Semi-groups of scalar type operators in Banach spaces

Panchapagesan, T. V.

doi:10.2140/pjm.1969.30.489

Cited by 12 publications

(13 citation statements)

References 17 publications

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“…with some ω ∈ , the collection of exponentials e tA t 0 { } ≥ is the C 0 -semigroup generated by A [15, Proposition 3.1] (cf. also [16,17]), and hence, if…”

Section: Providedmentioning

confidence: 95%

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

Markin

2020

Open Mathematics

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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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“…with some ω ∈ , the collection of exponentials e tA t 0 { } ≥ is the C 0 -semigroup generated by A [15, Proposition 3.1] (cf. also [16,17]), and hence, if…”

Section: Providedmentioning

confidence: 95%

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

Markin

2020

Open Mathematics

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“…A scalar type spectral C 0 -semigroup (i.e., a C 0 -semigroup of scalar type spectral operators) is generated by a scalar type spectral operator [3,23]. A scalar type spectral operator A in a complex Banach space X generates a (scalar type spectral)…”

Section: )mentioning

confidence: 99%

On asymptotics for $C_0$-semigroups

Markin

2021

Preprint

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We generalize to C 0 -semigroups of scalar type spectral operators on complex Banach spaces the spectral bound equal growth bound condition along with a generalized Lyapunov stability theorem, known to hold for C 0semigroups of normal operators on complex Hilbert spaces. For such semigroups, we obtain exponential estimates with the best stability constants. We also extend to a Banach space setting a celebrated characterization of uniform exponential stability for C 0 -semigroups on complex Hilbert spaces and thereby acquire a characterization of uniform exponential stability for scalar type spectral and eventually norm-continuous C 0 -semigroups.

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“…with some ω ∈ , the collection of exponentials e tA t 0 { } ≥ coincides with the C 0 -semigroup generated by A [12, Proposition 3.1] (see also [13,14]), and hence, if…”

Section: Preliminariesmentioning

confidence: 99%

“…with some ω ∈ , coincides with the C 0 -semigroup generated by A [12] (see also [13,14]). The spectrum of A lying on the imaginary axis i (i is the imaginary unit), we also show that non-hypercyclic is the strongly continuous group e tA t { } ∈ of bounded linear operators generated by A.…”

Section: Introductionmentioning

confidence: 99%

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

Markin

2020

Demonstratio Mathematica

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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.

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Semi-groups of scalar type operators in Banach spaces

Cited by 12 publications

References 17 publications

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

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

On asymptotics for $C_0$-semigroups

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

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