SOLUTION OF THE MPC PROBLEM IN TERMS OF THE SZ.-NAGY–FOIAŞ DILATION THEORY

2008 ◽  
Vol 11 (01) ◽  
pp. 73-96 ◽  
Author(s):  
F. GÓMEZ-CUBILLO
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Contraction Semigroup ◽  
Time Operator ◽  
General Context ◽  
Existence Of A Solution ◽  
Physical Problems ◽  
Dilation Theory ◽  
Injective Operator

Motivated by physical problems, Misra, Prigogine and Courbage (MPC) studied the following problem: given a one-parameter unitary group {Ut} on a separable Hilbert space [Formula: see text], find a Hilbert space [Formula: see text], a contraction semigroup {Wt} on [Formula: see text] and an injective operator [Formula: see text] with dense range which intertwines the actions of {Ut} and {Wt} (ΛWt = Ut Λ). More precisely, they studied the case where [Formula: see text] is an L2-space over a probability space and both {Ut} and {Wt} are Markovian (i.e. positivity and identity preserving). MPC gave a sufficient condition for the existence of a solution of the above problem, the existence of a time operator associated to {Ut}. In this paper we prove that, using the Sz.-Nagy–Foiaş dilation theory, it is possible to give a constructive characterization of all the solutions of the MPC problem in the general context. This criterium allows one to construct a solution of the MPC problem for which no time operator exists. When specialized to L2-spaces and Markovian {Ut} and {Wt}, the present criterium is applied to address the so-called inverse problem of Statistical Mechanics, namely to characterize the intrinsically random dynamics {Ut}.

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Nest Algebras ◽  
Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

scholarly journals Operator semistable probability measures on a Hilbert space

1980 ◽  
Vol 22 (3) ◽  
pp. 397-406 ◽  
Author(s):  
R.G. Laha ◽  
V.K. Rohatgi
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Probability Measures ◽  
Real Separable Hilbert Space ◽  
Semistable Probability

A characterization of the class of operator semistable probability measures on a real separable Hilbert space is given.

scholarly journals Isometries of Hilbert space valued function spaces

1996 ◽  
Vol 61 (2) ◽  
pp. 150-161 ◽  
Author(s):  
Beata Randrianantoanina
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Invariant Function ◽  
Function Spaces ◽  
Measurable Operator ◽  
Complex Rearrangement ◽  
Rearrangement Invariant Function Spaces ◽  
Surjective Isometry ◽  
Almost All

AbstractLet X be a (real or complex) rearrangement-invariant function space on Ω (where Ω = [0, 1] or Ω ⊆ N) whose norm is not proportional to the L2-norm. Let H be a separable Hilbert space. We characterize surjective isometries of X (H). We prove that if T is such an isometry then there exist Borel maps a: Ω → + K and σ: Ω → Ω and a strongly measurable operator map S of Ω into B (H) so that for almost all ω, S(ω) is a surjective isometry of H, and for any f ∈ X(H), T f(ω) = a(ω)S(ω)(f(σ(ω))) a.e. As a consequence we obtain a new proof of the characterization of surjective isometries in complex rearrangement-invariant function spaces.

A separable quasidiagonal C*-algebra with a non-quasidiagonal quotient by the compact operators

1991 ◽  
Vol 110 (1) ◽  
pp. 143-145 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Finite Rank ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
C Algebra ◽  
Alternative Characterization ◽  
Image Position ◽  
Algebra Of Operators

A C*-algebra A of operators on a separable Hilbert space H is said to be quasidiagonal if there is an increasing sequence E1, E2, … of finite-rank projections on H tending strongly to the identity and such thatas i → ∞ for T∈A. More generally a C*-algebra is quasidiagonal if there is a faithful *-representation π of A on a separable Hilbert space H such that π(A) is a quasidiagonal algebra of operators. When this is the case, there is a decomposition H = H1 ⊕ H2 ⊕ … where dim Hi < ∞ (i = 1, 2,…) such that each T∈π(A) can be written T = D + K where D= D1 ⊕ D2 ⊕ …, with Di∈L(Hi) (i = 1, 2,…), and K is a compact linear operator on H. As is well known (and readily seen), this is an alternative characterization of quasidiagonality.

scholarly journals CLOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT OF CERTAIN C*-ALGEBRAS

2020 ◽  
pp. 1-12
Author(s):  
BHARAT TALWAR ◽  
RANJANA JAIN
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hausdorff Space ◽  
Separable Hilbert Space ◽  
Locally Compact ◽  
C Algebras ◽  
Finite Sum ◽  
Locally Compact Hausdorff Space ◽  
Quotient Property

