Isometric flows in Hilbert space

1964 ◽  
Vol 60 (1) ◽  
pp. 45-49 ◽  
Author(s):  
Béla Sz.-Nagy
Keyword(s):  
Hilbert Space ◽  
Functional Calculus ◽  
Infinitesimal Generator ◽  
Continuous Semigroup ◽  
Contraction Operator ◽  
Strongly Continuous Semigroup ◽  
Linear Contraction ◽  
Image Position ◽  
Densely Defined

1. Let {Vi}i≥0 be a weakly (hence also strongly) continuous semigroup of (linear) contraction operators on a Hilbert space H, i.e. |Vt| ≤ 1 ( t ≥ 0). Let Z and W denote the corresponding infinitesimal generator and cogenerator, i.e.Z is in general non-bounded, but closed and densely defined, and W is a contraction operator (everywhere defined in H), such that 1 is not a proper value of W. Conversely, every contraction operator W not having the proper value 1 is the infinitesimal cogenerator of exactly one semigroup {Vi} of the above type; one has namelyin the sense of the functional calculus for contraction operators (4).

Free E0-Semigroups

1995 ◽  
Vol 47 (4) ◽  
pp. 744-785 ◽  
Author(s):  
Neal J. Fowler
Keyword(s):  
Hilbert Space ◽  
Free Product ◽  
Continuous Semigroup ◽  
Fock Space ◽  
Linear Operators ◽  
Free Flow ◽  
Strongly Continuous Semigroup ◽  
Bounded Linear Operators ◽  
Bounded Linear

AbstractGiven a strongly continuous semigroup of isometries ∪ acting on a Hilbert space ℋ, we construct an E0-semigroup α∪, the free E0-semigroup over ∪, acting on the algebra of all bounded linear operators on full Fock space over ℋ. We show how the semigroup αU⊗V can be regarded as the free product of α∪ and αV. In the case where U is pure of multiplicity n, the semigroup au, called the Free flow of rank n, is shown to be completely spatial with Arveson index +∞. We conclude that each of the free flows is cocycle conjugate to the CAR/CCR flow of rank +∞.

10.—An Abstract Relation for Multiparameter Eigenvalue Problems

1976 ◽  
Vol 74 ◽  
pp. 135-143 ◽  
Author(s):  
A. Källström ◽  
B. D. Sleeman
Keyword(s):  
Hilbert Space ◽  
Operator Equation ◽  
Eigenvalue Problems ◽  
Separable Hilbert Space ◽  
Bounded Operator ◽  
Image Position ◽  
Symmetric Operators ◽  
Densely Defined

SynopsisConsider the multiparameter systemwhere ut is an element of a separable Hilbert space Hi, i = 1, …, n. The operators Sij are assumed to be bounded symmetric operators in Hi and Ai is assumed self-adjoint. In addition consider the operator equationwhere B is densely defined and closed in a separable Hilbert space H and Tj, j = 1, …, n is a bounded operator in H. The problem treated in this paper is to seek an expression for a solution v of (**) in terms of the eigenfunctions of the system (*).

scholarly journals Common fixed points of one-parameter nonexpansive semigroups in strictly convex Banach spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-10 ◽  
Author(s):  
Tomonari Suzuki
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Continuous Semigroup ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Strongly Continuous Semigroup ◽  
Nonexpansive Semigroups ◽  
Semigroup Of Nonexpansive Mappings ◽  
Strictly Convex Banach Spaces

One of our main results is the following convergence theorem for one-parameter nonexpansive semigroups: letCbe a bounded closed convex subset of a Hilbert spaceE, and let{T(t):t∈ℝ+}be a strongly continuous semigroup of nonexpansive mappings onC. Fixu∈Candt1,t2∈ℝ+witht1<t2. Define a sequence{xn}inCbyxn=(1−αn)/(t2−t1)∫t1t2T(s)xnds+αnuforn∈ℕ, where{αn}is a sequence in(0,1)converging to0. Then{xn}converges strongly to a common fixed point of{T(t):t∈ℝ+}.

scholarly journals On a Certain Operator Related to Blackwell’s Markov Chain

2021 ◽  
Author(s):  
Ernest Nieznaj
Keyword(s):  
Markov Chain ◽  
Linear Operator ◽  
Continuous Semigroup ◽  
Basis Vector ◽  
Schauder Basis ◽  
Strongly Continuous Semigroup ◽  
Markov Operators ◽  
Densely Defined

AbstractWe present an example of a densely defined, linear operator on the $$l^{1}$$ l 1 space with the property that each basis vector of the standard Schauder basis of $$l^{1}$$ l 1 does not belong to its domain. Our example is based on the construction of a Markov chain with all states instantaneous given by D. Blackwell in 1958. In addition, it turns out that the closure of this operator is the generator of a strongly continuous semigroup of Markov operators associated with Blackwell’s chain.

