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.

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).

Formally Normal Operators Having no Normal Extensions

1965 ◽  
Vol 17 ◽  
pp. 1030-1040 ◽  
Author(s):  
Earl A. Coddington
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Null Space ◽  
Normal Operator ◽  
Closed Linear Operator ◽  
Normal Operators ◽  
The Given ◽  
Densely Defined ◽  
Made In

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

scholarly journals Stability of essential approximate point spectrum and essential defect spectrum of linear operator

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1983-1994
Author(s):  
Aymen Ammar ◽  
Mohammed Dhahri ◽  
Aref Jeribi
Keyword(s):  
Linear Operator ◽  
Measure Of Noncompactness ◽  
Point Spectrum ◽  
Fredholm Operators ◽  
Approximate Point Spectrum ◽  
Densely Defined

In the present paper, we use the notion of measure of noncompactness to give some results on Fredholm operators and we establish a fine description of the essential approximate point spectrum and the essential defect spectrum of a closed densely defined linear operator.

scholarly journals Dirac structures on Hilbert spaces

1999 ◽  
Vol 22 (1) ◽  
pp. 97-108 ◽  
Author(s):  
A. Parsian ◽  
A. Shafei Deh Abad
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Smooth Manifold ◽  
Real Hilbert Space ◽  
Dirac Structure ◽  
Dirac Structures ◽  
Basic Properties ◽  
Hilbert Subspace ◽  
Densely Defined

For a real Hilbert space(H,〈,〉), a subspaceL⊂H⊕His said to be a Dirac structure onHif it is maximally isotropic with respect to the pairing〈(x,y),(x′,y′)〉+=(1/2)(〈x,y′〉+〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures onHare in one-to-one correspondence with isometries onH, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structureLonH, everyz∈His uniquely decomposed asz=p1(l)+p2(l)for somel∈L, wherep1andp2are projections. Whenp1(L)is closed, for any Hilbert subspaceW⊂H, an induced Dirac structure onWis introduced. The latter concept has also been generalized.

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.

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

scholarly journals Derivations, local derivations and atomic boolean subspace lattices

2002 ◽  
Vol 66 (3) ◽  
pp. 477-486 ◽  
Author(s):  
Pengtong Li ◽  
Jipu Ma
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Closed Linear Operator ◽  
Subspace Lattice ◽  
Local Derivation ◽  
Densely Defined ◽  
Subspace Lattices

Let ℒ be an atomic Boolean subspace lattice on a Banach space X. In this paper, we prove that if ℳ is an ideal of Alg ℒ then every derivation δ from Alg ℒ into ℳ is necessarily quasi-spatial, that is, there exists a densely defined closed linear operator T: 𝒟(T) ⊆ X → X with its domain 𝒟(T) invariant under every element of Alg ℒ, such that δ(A) x = (TA – AT) x for every A ∈ Alg ℒ and every x ∈ 𝒟(T). Also, if ℳ ⊆ ℬ(X) is an Alg ℒ-module then it is shown that every local derivation from Alg ℒ into ℳ is necessary a derivation. In particular, every local derivation from Alg ℒ into ℬ(X) is a derivation and every local derivation from Alg ℒ into itself is a quasi-spatial derivation.

scholarly journals Closed linear operators with domain containing their range

1984 ◽  
Vol 27 (2) ◽  
pp. 229-233 ◽  
Author(s):  
Schôichi Ôta
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Linear Operators ◽  
Closed Linear Operator ◽  
Unbounded Operators ◽  
Densely Defined ◽  
Closed Linear Operators

In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T* and is globally invariant under T and T*,then T is bounded. In the case of a Banach space (in particular, a C*-algebra) weshowed in [6] that a densely defined closed derivation in a C*-algebra with domaincontaining its range is automatically bounded (see the references in [6] and [7] for thetheory of derivations in C*-algebras).

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