Unitary Parts Research Articles

Brownian Type Parts of Operators in Hilbert Spaces

We obtain some reducing (or just invariant) subspaces for a Hilbert space operator on which it acts as a Brownian type 2-isometry. More exactly Brownian isometric (unitary) parts as well as more general quasi-Brownian isometric (unitary) parts of an operator are investigated. They are explicitely described for a 2-isometry.

Non-Markovian decoherence dynamics in nonequilibrium environments

We theoretically investigate the non-Markovian dynamical decoherence of a quantum system coupled to nonequilibrium environments with nonstationary statistical properties. We show the time evolution of the decoherence factor in real-imaginary space to study the environment-induced energy renormalization and backaction of coherence which are associated with the unitary and nonunitary parts of the quantum master equation, respectively. It is also shown that the nonequilibrium decoherence dynamics displays a transition between Markovian and non-Markovian and the transition boundary depends on the environmental parameters. The results are helpful for further understanding non-Markovian dynamics and coherence backaction on an open quantum system from environments.

Research on Updating of Body Schema Using AR Limb and Measurement of the Updated Value

The body schema, an internal model in the brain used to recall the body position, is known to be updated by adapting to the changes in the body figure due to the body growth. Furthermore, the recent research has reported the active updating of the body schema by misunderstanding one's body position using the illusion phenomenon. However, the current method of actively updating the body schema has limited effects and does not handle selective updating of each body part. Therefore, in this paper, a new method using an augmented reality (AR) limb is proposed for the selective updating of unitary parts of the body image in a short period of time. The AR limb is the virtual limb composed by actively extendable and retractable parts, projected by head-mounted display. The basic concept is that the body schema is updated to match the extended and retracted part of the AR limb based on the body schema learning mechanism in the sensory integration of visual and somatic information. The potential effectiveness of the method was verified by actual experiment applying the body schema update to the right forearm.

Operators intertwining with isometries and Brownian parts of 2-isometries

For two operators A and T (A≥0) on a Hilbert space H satisfying T⁎AT=A and the A-regularity condition AT=A1/2TA1/2 we study the subspace N(A−A2) in connection with N(AT−TA), for T belonging to different classes. Our results generalize those due to C. Kubrusly concerning the case when T is a contraction and A=ST is the asymptotic limit of T. Also, the particular case of a 2-isometry in the sense of S. Richter as well as J. Agler and M. Stankus is considered. For such operators, under the same regularity condition we completely describe the reducing Brownian unitary and isometric parts, as well as the invariant Brownian isometric part. Some examples are provided in order to illustrate the limits of the theoretical results.

Using intra‐individual variation in shrub architecture to explain population cover

Plant architecture is related to the performance of long‐lived plants; its role in promoting species coexistence and in successional patterns is now widely recognized. However, because plant architecture involves branching processes, it is highly variable at the intra‐specific level. In this paper, we address two questions: what is the best way to describe plant architecture to obtain meaningful information for explaining population cover: at the whole‐plant level, or at the level of its unitary constituent parts? Further, are there architectural designs related to populations’ success? We evaluated the relative impact of ontogeny and whole‐plant traits on the cover achieved by the populations of five shrub species developing on 25 abandoned farmlands in southwestern Québec (Canada). We compared four ways of analyzing plant architecture: 1–2) using morphological traits described at the scale of a module (an elementary architectural unit made up of all the different types of shoots), with or without taking into account the ontogeny of the whole organism, 3) using the rate of changes during ontogeny as traits, and 4) using whole‐plant traits describing branching processes at a scale larger than modules. We then used variation partitioning to discriminate the actual effects of these traits on percent cover of the species from hidden effects due to plant ontogenesis and population spatial structure. Our results suggest that the predominant variables that effectively describe population cover vary from one species to another. At the same time, whole‐plant architectural traits and the rate of change of morphological traits during ontogeny both have an important effect on population cover. These findings suggest that acknowledging the developmental pattern of woody species can clarify the impact of intra‐specific trait variation on population cover.

Quasi-similar k-paranormal operators

It is proved in this paper that k-paranormal operators satisfy (Bishop’s) property (β) ; and also that if S and T are k -paranormal contractions such that the completely non-unitary part Sc of S has finite multiplicity, then S is quasi-similar to T if and only if their unitary parts are unitarily equivalent and their completely non-unitary parts are quasi-similar. This generalizes a result of W.W. Hastings [4] on subnormal operators and P.Y. Wu [11] on hyponormal operators. Mathematics subject classification (2010): Primary 47A45, Secondary 47B20.