A Uniqueness Theorem on the Degenerate Cauchy Problem(1)

1975 ◽  
Vol 18 (3) ◽  
pp. 417-421 ◽  
Author(s):  
Chung-Lie Wang
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Uniqueness Theorem ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Continuous Linear ◽  
Linear Maps ◽  
Image Position ◽  
Degenerate Cauchy Problem ◽  
Densely Defined

In [4] Carroll and the author have treated the following problem(1)where Λ is a closed densely defined self-adjoint operator in a separable Hilbert space H with (Λu, u) ≥ c ‖u‖2, c > 0, Λ-1 ∊ L(H) (L(E, F) is the space of continuous linear maps from E to F; in particular, L(H) = L(H, H)), a(t) > 0 for t > 0 a(0) = 0 and S(t), R(t), B(t) ∈ L(H).

scholarly journals Generation of generators of holomorphic semigroups

1993 ◽  
Vol 55 (2) ◽  
pp. 246-269 ◽  
Author(s):  
Christian Berg ◽  
Khristo Boyadzhiev ◽  
Ralph Delaubenfels
Keyword(s):  
Functional Calculus ◽  
Continuous Semigroup ◽  
Monotone Function ◽  
Linear Operators ◽  
Bernstein Function ◽  
Strongly Continuous Semigroup ◽  
Resolvent Set ◽  
The Relationship ◽  
Bounded Holomorphic ◽  
Unbounded Linear Operators

AbstractWe construct a functional calculus, g → g(A), for functions, g, that are the sum of a Stieltjes function and a nonnegative operator monotone function, and unbounded linear operators, A, whose resolvent set contains (−∞, 0), with {‖r(r + A)−1‖ ¦ r > 0} bounded. For such functions g, we show that –g(A) generates a bounded holomorphic strongly continuous semigroup of angle θ, whenever –A does.We show that, for any Bernstein function f, − f(A) generates a bounded holomorphic strongly continuous semigroup of angle π/2, whenever − A does.We also prove some new results about the Bochner-Phillips functional calculus. We discuss the relationship between fractional powers and our construction.

Quasicyclic Subnormal Semigroups

1977 ◽  
Vol 29 (6) ◽  
pp. 1230-1246 ◽  
Author(s):  
Richard Frankfurt
Keyword(s):  
Hilbert Space ◽  
Continuous Semigroup ◽  
Separable Hilbert Space ◽  
Strongly Continuous Semigroup ◽  
Bounded Operators

Let T(s), s ≧ 0, be a strongly continuous semigroup of bounded operators on a separable Hilbert space . T(s) is said to be quasicyclic if there is a continuum of vectors such that T(s)xt = xs+t for all s, t > 0 and

scholarly journals SECOND QUANTIZATION AND THE Lp-SPECTRUM OF NONSYMMETRIC ORNSTEIN–UHLENBECK OPERATORS

2005 ◽  
Vol 08 (03) ◽  
pp. 473-495 ◽  
Author(s):  
J. M. A. M VAN NEERVEN
Keyword(s):  
Hilbert Space ◽  
Continuous Semigroup ◽  
Contraction Semigroup ◽  
Strongly Continuous Semigroup ◽  
Second Quantization ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Cylindrical Wiener Process ◽  
Finite Dimensional Case ◽  
Strong Feller

The spectra of the second quantization and the symmetric second quantization of a strict Hilbert space contraction are computed explicitly and shown to coincide. As an application, we compute the spectrum of the nonsymmetric Ornstein–Uhlenbeck operator L associated with the infinite-dimensional Langevin equation [Formula: see text] where A is the generator of a strongly continuous semigroup on a Banach space E and W is a cylindrical Wiener process in E. Assuming the existence of an invariant measure μ for L, under suitable assumptions on A we show that the spectrum of L in the space Lp (E, μ) (1< p< ∞) is given by [Formula: see text] where Aμ is the generator of a Hilbert space contraction semigroup canonically associated with A and μ. We prove that the assumptions on A are always satisfied in the strong Feller case and in the finite-dimensional case. In the latter case we recover the recent Metafune–Pallara–Priola formula for σ(L).

scholarly journals Dynamical semigroups commuting with compact abelian actions

1983 ◽  
Vol 3 (2) ◽  
pp. 187-217 ◽  
Author(s):  
Ola Bratteli ◽  
David E. Evans
Keyword(s):  
Fixed Point ◽  
Continuous Semigroup ◽  
Probability Measures ◽  
Matrix Inequality ◽  
Strongly Continuous Semigroup ◽  
Completely Positive ◽  
Finite Sequences ◽  
The Matrix ◽  
Image Position ◽  
Fixed Point Algebra

AbstractLet be a C*-algebra and τ:G → Aut a compact abelian action such that the fixed point algebra τ is simple. Denote by F the *-subalgebra of G-finite elements. Let H: F → be a *-operator commuting with τ such that and the matrix inequalityholds for all finite sequences X1, …, Xn in F. Then H is closable, and the closure is the generator of a strongly continuous semigroup {exp (−t): t ≥ 0} of completely positive contractions. Furthermore, there exists a convolution semigroup {μt: t ≥ 0} of probability measures on G such that.This result has various extensions and refinements.

scholarly journals On Copula-Itô processes

2019 ◽  
Vol 7 (1) ◽  
pp. 322-347
Author(s):  
Piotr Jaworski
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Infinitesimal Generator ◽  
Continuous Semigroup ◽  
Tangent Vector ◽  
Parabolic Partial Differential Equation ◽  
Strongly Continuous Semigroup ◽  
Tangent Vector Field ◽  
The Family ◽  
Semigroup Of Transformations

AbstractWe study the dynamics of the family of copulas {Ct}t≥0 of a pair of stochastic processes given by stochastic differential equations (SDE). We associate to it a parabolic partial differential equation (PDE). Having embedded the set of bivariate copulas in a dual of a Sobolev Hilbert space H1 (ℝ2)* we calculate the derivative with respect to t and the *weak topology i.e. the tangent vector field to the image of the curve t → Ct. Furthermore we show that the family {Ct}t≥0 is an orbit of a strongly continuous semigroup of transformations and provide the infinitesimal generator of this semigroup.

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