classified by the associated reduced subspaces contained in the kernel of synthesis operator for the universal dilation.

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arXiv2014

Dilations of frames, operator valued measures and bounded linear maps

We will give an outline of the main results in our recent AMS Memoir, and include some new results, exposition and open problems. In that memoir we developed a general dilation theory for operator valued measures acting on Banach spaces where operator-valued measures (or maps) are not necessarily completely bounded. The main results state that any operator-valued measure, not necessarily completely bounded, always has a dilation to a projection-valued measure acting on a Banach space, and every bounded linear map, again not necessarily completely bounded, on a Banach algebra has a bounded homomorphism dilation acting on a Banach space. Here the dilation space often needs to be a Banach space even if the underlying space is a Hilbert space, and the projections are idempotents that are not necessarily self-adjoint. These results lead to some new connections between frame theory and operator algebras, and some of them can be considered as part of the investigation about "noncommutative" frame theory.

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arXiv2014

Dilations for Systems of Imprimitivity acting on Banach Spaces

Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in general can not be taken as a Hilbert space. However, it can be taken as a Hilbert space for positive operator valued systems of imprimitivity. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.

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arXiv2014

Signal Reconstruction from Frame and Sampling Erasures

We give some new methods for perfect reconstruction from frame and sampling erasures in finitely many steps. By bridging an erasure set we mean replacing the erased Fourier coefficients of a function with respect to a frame by appropriate linear combinations of the non-erased coefficients. We prove that if a minimal redundancy condition is satisfied bridging can always be done to make the reduced error operator nilpotent of index 2 using a bridge set of indices no larger than the cardinality of the erasure set. This results in perfect reconstruction of the erased coefficients in one final matricial step. We also obtain a new formula for the inverse of an invertible partial reconstruction operator. This leads to a second method of perfect reconstruction from frame and sampling erasures in finitely many steps. This gives an alternative to the bridging method for many (but not all) cases. The methods we use employ matrix techniques only of the order of the cardinality of the erasure set, and are applicable to rather large finite erasure sets for infinite frames and sampling schemes as well as for finite frame theory. Some new classification theorems for frames are obtained and some new methods of measuring redundancy are introduced based on our bridging theory.

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arXiv2013

Framings and dilations

The notion of framings, recently emerging in P. G. Casazza, D. Han, and D. R. Larson, Frames for Banach spaces, in {\em The functional and harmonic analysis of wavelets and frames} (San Antonio, TX, 1999), {\em Contemp. Math}. {\bf 247} (1999), 149-182 as generalization of the reconstraction formula generated by pairs of dual frames, is in this note extended substantially. This calls on refining the basic dilation results which still being in the flavor of {\em théorème principal} of B. Sz-Nagy go much beyond it.

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arXiv2012

Operator-Valued Measures, Dilations, and the Theory of Frames

We develop elements of a general dilation theory for operator-valued measures and bounded linear maps between operator algebras that are not necessarily completely-bounded. We prove our main results by extending and generalizing some known results from the theory of frames and framings.

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arXiv2010

Irreducible wavelet representations and ergodic automorphisms on solenoids

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in \cite{DuJo06b}, and we present several equivalent formulations of the problem.

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arXiv2007

Interpolation maps and congruence domains for wavelet sets

It is proven that if an interpolation map between two wavelet sets preserves the union of the sets, then the pair must be an interpolation pair. We also construct an example of a pair of wavelet sets for which the congruence domains of the associated interpolation map and its inverse are equal, and yet the pair is not an interpolation pair. The first result solves affirmatively a problem that the second author had posed several years ago, and the second result solves an intriguing problem of D. Han. The key to this counterexample is a special technical lemma on constructing wavelet sets. Several other applications of this result are also given. In addition, some problems are posed. We also take the opportunity to give some general exposition on wavelet sets and operator-theoretic interpolation of wavelets.

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arXiv2006

Unitary systems, wavelet sets, and operator-theoretic interpolation of wavelets and frames

A wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a collection, or system, of unitary operators. We will describe the operator-interpolation approach to wavelet theory using the local commutant of a system. This is really an abstract application of the theory of operator algebras to wavelet theory. The concrete applications of this method include results obtained using specially constructed families of wavelet sets. A frame is a sequence of vectors in a Hilbert space which is a compression of a basis for a larger space. This is not the usual definition in the frame literature, but it is easily equivalent to the usual definition. Because of this compression relationship between frames and bases, the unitary system approach to wavelets (and more generally: wandering vectors) is perfectly adaptable to frame theory. The use of the local commutant is along the same lines as in the wavelet theory. Finally, we discuss constructions of frames with special properties using targeted decompositions of positive operators, and related problems.

