scholarly journals Necessary and sufficient conditions for spectral sets

1981 ◽  
Vol 24 (3) ◽  
pp. 349-355
Author(s):  
Takayuki Furuta ◽  
Muneo Chō
Keyword(s):  
Hilbert Space ◽  
Complex Plane ◽  
Sufficient Conditions ◽  
Complex Hilbert Space ◽  
Necessary And Sufficient Conditions ◽  
Spectral Sets ◽  
Closed Set ◽  
Spectral Set ◽  
Necessary And Sufficient

We shall show necessary and sufficient conditions for which closed set X in the complex plane is a spectral set of an operator T on a complex Hilbert space.

scholarly journals Two characterisations of additive *-automorphisms of B(H)

1996 ◽  
Vol 53 (3) ◽  
pp. 391-400 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Complex Hilbert Space ◽  
Necessary And Sufficient Conditions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Preserver Problems ◽  
Necessary And Sufficient

Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. In this paper we give two necessary and sufficient conditions for an additive bijection of B(H) to be a *-automorphism. Both of the results in the paper are related to the so-called preserver problems.

scholarly journals Admissibility of Fractional Order Descriptor Systems Based on Complex Variables: An LMI Approach

2020 ◽  
Vol 4 (1) ◽  
pp. 8
Author(s):  
Xuefeng Zhang ◽  
Yuqing Yan
Keyword(s):  
Fractional Order ◽  
Complex Plane ◽  
Sufficient Conditions ◽  
Complex Variables ◽  
Necessary And Sufficient Conditions ◽  
Descriptor Systems ◽  
Fractional Order Systems ◽  
Lmi Approach ◽  
Numerical Examples ◽  
Necessary And Sufficient

This paper is devoted to the admissibility issue of singular fractional order systems with order α ∈ ( 0 , 1 ) based on complex variables. Firstly, with regard to admissibility, necessary and sufficient conditions are obtained by strict LMI in complex plane. Then, an observer-based controller is designed to ensure system admissible. Finally, numerical examples are given to reveal the validity of the theoretical conclusions.

scholarly journals When do Projections Commute?

1980 ◽  
Vol 35 (4) ◽  
pp. 437-441 ◽  
Author(s):  
W. Rehder
Keyword(s):  
Hilbert Space ◽  
Conditional Probability ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantum Mechanical ◽  
Probability Operator ◽  
Quantum Mechanical Systems ◽  
Necessary And Sufficient ◽  
New Criteria

Abstract Necessary and sufficient conditions for commutativity of two projections in Hilbert space are given through properties of so-called conditional connectives which are derived from the conditional probability operator PQP. This approach unifies most of the known proofs, provides a few new criteria, and permits certain suggestive interpretations for compound properties of quantum-mechanical systems.

scholarly journals Equivalent necessary and sufficient conditions on noise sequences for stochastic approximation algorithms

1996 ◽  
Vol 28 (3) ◽  
pp. 784-801 ◽  
Author(s):  
I-Jeng Wang ◽  
Edwin K. P. Chong ◽  
Sanjeev R. Kulkarni
Keyword(s):  
Hilbert Space ◽  
Approximation Algorithms ◽  
Stochastic Approximation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Noise Sequences

We consider stochastic approximation algorithms on a general Hilbert space, and study four conditions on noise sequences for their analysis: Kushner and Clark's condition, Chen's condition, a decomposition condition, and Kulkarni and Horn's condition. We discuss various properties of these conditions. In our main result we show that the four conditions are all equivalent, and are both necessary and sufficient for convergence of stochastic approximation algorithms under appropriate assumptions.

scholarly journals Some additive and multiplicative results for generalized inverses

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 2049-2057
Author(s):  
Jovana Nikolov-Radenkovic
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Reverse Order ◽  
Reverse Order Law ◽  
Necessary And Sufficient ◽  
Regular Operators

In this paper we give necessary and sufficient conditions for A1{1,3} + A2{1, 3}+ ... + Ak{1,3} ? (A1 + A2 + ... + Ak){1,3} and A1{1,4} + A2{1,4} + ... + Ak{1,4} ? (A1 + A2 + ... + Ak){1,4} for regular operators on Hilbert space. We also consider similar inclusions for {1,2,3}- and {1,2,4}-i inverses. We give some new results concerning the reverse order law for reflexive generalized inverses.

