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 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.

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

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 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.

On prespectrality of generalised derivations

1986 ◽  
Vol 104 (1-2) ◽  
pp. 93-106 ◽  
Author(s):  
Milan Hladnik
Keyword(s):  
Direct Methods ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Spectral Operator ◽  
Bounded Linear ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Scalar Type ◽  
Necessary And Sufficient

SynopsisIn this paper it is proved that, for scalar-type operators a and b on an infinite dimensional separable complex Hilbert space H, the generalised derivation Δa,b, defined for bounded linear operators x onℋ by the equation Δa,bx = ax − xb, is a (scalar-type) prespectral operator of the class (the trace class operators on ℋ) if and only if at least one of the spectra σ(a)or σ(b)is finite. It is shown also that the same condition is necessary and sufficient for Δa,b restricted to any one of the von Neumann-Schatten classes(p≠2) to be a spectral operator (of scalar type). Our results may be compared with those of J. Anderson and C. Foiaş, who established in [1] that, for scalar-type a, b, Δa,b is a (scalar-type) spectral operator if and only if both spectra, σ(a) and σ(b), are finite. However, we use different and more direct methods to show the existence or nonexistence of the spectral resolution of identity for Δa,b.

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