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 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.

scholarly journals On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one

2019 ◽  
Vol 17 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Abstract Evolution Equation

Abstract It is shown that, if all weak solutions of the evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array} $$ with a scalar type spectral operator A in a complex Banach space are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded.

scholarly journals On an abstract evolution equation with a spectral operator of scalar type

2002 ◽  
Vol 32 (9) ◽  
pp. 555-563 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Operational Calculus ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Initial Values ◽  
Scalar Type ◽  
Abstract Evolution Equation

It is shown that the weak solutions of the evolution equationy′(t)=Ay(t),t∈[0,T) (0<T≤∞), whereAis a spectral operator of scalar type in a complex Banach spaceX, defined by Ball (1977), are given by the formulay(t)=e tAf,t∈[0,T), with the exponentials understood in the sense of the operational calculus for such operators and the set of the initial values,f's, being∩ 0≤t<TD(e tA), that is, the largest possible such a set inX.

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

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Marat V. Markin
Keyword(s):  
Evolution Equation ◽  
Weak Solutions ◽  
Normal Operator ◽  
A Priori ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Improvement Effect ◽  
Abstract Evolution Equation

Given the abstract evolution equation y′(t)=Ay(t),  t∈R, 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 infinite differentiable on R. The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R.

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 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 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 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.

scholarly journals A characterization of the generators of analyticC0-semigroups in the class of scalar type spectral operators

2004 ◽  
Vol 2004 (12) ◽  
pp. 1007-1018 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Scalar Type

In the class of scalar type spectral operators in a complex Banach space, a characterization of the generators of analyticC0-semigroups in terms of the analytic vectors of the operators is found.

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

2020 ◽  
Vol 18 (1) ◽  
pp. 1952-1976
Author(s):  
Marat V. Markin
Keyword(s):  
Evolution Equation ◽  
Weak Solutions ◽  
Normal Operator ◽  
A Priori ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Abstract Evolution Equation ◽  
Necessary And Sufficient

Abstract Given the abstract evolution equation y ′ ( t ) = A y ( t ) , t ∈ ℝ , y^{\prime} (t)=Ay(t),t\in {\mathbb{R}}, with a scalar type spectral operator A in a complex Banach space, we find conditions on A, formulated exclusively in terms of the location of its spectrum in the complex plane, 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 \beta \ge 1 , in particular analytic or entire, on ℝ {\mathbb{R}} . We also reveal certain inherent smoothness improvement effects and show that if all weak solutions of the equation are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded. The important particular case of the equation with a normal operator A in a complex Hilbert space follows immediately.

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