Weighted Schur-class functions and functional models

We generalize the class of contractive-valued functions analytic in the unit disk (Schur class) to the case of finitely connected domains with operator-valued weights on the boundary. The two fundamental decompositions (the inner-outer factorization and the decomposition into the orthogonal sum of the pure and unitary parts) are extended to our setting. This generalization allows one to use a uniform approach to functional models developed recently in papers of the author and D. Yakubovich. Moreover, it gives an alternative language to handle multi-valued functions in the multiply connected case.

On perturbations of the group of shifts on the line by unitary cocycles

It is shown that the class of perturbations of the semigroup of shifts on L 2 (R + ) by unitary cocycles V with the property V t - I ∈ s 2 , t > 0 (where s 2 is the Hilbert-Schmidt class) contains strongly continuous semigroups of isometric operators, whose unitary parts possess spectral decompositions with the measure being singular with respect to the Lebesgue measure. Thus, we describe also the subclass of strongly continuous groups of unitary operators that are perturbations of the group of shifts on L 2 (R) by Markovian cocycles W with the property W t - I ∈ s 2 , t ∈ R.

Unitary equivalence of operators and dilations

Given two contractions $T$ and $T'$ such that $T'-T$ is an operator of finite rank, we prove, under some conditions, the unitary equivalence of the unitary parts of the minimal isometric dilations (respectively minimal co-isometric extensions) of $T$ and&

Compact and finite rank operators satisfying a Hankel type equationT 2X=XT 1 *

In 1997, V. Ptak defined the notion of generalized Hankel operator as follows: Given two contractions $$\mathcal{B}(\mathcal{H}_1 )$$ and $$\mathcal{B}(\mathcal{H}_2 )$$ , an operatorX: $$X:\mathcal{H}_1 \to \mathcal{H}_2 $$ is said to be a generalized Hankel operator ifT 2 X=XT 1 * andX satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations ofT 1 andT 2. The purpose behind this kind of generalization is to study which properties of classical Hankel operators depend on their characteristic intertwining relation rather than on the theory of analytic functions. Following this spirit, we give appropriate versions of a number of results about compact and finite rank Hankel operators that hold within Ptak's generalized framework. Namely, we extend Adamyan, Arov and Krein's estimates of the essential norm of a Hankel operator, Hartman's characterization of compact Hankel operators and Kronecker's characterization of finite rank Hankel operators.

Time evolution of tunneling and decoherence

Decoherence effects associated to the damping of a tunneling two-level system are shown to dominate the tunneling probability at short times in strong-coupling regimes in the context of a soluble model. A general decomposition of tunneling rates in dissipative and unitary parts are implemented. Master equation treatments fail to describe the model system correctly when more than a single relaxation time is involved.

Scattering operator and associated semigroup in the generalized Lax-Phillips scheme

Properties of the scattering operator and associated semigroup generated by the generalized Lax-Phillips scheme are studied. In particular, it is proved that the scattering operator admits decomposition into an orthogonal sum S=S1 ⊕ S2, where S1 is a unitary scattering operator in the sense of Lax-Phillips, S2 is a completely nonunitary operator of the type studied by C. Foias. For the associated semigroup T(t) up to the unitary component one gets the formula T(t)=T1(t) ⊕ T2(t), where T1(t) is the associated semigroup of class C00 and there exist subspaces such that a) The study of the structure of the original space and the properties of the operator S and of the semigroup T(t) let one decompose the scattering matrix into the orthogonal sum of pure and constant unitary parts.

Remark on "Tridiagonal Matrix Representations of Cyclic Self-Adjoint Operators"

