Partial orders on partial isometries

Ross, William T.; Garcia, Stephan Ramon; W., Martin, Robert T.

doi:10.7900/jot.2015may20.2062

Cited by 7 publications

(13 citation statements)

References 14 publications

(19 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This analysis, having its original source in a classical paper of Livsic [7], also draws on subsequent work in [1,5,8]. Among other things, three preorder relations are introduced in [3]; each of these induces an equivalence relation, and one of the questions addressed is when the classes of equivalence are precisely the classes of unitary equivalence. It is shown that the question can be rephrased as the existence of certain multipliers between model spaces (in the sense of [2] or [10]), and partial results are obtained.…”

Section: Introductionmentioning

confidence: 99%

“…The starting point for this note is the paper [3], which makes a detailed analysis of the class of completely nonunitary partial isometries on a complex separable Hilbert space with equal defect indices (finite or infinite). This analysis, having its original source in a classical paper of Livsic [7], also draws on subsequent work in [1,5,8].…”

Section: Introductionmentioning

confidence: 99%

“…The relation has an obvious extension to all contractions (even to general bounded operators, but we will stick to contractions). One of the tools in [3] is the characteristic function of a partial isometry, and a comprehensive theory of Sz.-Nagy-Foias [10] extends the notion of characteristic function to all completely non unitary contractions. On the other hand, the structure of unitary operators is well known by multiplicity theory [4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a preorder relation for contractions

Timotin¹

2016

Acta Sci. Math.

View full text Add to dashboard Cite

An order relation for contractions on a Hilbert space can be introduced by stating that A B if and only if A is unitarily equivalent to the restriction of B to an invariant subspace. We discuss the equivalence classes associated to this relation, and identify cases in which they coincide with classes of unitary equivalence. The results extend those for completely nonunitary partial isometries obtained by Garcia, Martin, and Ross.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On a preorder relation for contractions

Timotin¹

2016

Acta Sci. Math.

View full text Add to dashboard Cite

show abstract

“…Recall here, that the classes of partial isometries and symmetric linear transformations on Hilbert space are equivalent under a natural bijection, the Cayley Transform, a fractional linear transformation of the closed complex upper half-plane onto the closed unit disk (minus the point {1}) [1,Chapter VII]. In [2,24,40], several key aspects of these approaches were combined and slightly generalized with the introduction of the concept of a model map for a symmetric linear transformation or partial isometry with equal defect indices [2, Section 2.1], [24,Section 3], [40,Definition 5.1].…”

Section: Introductionmentioning

confidence: 99%

A Gleason Solution Model for Row Contractions

Ramanantoanina

2019

Interpolation and Realization Theory With Applications to Control Theory

View full text Add to dashboard Cite

In the de Branges-Rovnyak functional model for contractions on Hilbert space, any completely non-coisometric (CNC) contraction is represented as the adjoint of the restriction of the backward shift to a de Branges-Rovnyak space, H (b), associated to a contractive analytic operator-valued function, b, on the open unit disk. We extend this model to a large class of CNC contractions of several copies of a Hilbert space into itself (including all CNC row contractions with commuting component operators). Namely, we completely characterize the set of all CNC row contractions, T , which are unitarily equivalent to an extremal Gleason solution for a de Branges-Rovnyak space, H (b T ), contractively contained in a vector-valued Drury-Arveson space of analytic functions on the open unit ball in several complex dimensions. Here, a Gleason solution is the appropriate several-variable analogue of the adjoint of the restricted backward shift and the characteristic function, b T , belongs to the several-variable Schur class of contractive multipliers between vector-valued Drury-Arveson spaces. The characteristic function, b T , is a unitary invariant, and we further characterize a natural sub-class of CNC row contractions for which it is a complete unitary invariant. 1 2 R.T.W. MARTIN AND A. RAMANANTOANINARecall that the shift, S : H 2 (D) → H 2 (D) is the canonical isometry of multiplication by z and its adjoint, the backward shift, S * , acts as the difference quotient:The shift plays a central role in the classical theory of Hardy spaces [29,42,25]. If T is any CNC contraction, there is a (essentially unique) contractive, operator-valued analytic function, bT , on the unit disk, bT (z) ∈ L (H, K) (i.e. a member of the operatorvalued Schur class), so that T is unitarily equivalent to X where X * := S * | H (b T ) is the restriction of the backward shift of the vector-valued Hardy space H 2 (D) ⊗ K to the de Branges-Rovnyak space H (bT ) [7,11]. (Any de Branges-Rovnyak space H (b) associated to a contractive, operator-valued analytic function, b, on the disk is always contractively contained in vector-valued Hardy space and is always co-invariant for the shift [48].) This provides a natural model for CNC contractions as adjoints of restrictions of the backward shift to de Branges-Rovnyak reproducing kernel Hilbert spaces, and this is the model we extend to several variables in this paper. A canonical several-variable extension of the Hardy space of the disk is the Drury-Arveson space, H 2 d , the unique RKHS of analytic functions on the open unit ball B d := (C d )1 corresponding to the several-variable Szegö kernel. (If Y is a Banach space, let (Y )1 denote the open unit ball and let [Y ]1 denote the closed unit ball.) The Schur classes of contractive, operator-valued functions on the disk are promoted to the multi-variable Schur classes, S d (J, K), of contractive, operator-valued multipliers between vector-valued Drury-Arveson spaces H 2 d ⊗ J and H 2 d ⊗ K (see Subsection 2.1), and the appropriate analogue of the adjoint o...

show abstract

“…A linear transformation T belongs to S if and only if T ≃ T Γ for some measure Γ obeying the Herglotz condition(23).…”

mentioning

confidence: 99%

Function spaces obeying a time-varying bandlimit

Kempf

2018

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Motivated by applications to signal processing and mathematical physics, recent work on the concept of time-varying bandwidth has produced a class of function spaces which generalize the Paley-Wiener spaces of bandlimited functions: any regular simple symmetric linear transformation with deficiency indices (1, 1) is naturally represented as multiplication by the independent variable in one of these spaces. We explicitly demonstrate the equivalence of this model for such linear transformations to several other functional models based on the theories of meromorphic model spaces of Hardy space and purely atomic Herglotz measures on the real line, respectively. This theory provides a precise notion of a time-varying or local bandwidth, and we describe how it may be applied to construct signal processing techniques that are adapted to signals obeying a time-varying bandlimit.

show abstract

Partial orders on partial isometries

Cited by 7 publications

References 14 publications

On a preorder relation for contractions

On a preorder relation for contractions

A Gleason Solution Model for Row Contractions

Function spaces obeying a time-varying bandlimit

Contact Info

Product

Resources

About