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arXiv2006

Unitary systems and wavelet sets

We give a detailed description of the local commutant approach to wavelet theory using operator algebraic methods. We include a new result on interpolation pairs of wavelet sets: Every pair in the generalized Journe family of wavelet sets is an interpolation pair. In the last section, we give a discussion of several open problems on wavelets and frame-wavelets.

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arXiv2006

Robertson-type Theorems for Countable Groups of Unitary Operators

Let $\mathcal{G}$ be a countably infinite group of unitary operators on a complex separable Hilbert space $H$. Let $X = \{x_{1},...,x_{r}\}$ and $Y = \{y_{1},...,y_{s}\}$ be finite subsets of $H$, $r < s$, $V_{0} = \bar{span} \mathcal{G}(X), V_1 = \bar{span} \mathcal{G}(Y)$ and $ V_{0} \subset V_{1} $. We prove the following result: Let $W_0$ be a closed linear subspace of $V_1$ such that $V_0 \oplus W_0 = V_1$ (i.e., $V_0 + W_0 = V_1$ and $V_0 \cap W_0 = \{0 \}$). Suppose that $\mathcal{G}(X)$ and $\mathcal{G}(Y)$ are Riesz bases for $V_{0}$ and $V_{1}$ respectively. Then there exists a subset $Γ=\{z_1,..., z_{s-r}\}$ of $W_0$ such that $\mathcal{G}(Γ)$ is a Riesz basis for $W_0$ if and only if $ g(W_0) \subseteq W_0 $ for every $g$ in $\mathcal{G}$. We first handle the case where the group is abelian and then use a cancellation theorem of Dixmier to adapt this to the non-abelian case. Corresponding results for the frame case and the biorthogonal case are also obtained.

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arXiv2000

A module frame concept for Hilbert C*-modules

The goal of the present paper is a short introduction to a general module frame theory in C*-algebras and Hilbert C*-modules. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal bases, and of reconstruction of the frames by projections and other bounded module operators with suitable ranges. We obtain frame representation and decomposition theorems, as well as similarity and equivalence results. The relative position of two and more frames in terms of being complementary or disjoint is investigated in detail. In the last section some recent results by P. G. Casazza are generalized to our setting. The Hilbert space situation appears as a special case. For detailled proofs we refer to another paper also contained in the ArXiv.

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arXiv2000

Frames in Hilbert C-modules and C-algebras

We present a general approach to a modular frame theory in C*-algebras and Hilbert C*-modules. The investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and by other bounded modular operators with suitable ranges. We obtain frame representations and decomposition theorems, as well as similarity and equivalence results for frames. Hilbert space frames and quasi-bases for conditional expectations of finite index on C*-algebras appear as special cases. Using a canonical categorical equivalence of Hilbert C*-modules over commutative C*-algebras and (F)Hilbert bundles the results find a reintepretation for frames in vector and (F)Hilbert bundles. Fields of applications are investigations on Cuntz-Krieger-Pimsner algebras, on conditional expectations of finite index, on various ranks of C*-algebras, on classical frame theory of Hilbert spaces (wavelet and Gabor frames), and others. 2001: In the introduction we refer to related publications in detail.

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arXiv2000

Modular frames for Hilbert C*-modules and symmetric approximation of frames

We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related modular operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modulesover unital C*-algebras that admit an orthonormal Riesz basis. Interrelations and applications to classical linear frame theory are indicated. As an application we describe the nature of families of operators {S_i} such that SUM_i S*_iS_i=id_H, where H is a Hilbert space. Resorting to frames in Hilbert spaces we discuss some measures for pairs of frames to be close to one another. Most of the measures are expressed in terms of norm-distances of different kinds of frame operators. In particular, the existence and uniqueness of the closest (normalized) tight frame to a given frame is investigated. For Riesz bases with certain restrictions the set of closetst tight frames often contains a multiple of its symmetric orthogonalization (i.e. L\"owdin orthogonalization).

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