On the J-unitary dilation of a bounded operator

1975 ◽  
Vol 8 (3) ◽  
pp. 265-267
Author(s):  
L. A. Sakhnovich
Keyword(s):  
Bounded Operator ◽  
Unitary Dilation

On the solvability of the degenerate Noetherian difference-algebraic boundary value problem

2019 ◽  
Vol 33 ◽  
pp. 204-217
Author(s):  
Sergei Chuiko ◽  
Yaroslav Kalinichenko ◽  
Nikita Popov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Nonlinear Oscillations ◽  
Bounded Operator ◽  
Boundary Value ◽  
Homogeneous Part ◽  
Solvability Conditions ◽  
Engineering Control ◽  
Generalized Green Operator

The original conditions of solvability and the scheme of finding solutions of a linear Noetherian difference-algebraic boundary-value problem are proposed in the article, while the technique of pseudoinversion of matrices by Moore-Penrose is substantially used. The problem posed in the article continues to study the conditions for solvability of linear Noetherian boundary value problems given in the monographs of A.M. Samoilenko, A.V. Azbelev, V.P. Maximov, L.F. Rakhmatullina and A.A. Boichuk. The study of differential-algebraic boundary-value problems is closely related to the investigation of boundary-value problems for difference equations, initiated in the works of A.A. Markov, S.N. Bernstein, Y.S. Bezikovych, O.O. Gelfond, S.L. Sobolev, V.S. Ryabenkyi, V.B. Demidovych, A. Halanai, G.I. Marchuk, A.A. Samarskyi, Yu.A. Mytropolskyi, D.I. Martyniuk, G.M. Vainiko, A.M. Samoilenko and A.A. Boichuk. On the other hand, the study of boundary-value problems for difference equations is related to the study of differential-algebraic boundary-value problems initiated in the papers of K. Weierstrass, N.N. Lusin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, N.A. Perestiyk, V.P. Yakovets, A.A. Boichuk, A. Ilchmann and T. Reis. The study of differential-algebraic boundary value problems is also associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, control theory, motion stability theory. The general case of a linear bounded operator corresponding to the homogeneous part of a linear Noetherian difference-algebraic boundary value problem has no inverse is investigated. The generalized Green operator of a linear difference-algebraic boundary value problem is constructed in the article. The relevance of the study of solvability conditions, as well as finding solutions of linear Noetherian difference-algebraic boundary-value problems, is associated with the widespread use of difference-algebraic boundary-value problems obtained by linearizing nonlinear Noetherian boundary-value problems for systems of ordinary differential and difference equations. Solvability conditions are proposed, as well as the scheme of finding solutions of linear Noetherian difference-algebraic boundary value problems are illustrated in detail in the examples.

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

scholarly journals Unitary dilation for polar decompositions of p-hyponormal operators

2005 ◽  
Author(s):  
Muneo Chō ◽  
Tadasi Huruya ◽  
Kôtarô Tanahashi
Keyword(s):  
Hyponormal Operators ◽  
Unitary Dilation

scholarly journals Approximative K-atomic decompositions and frames in Banach spaces

2018 ◽  
Vol 26 (1/2) ◽  
pp. 153-166
Author(s):  
Shah Jahan
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Atomic Decomposition ◽  
Necessary Conditions ◽  
Bounded Operator ◽  
Special Kind ◽  
Frame Theory ◽  
Bessel Sequence ◽  
Atomic Decompositions ◽  
Counter Example

L. Gǎvruţa (2012) introduced a special kind of frames, named K-frames, where K is an operator, in Hilbert spaces, which is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative K-atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative K-atomic decompositions in Banach spaces. Also some results on the existence of approximative K-atomic decompositions are obtained. We discuss several methods to construct approximative K-atomic decomposition for Banach Spaces. Further, approximative d-frame and approximative d-Bessel sequence are introduced and studied. Two necessary conditions are given under which an approximative d-Bessel sequence and approximative d-frame give rise to a bounded operator with respect to which there is an approximative K-atomic decomposition. Example and counter example are provided to support our concept. Finally, a possible application is given.

scholarly journals On power-bounded prespectral operators

1976 ◽  
Vol 20 (2) ◽  
pp. 173-175
Author(s):  
H. R. Dowson
Keyword(s):  
Growth Condition ◽  
Bounded Operator ◽  
Spectral Operator ◽  
Scalar Type ◽  
Image Position ◽  
Power Bounded Operator ◽  
Power Bounded

