Differentiation of Multiparameter Superadditive Processes

1985 ◽  
Vol 37 (3) ◽  
pp. 385-404
Author(s):  
Doğan Çömez
Keyword(s):  
Continuous Semigroup ◽  
Strongly Continuous Semigroup ◽  
Semigroup Of Operators ◽  
Markovian Semigroup ◽  
Extra Assumption ◽  
Differentiation Theorem

In this article our purpose is to prove a differentiation theorem for multiparameter processes which are strongly superadditive with respect to a strongly continuous semigroup of positive L1 contractions (see Section 1 for definitions).Recently, the differentiation theorem for superadditive processes with respect to a one-parameter semigroup of positive L1-contractions has been proved by D. Feyel [9]. Another proof is given by M. A. Akçoğlu [1]. R. Emilion and B. Hachem [7] also proved the same theorem, but with an extra assumption on the process (see also [1]). The proof of this theorem for superadditive processes with respect to a Markovian semigroup of operators on L1 is given by M. A. Akçoğlu and U. Krengel [4]. Thus [1] and [9] extend the result of [4] to the sub-Markovian setting. Here we will obtain the multiparameter sub-Markovian version of this theorem, namely Theorem 3.17 below

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 +∞.

Dilations of one Parameter Semigroups of Positive Contractions on Lp Spaces

1997 ◽  
Vol 49 (4) ◽  
pp. 736-748 ◽  
Author(s):  
Gero Fendler
Keyword(s):  
Continuous Semigroup ◽  
Continuous Group ◽  
Strongly Continuous Semigroup ◽  
Von Neumann ◽  
Lp Spaces ◽  
Discrete Semigroup ◽  
Neumann Type ◽  
Transfer Map

AbstractIt is proved in this note, that a strongly continuous semigroup of (sub)positive contractions acting on an Lp-space, for 1 < p < ∞ p ≠ 2, can be dilated by a strongly continuous group of (sub)positive isometries in a manner analogous to the dilation M. A. Akçoglu and L. Sucheston constructed for a discrete semigroup of (sub)positive contractions. From this an improvement of a von Neumann type estimation, due to R. R.Coifman and G.Weiss, on the transfer map belonging to the semigroup is deduced.

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.

scholarly journals Exponential dichotomy of strongly discontinuous semigroups

1984 ◽  
Vol 30 (3) ◽  
pp. 435-448 ◽  
Author(s):  
P. Preda ◽  
M. Megan
Keyword(s):  
Banach Space ◽  
Exponential Stability ◽  
General Class ◽  
Exponential Dichotomy ◽  
Continuous Semigroup ◽  
Sufficient Conditions ◽  
Semigroups Of Operators ◽  
Strongly Continuous Semigroup ◽  
Strongly Continuous Semigroups ◽  
Necessary And Sufficient

In this paper we give necessary and sufficient conditions for exponential dichotomy of a general class of strongly continuous semigroups of operators defined on a Banach space. As a particular case we obtain a Datko theorem for exponential stability of a strongly continuous semigroup of class C0 defined on a Banach space.

scholarly journals On the Adjoint of a Strongly Continuous Semigroup

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Diómedes Bárcenas ◽  
Luis Gerardo Mármol
Keyword(s):  
Banach Spaces ◽  
Continuous Semigroup ◽  
Isomorphic Copy ◽  
Strongly Continuous Semigroup ◽  
Strong Continuity ◽  
Strongly Continuous Semigroups ◽  
Vector Integration ◽  
Finite Dimensional ◽  
Weak Measurability

Using some techniques from vector integration, we prove the weak measurability of the adjoint of strongly continuous semigroups which factor through Banach spaces without isomorphic copy ofl1; we also prove the strong continuity away from zero of the adjoint if the semigroup factors through Grothendieck spaces. These results are used, in particular, to characterize the space of strong continuity of{T**(t)}t≥0, which, in addition, is also characterized for abstractL- andM-spaces. As a corollary, it is proven that abstractL-spaces with no copy ofl1are finite-dimensional.

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.

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).

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