scholarly journals Application of Scalar Type Operators to Decomposability

2021 ◽  
pp. 38-44
Author(s):  
Adicka Daniel Onyango
Keyword(s):  
Banach Space ◽  
Continuous Group ◽  
Scalar Type

In this paper, we give some application of scalar type operators to Decomposibility. In particular, we show that if H is of (α, α + 1) type R and that it generates a strongly continuous group on a Banach space, then its resolvent is Decomposable hence scalar type.

scholarly journals Lipschitz continuity of spectral measures

1999 ◽  
Vol 59 (3) ◽  
pp. 369-373
Author(s):  
Werner J. Ricker
Keyword(s):  
Banach Space ◽  
Lipschitz Condition ◽  
Lipschitz Continuity ◽  
Analogous Result ◽  
Spectral Measures ◽  
Scalar Type ◽  
Algebra Of Sets

A characterisation is given of all (finitely additive) spectral measures in a Banach space (and defined on an algebra of sets) which satisfy a Lipschitz condition. This also corrects (slightly) an analogous result in the more specialised setting of resolutions of the identity of scalar-type spectral operators (due to C.A. McCarthy).

scholarly journals On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

2019 ◽  
Vol 17 (1) ◽  
pp. 1082-1112
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
A Priori ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Improvement Effect ◽  
Abstract Evolution Equation ◽  
Necessary And Sufficient

Abstract Given the abstract evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array}$$ with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

Uniform operator σ-additivity of indefinite integrals induced by scalar-type spectral operators

1985 ◽  
Vol 101 (1-2) ◽  
pp. 141-146 ◽  
Author(s):  
S. Okada ◽  
W. Ricker
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Borel Sets ◽  
Scalar Type ◽  
Resolution Of The Identity ◽  
Spectral Integral ◽  
Set Function

SynopsisThis note characterises those Banach space valued, scalar-type spectral operators T = ∫ z dP(z), where P is the resolution of the identity for T, whose indefinite spectral integral E→∫EzdP(z) as a set function of the Borel sets of the complex plane is countably additive with respect to the uniform operator topology.

scholarly journals On scalar type spectral operators, infinite differentiable and Gevrey ultradifferentiableC0-semigroups

2004 ◽  
Vol 2004 (45) ◽  
pp. 2401-2422 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Spectral Operator ◽  
Scalar Type ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a scalar type spectral operator in a Banach space to be a generator of an infinite differentiable or a Gevrey ultradifferentiableC0-semigroup are found, the latter formulated exclusively in terms of the operator's spectrum.

scholarly journals On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

2020 ◽  
Vol 53 (1) ◽  
pp. 352-359
Author(s):  
Marat V. Markin
Keyword(s):  
Hilbert Space ◽  
Normal Operator ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Continuous Group ◽  
Left Translation ◽  
Bounded Linear ◽  
Scalar Type ◽  
Operator Group

Abstract Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a scalar type spectral operator A in a complex Banach space as well as of the collection { e t A } t ≥ 0 {\{{e}^{tA}\}}_{t\ge 0} of its exponentials, which, under a certain condition on the spectrum of the operator A, coincides with the C 0 {C}_{0} -semigroup generated by A. The spectrum of A lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group { e t A } t ∈ ℝ {\{{e}^{tA}\}}_{t\in {\mathbb{R}}} of bounded linear operators generated by A. From the general results, we infer that, in the complex Hilbert space L 2 ( ℝ ) {L}_{2}({\mathbb{R}}) , the anti-self-adjoint differentiation operator A ≔ d d x A:= \tfrac{\text{d}}{\text{d}x} with the domain D ( A ) ≔ W 2 1 ( ℝ ) D(A):= {W}_{2}^{1}({\mathbb{R}}) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A.

scholarly journals On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Spectral Operator ◽  
Scalar Type ◽  
Abstract Evolution Equation ◽  
Necessary And Sufficient ◽  
Space Conditions

For the evolution equation with a scalar type spectral operator in a Banach space, conditions on are found that are necessary and sufficient for all weak solutions of the equation on to be strongly infinite differentiable on or . Certain effects of smoothness improvement of the weak solutions are analyzed.

scholarly journals On a higher-order evolution equation with a Stepanov-bounded solution

2004 ◽  
Vol 2004 (72) ◽  
pp. 3959-3964
Author(s):  
Aribindi Satyanarayan Rao
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Infinitesimal Generator ◽  
Bounded Solution ◽  
Strong Solutions ◽  
Higher Order ◽  
Continuous Group ◽  
Almost Periodicity ◽  
Almost Periodic ◽  
General Banach Space

We study strong solutionsu:ℝ→X, a Banach spaceX, of thenth-order evolution equationu(n)−Au(n−1)=f, an infinitesimal generator of a strongly continuous groupA:D(A)⊆X→X, and a given forcing termf:ℝ→X. It is shown that ifXis reflexive,uandu(n−1)are Stepanov-bounded, andfis Stepanov almost periodic, thenuand all derivativesu′,…,u(n−1)are strongly almost periodic. In the case of a general Banach spaceX, a corresponding result is obtained, proving weak almost periodicity ofu,u′,…,u(n−1).

scholarly journals On the Carleman classes of vectors of a scalar type spectral operator

2004 ◽  
Vol 2004 (60) ◽  
pp. 3219-3235 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Resolution Of The Identity ◽  
Wiener Type

The Carleman classes of a scalar type spectral operator in a reflexive Banach space are characterized in terms of the operator's resolution of the identity. A theorem of the Paley-Wiener type is considered as an application.

scholarly journals Comment on “On the Carleman Classes of Vectors of a Scalar Type Spectral Operator”

2018 ◽  
Vol 2018 ◽  
pp. 1-3
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Gevrey Classes ◽  
Scalar Type

The results of three papers, in which the author inadvertently overlooks certain deficiencies in the descriptions of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a complex Banach space established in “On the Carleman Classes of Vectors of a Scalar Type Spectral Operator,” Int. J. Math. Math. Sci. 2004 (2004), no. 60, 3219–3235, are observed to remain true due to more recent findings.

scholarly journals Boolean algebras of projections and resolutions of the identity of scalar-type spectral operators

1997 ◽  
Vol 40 (3) ◽  
pp. 425-435 ◽  
Author(s):  
B. de Pagter ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Boolean Algebras ◽  
Spectral Operator ◽  
Strong Operator ◽  
Scalar Type ◽  
Resolution Of The Identity ◽  
Separable Spaces ◽  
Operator Closure ◽  
Positive Results

Let Μ be a Bade complete (or σ-complete) Boolean algebra of projections in a Banach space X. This paper is concerned with the following questions: When is Μ equal to the resolution of the identity (or the strong operator closure of the resolution of the identity) of some scalar-type spectral operator T (with σ(T) ⊆ ℝ) in X? It is shown that if X is separable, then Μ always coincides with such a resolution of the identity. For certain restrictions on Μ some positive results are established in non-separable spaces X. An example is given for which Μ is neither a resolution of the identity nor the strong operator closure of a resolution of the identity.

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