Foguel (8) and Fixman (7) independently proved that an invertible spectral operator, which is power-bounded, is of scalar type. Their proofs rely heavily on a result of Dunford on spectral operators whose resolvents satisfy a growth condition. (See Lemma 3.16 of (6, p. 609).) Observe that the resolvent of an invertible power-bounded operator T satisfies an inequality of the form

scholarly journals Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
Esteban Castro-Ruiz ◽  
Flaminia Giacomini ◽  
Alessio Belenchia ◽  
Časlav Brukner
Keyword(s):  
Reference Frame ◽  
Reference Frames ◽  
Quantum Systems ◽  
Indefinite Metric ◽  
Causal Order ◽  
Quantum Operations ◽  
Unitary Dilation ◽  
Local Quantum ◽  
Fixed Space ◽  
Quantum Clocks

AbstractThe standard formulation of quantum theory relies on a fixed space-time metric determining the localisation and causal order of events. In general relativity, the metric is influenced by matter, and is expected to become indefinite when matter behaves quantum mechanically. Here, we develop a framework to operationally define events and their localisation with respect to a quantum clock reference frame, also in the presence of gravitating quantum systems. We find that, when clocks interact gravitationally, the time localisability of events becomes relative, depending on the reference frame. This relativity is a signature of an indefinite metric, where events can occur in an indefinite causal order. Even if the metric is indefinite, for any event we can find a reference frame where local quantum operations take their standard unitary dilation form. This form is preserved when changing clock reference frames, yielding physics covariant with respect to quantum reference frame transformations.

scholarly journals ON THE INDEX AND DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS

1999 ◽  
Vol 10 (07) ◽  
pp. 791-823 ◽  
Author(s):  
WILLIAM ARVESON
Keyword(s):  
Bounded Operator ◽  
Positive Semigroup ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Semigroups ◽  
Matrix Algebras ◽  
Positive Maps ◽  
Completely Positive Semigroups ◽  
Numerical Index ◽  
Completely Positive Semigroup

It is known that every semigroup of normal completely positive maps P = {Pt:t≥ 0} of ℬ(H), satisfying Pt(1) = 1 for every t ≥ 0, has a minimal dilation to an E0 acting on ℬ(K) for some Hilbert space K⊇H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator [Formula: see text] in terms of natural structures associated with the generator. This includes all unital CP semigroups acting on matrix algebras. We also show that the minimal dilation of the semigroup P={ exp tL:t≥ 0} to an E0-semigroup is is cocycle conjugate to a CAR/CCR flow.

Estimates of the Distance to the Set of Solenoidal Vector Fields and Applications to A Posteriori Error Control

2015 ◽  
Vol 15 (4) ◽  
pp. 515-530 ◽  
Author(s):  
Sergey Repin
Keyword(s):  
Error Control ◽  
Vector Fields ◽  
Bounded Operator ◽  
Posteriori Error ◽  
Stability Theorem ◽  
A Posteriori ◽  
Solenoidal Vector ◽  
Bounded Lipschitz Domain ◽  
Incompressible Viscous Fluids ◽  
The Stability

AbstractThe paper is concerned with computable estimates of the distance between a vector-valued function in the Sobolev space$W^{1,\gamma }(\Omega ,\mathbb {R}^d)$(where${\gamma \in (1,+\infty )}$and Ω is a bounded Lipschitz domain in ℝd) and the subspace${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$containing all divergence-free (solenoidal) vector functions. Derivation of these estimates is closely related to the stability theorem that establishes existence of a bounded operator inverse to the operator${\operatorname{div}}$. The constant in the respective stability inequality arises in the estimates of the distance to the set${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$. In general, it is difficult to find a guaranteed and realistic upper bound of this global constant. We suggest a way to circumvent this difficulty by using weak (integral mean) solenoidality conditions and localized versions of the stability theorem. They are derived for the case where Ω is represented as a union of simple subdomains (overlapping or non-overlapping), for which estimates of the corresponding stability constants are known. These new versions of the stability theorem imply estimates of the distance to${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$that involve only local constants associated with subdomains. Finally, the estimates are used for deriving fully computable a posteriori estimates for problems in the theory of incompressible viscous fluids.

Operators with Closed Range, Pseudo-Inverses, and Perturbation of Frames for a Subspace

1999 ◽  
Vol 42 (1) ◽  
pp. 37-45 ◽  
Author(s):  
Ole Christensen
Keyword(s):  
Recent Work ◽  
Hilbert Spaces ◽  
Bounded Operator ◽  
Closed Range ◽  
The Stability

AbstractRecent work of Ding and Huang shows that if we perturb a bounded operator (between Hilbert spaces) which has closed range, then the perturbed operator again has closed range. We extend this result by introducing a weaker perturbation condition, and our result is then used to prove a theorem about the stability of frames for a subspace.

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