On extensions of the Riemann and Lebesgue Integrals by Nets

1975 ◽  
Vol 18 (1) ◽  
pp. 7-17 ◽  
Author(s):  
O. S. Bellamy ◽  
H. W. Ellis
Keyword(s):  
General Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closed Set ◽  
Denjoy Integral ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Open Extension

In this note our principal interest is in using nets to give spaces of non-absolutely convergent integrals as extensions of the spaces of absolutely convergent Riemann and Lebesgue integrals. For this purpose we develop a general theory of extensions, by nets, of functions defined on the open intervals with closures in the complement of a fixed closed set, the nets being directed by inclusion for finite disjoint collections of such intervals. Two cases are considered leading to open extension (OE-) and conditional open extension (COE-) nets, the latter being subnets of the former. Necessary and sufficient conditions for the convergence of the OE- and COE-nets are given, those for the COE-nets being similar to conditions that arise in the definition of the restricted Denjoy integral. Properties of inner continuity, weak additivity and the existence of a continuous integral are defined and studied. These relate to the more specialized nets that are suitable for the extension of integrals.

On the Exponential Stability of the Singular Distributed Parameter Systems

2015 ◽  
Vol 719-720 ◽  
pp. 496-503
Author(s):  
Zhao Qiang Ge
Keyword(s):  
Hilbert Space ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Distributed Parameter Systems ◽  
Necessary And Sufficient Conditions ◽  
Distributed Parameter ◽  
Necessary And Sufficient

Exponential stability for the singular distributed parameter systems is discussed in the light of the theory of GE0-semigroup in Hilbert space. The necessary and sufficient conditions concerning the exponential stability are given.

Unitary equivalence and reductibility or invertibly weighted shifts

1971 ◽  
Vol 5 (2) ◽  
pp. 157-173 ◽  
Author(s):  
Alan Lambert
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Sequence ◽  
Complex Hilbert Space ◽  
Weighted Shifts ◽  
Infinite Dimensional ◽  
Reducing Subspaces ◽  
Positive Weights ◽  
Completely Reducible ◽  
Necessary And Sufficient

Let H be a complex Hilbert space and let {A1, A2, …} be a uniformly bounded sequence of invertible operators on H. The operator S on l2(H) = H ⊕ H ⊕ … given by S〈x0, x1, …〉 = 〈0, A1x0, A2x1, …〉 is called the invertibly veighted shift on l2(H) with weight sequence {An }. A matricial description of the commutant of S is established and it is shown that S is unitarily equivalent to an invertibly weighted shift with positive weights. After establishing criteria for the reducibility of S the following result is proved: Let {B1, B2, …} be any sequence of operators on an infinite dimensional Hilbert space K. Then there is an operator T on K such that the lattice of reducing subspaces of T is isomorphic to the corresponding lattice of the W* algebra generated by {B1, B2, …}. Necessary and sufficient conditions are given for S to be completely reducible to scalar weighted shifts.

scholarly journals Bell-CHSH violation under global unitary operations: Necessary and sufficient conditions

2018 ◽  
Vol 16 (04) ◽  
pp. 1850040 ◽  
Author(s):  
Nirman Ganguly ◽  
Amit Mukherjee ◽  
Arup Roy ◽  
Some Sankar Bhattacharya ◽  
Biswajit Paul ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Comparative Study ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Criterion ◽  
Density Matrices ◽  
Cartan Decomposition ◽  
Chsh Inequality ◽  
Necessary And Sufficient ◽  
Local Filtering

The relation between Bell-CHSH violation and factorization of Hilbert space is considered here. That is, a state which is local in the sense of the Bell-CHSH inequality under a certain factorization of the underlying Hilbert space can be Bell-CHSH nonlocal under a different factorization. While this question has been addressed with respect to separability, the relation of the factorization with Bell-CHSH violation has remained hitherto unexplored. We find here that there is a set containing density matrices, which do not exhibit Bell-CHSH violation under any factorization of the Hilbert space brought about by global unitary operations. Using the Cartan decomposition of [Formula: see text], we characterize the set in terms of a necessary and sufficient criterion based on the spectrum of density matrices. Sufficient conditions are obtained to characterize such density matrices based on their bloch representations. For some classes of density matrices, necessary and sufficient conditions are derived in terms of bloch parameters. Furthermore, an estimation of the volume of such density matrices is achieved in terms of purity. The criterion is applied to some well-known class of states in two qubits. Since both local filtering and global unitary operations influence the Bell-CHSH violation of a state, a comparative study is made between the two operations. The inequivalence of the two operations (in terms of increasing Bell-CHSH violation) is exemplified through their action on some classes of states.

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