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Analytic semigroups generated on a functional extrapolation space by variational elliptic equations

Vincenzo Vespri (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove that any elliptic operator of second order in variational form is the infinitesimal generator of an analytic semigroup in the functional space C - 1 , α ⁢ ( Ω ) consinsting of all derivatives of hölder-continuous functions in Ω where Ω is a domain in ℝ n not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space C - 1 , α ⁢ ( Ω ) . We prove also that the spaces C - 1 , α ⁢ ( Ω ) can be considered as extrapolation spaces relative to suitable non-variational operators....

Analytic semigroups on vector valued noncommutative L p -spaces

Cédric Arhancet (2013)

Studia Mathematica

We give sufficient conditions on an operator space E and on a semigroup of operators on a von Neumann algebra M to obtain a bounded analytic or R-analytic semigroup ( ( T ⊗ I d E ) t ≥ 0 on the vector valued noncommutative L p -space L p ( M , E ) . Moreover, we give applications to the H ∞ ( Σ θ ) functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu.

Analyticity for some degenerate one-dimensional evolution equations

G. Metafune (1998)

Studia Mathematica

We study the analyticity of the semigroups generated by some degenerate second order differential operators in the space C([α,β]), where [α,β] is a bounded real interval. The asymptotic behaviour and regularity with respect to the space variable are also investigated.

Analyticity of transition semigroups and closability of bilinear forms in Hilbert spaces

Marco Fuhrman (1995)

Studia Mathematica

We consider a semigroup acting on real-valued functions defined in a Hilbert space H, arising as a transition semigroup of a given stochastic process in H. We find sufficient conditions for analyticity of the semigroup in the L 2 ( μ ) space, where μ is a gaussian measure in H, intrinsically related to the process. We show that the infinitesimal generator of the semigroup is associated with a bilinear closed coercive form in L 2 ( μ ) . A closability criterion for such forms is presented. Examples are also given....

Approximate solution of an inhomogeneous abstract differential equation

Emil Vitásek (2012)

Applications of Mathematics

Recently, we have developed the necessary and sufficient conditions under which a rational function F ( h A ) approximates the semigroup of operators exp ( t A ) generated by an infinitesimal operator A . The present paper extends these results to an inhomogeneous equation u ' ( t ) = A u ( t ) + f ( t ) .

Approximate solutions of abstract differential equations

Emil Vitásek (2007)

Applications of Mathematics

The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy.

Approximation theorem for evolution operators

Rinka Azuma (2003)

Studia Mathematica

This paper is devoted to the study of the approximation problem for the abstract hyperbolic differential equation u'(t) = A(t)u(t) for t ∈ [0,T], where A(t):t ∈ [0,T] is a family of closed linear operators, without assuming the density of their domains.

Around the Kato generation theorem for semigroups

Jacek Banasiak, Mirosław Lachowicz (2007)

Studia Mathematica

We show that the result of Kato on the existence of a semigroup solving the Kolmogorov system of equations in l₁ can be generalized to a larger class of the so-called Kantorovich-Banach spaces. We also present a number of related generation results that can be proved using positivity methods, as well as some examples.

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