In [2], Dombrowski used a polynomials technique to obtain a sufficient condition (in terms of the weights) for the existence of an absolutely continuous subspace for real parts of unilateral weighted shifts. The purpose of this note is to present a technique that produces a trace criterion for the existence of an absolutely continuous subspace for real parts as well as unitary parts of (bounded) operators. Let B(H) denote the class of all (bounded) linear operators acting on a complex separable Hilbert space H. For T e B(H), we define ReT = (T + T*)/2, [T*, T] = T*TTT* and TI = (T*T)1/2. For the hermitian operator A, we let A+= (IAI ? A)/2 and Ha(A) = {.x I IIE(X)x1I2 is absolutely continuous as a function of X, where E is the spectral measure of A). Observe that Ha(A) is a reducing subspace for A (see, e.g., [5, p. 104]). Ha(A) is called the absolutely continuous subspace of A; its orthogonal complement, denoted by H,(A), is called the singular subspace of A. Similarly, we can define Ha(U) and HJ(U) as the absolutely continuous subspace and singular subspace, respectively, of a unitary operator U. In [2], Dombrowski used a polynomials technique to obtain a sufficient condition (in terms of the weights) for the existence of an absolutely continuous subspace for real parts of unilateral weighted shifts. The purpose of this note is to present a technique that produces a trace criterion for the existence of an absolutely continuous subspace for real parts as well as unitary parts of (bounded) operators. Different as it may look, this criterion turns out to be equivalent to that of Dombrowski's when the operators are unilateral weighted shifts; besides, in the case of bilateral weighted shifts, it can easily be translated into a Dombrowski type condition. Furthermore, the technique leads to a refinement of a result of Putnam [6, Theorem 2.4.1 (ii)]. 1. The main results. Before proceeding to the main results, we list the following tools for convenience. (I) Every singular hermitian operator can be written as a sum of a diagonal operator and a trace class operator; see Carey and Pincus [1, p. 484]. Received by the editors September 17, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 47A65, 47B37.

Remark on: “Tridiagonal matrix representations of cyclic selfadjoint operators” [Pacific J. Math. 114 (1984), no. 2, 325–334; MR0757504 (85h:47033)] by J. Dombrowski

In [2], Dombrowski used a "orthogonal polynomials" technique to obtain a sufficient condition (in terms of the weights) for the existence of an absolutely continuous subspace for real parts of unilateral weighted shifts. The purpose of this note is to present a "diagonal" technique that produces a trace criterion for the existence of an absolutely continuous subspace for real parts as well as unitary parts of (bounded) operators.

Some ways to use thermodynamic information to characterize linkage systems

In systems of coupled reactions, processes linked to the measured process whether they be free-swinging equilibria such as ionization of protein groups, or vehicles for site to site linkage such as conformation changes, are detectable and made available for quantitative study through compensation relationships between enthalpy and entropy often surprisingly linear. In protein systems thus far studied the most common effector of such behavior is a change of pH. This perturbs a measured protein process with only small net change in chemical potentials of the ionizing groups and the proton buffer species involved since cratic and unitary parts nearly balance. However, enthalpy and entropy changes can be large and will tend toward a linear relationship on pH variation with a proportionality constant near the mean experimental temperature. The ionized and un-ionized states of each acidic or basic group form a two-state system linked to the measured process. Other free-swinging processes such as the change in state populations of water produce similar results but the water process is complicated by the fact that the standard chemical potentials (unitary parts) of the macro states of bulk water are equal near 25/sup 0/C, the usual mean experimental temperature. Coupled multistate systems such asmore » these are a common source of enthalpy-entropy compensation behavior, but a more fundamental souce lies in the second-law requirement that those parts of ..delta..H and ..delta..S reserved to accommodate enthalpy and entropy fluctuations between system and surroundings at constant T and thus not able to contribute to ..delta..G in isothermal, isobaric processes, must cancel each other through the relationship ..delta..H/sub compensation/ = T/sub mean/ ..delta..S/sub compensation/.« less

Unitary parts of generalized Hankel operators

Unitary parts of generalized Hankel operators

Unitary Parts of Generalized Toeplitz Operators

For a generalized Toeplitz operator acting on a Hilbert space with unilateral shift, necessary and sufficient conditions are obtained for the existence of a nontrivial unitary part, and an explicit description of this unitary part is given.

Unitary Parts of Contractive Hankel Matrices

Unitary Parts of Contractive Hankel Matrices

Unitary parts of generalized Toeplitz operators

For a generalized Toeplitz operator acting on a Hilbert space with unilateral shift, necessary and sufficient conditions are obtained for the existence of a nontrivial unitary part, and an explicit description of this unitary part is given.

Unitary parts of contractive Hankel matrices

For a Hankel matrix H = ( c j + k ) H = ({c_{j + k}}) which is a contraction, necessary and sufficient conditions are obtained for the existence of a nontrivial unitary part, and an explicit description of this unitary part is given.