Abstract For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗minA is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.

scholarly journals Characterization of the Solvability of Generalized Constrained Variational Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-36 ◽  
Author(s):  
Manuel Ruiz Galán
Keyword(s):  
Elliptic Boundary ◽  
Impulsive Differential Equations ◽  
General Context ◽  
Closed Unit Ball ◽  
Existence Of A Solution ◽  
Variational Equations ◽  
Elliptic Boundary Value ◽  
Locally Convex ◽  
Linear Elliptic Boundary

In a general context, that of the locally convex spaces, we characterize the existence of a solution for certain variational equations with constraints. For the normed case and in the presence of some kind of compactness of the closed unit ball, more specifically, when we deal with reflexive spaces or, in a more general way, with dual spaces, we deduce results implying the existence of a unique weak solution for a wide class of linear elliptic boundary value problems that do not admit a classical treatment. Finally, we apply our statements to the study of linear impulsive differential equations, extending previously stated results.

scholarly journals ON THE COMPONENTS OF SPACES OF CURVES ON THE 2-SPHERE WITH GEODESIC CURVATURE IN A PRESCRIBED INTERVAL

2013 ◽  
Vol 24 (14) ◽  
pp. 1350101 ◽  
Author(s):  
NICOLAU C. SALDANHA ◽  
PEDRO ZÜHLKE
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Geodesic Curvature ◽  
Connected Components ◽  
Closed Curves ◽  
Simple Characterization ◽  
Class C

Let [Formula: see text] denote the set of all closed curves of class Cron the sphere S2whose geodesic curvatures are constrained to lie in (κ1, κ2), furnished with the Crtopology (for some r ≥ 2 and possibly infinite κ1< κ2). In 1970, J. Li ttle proved that the space [Formula: see text] of closed curves having positive geodesic curvature has three connected components. Let ρi= arccot κi(i = 1, 2). We show that [Formula: see text] has n connected components [Formula: see text] where [Formula: see text] and [Formula: see text] contains circles traversed j times (1 ≤ j ≤ n). The component [Formula: see text] also contains circles traversed (n - 1) + 2k times, and [Formula: see text] also contains circles traversed n + 2k times, for any k ∈ N. Further, each of [Formula: see text](n ≥ 3) is homeomorphic to SO3× E, where E is the separable Hilbert space. We also obtain a simple characterization of the components in terms of the properties of a curve and prove that [Formula: see text] is homeomorphic to [Formula: see text] whenever [Formula: see text].

scholarly journals A note on local asymptotic behaviour for Brownian motion in Banach spaces

1979 ◽  
Vol 2 (4) ◽  
pp. 669-676
Author(s):  
Mou-Hsiung Chang
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Brownian Motion ◽  
Gaussian Measure ◽  
Separable Hilbert Space ◽  
Integral Test ◽  
Real Separable Banach Space ◽  
Real Separable Hilbert Space ◽  
Special Case

In this paper we obtain an integral characterization of a two-sided upper function for Brownian motion in a real separable Banach space. This characterization generalizes that of Jain and Taylor [2] whereB=ℝn. The integral test obtained involves the index of a mean zero Gaussian measure on the Banach space, which is due to Kuelbs [3]. The special case that whenBis itself a real separable Hilbert space is also illustrated.

scholarly journals On a Characterization of Convergence in Banach Spaces with a Schauder Basis

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Marat V. Markin ◽  
Olivia B. Soghomonian
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Schauder Basis ◽  
Infinite Dimensional ◽  
Summable Sequence ◽  
Convergent Sequences

We extend the well-known characterizations of convergence in the spaces l p ( 1 ≤ p < ∞ ) of p -summable sequence and c 0 of vanishing sequences to a general characterization of convergence in a Banach space with a Schauder basis and obtain as instant corollaries characterizations of convergence in an infinite-dimensional separable Hilbert space and the space c of convergent sequences.“The method in the present paper is abstract and is phrased in terms of Banach spaces, linear operators, and so on. This has the advantage of greater simplicity in proof and greater generality in applications.” Jacob T. Schwartz

scholarly journals Absorption in Invariant Domains for Semigroups of Quantum Channels

2021 ◽  
Author(s):  
Raffaella Carbone ◽  
Federico Girotti
Keyword(s):  
Hilbert Space ◽  
Ergodic Theory ◽  
Markov Processes ◽  
Separable Hilbert Space ◽  
Quantum Channels ◽  
Quantum Markov Processes ◽  
Positive Recurrent ◽  
Markov Evolution ◽  
Absorption Probabilities ◽  
Invariant Domains

AbstractWe introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

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