1,721,317 research outputs found
POLYNOMIAL OPERATOR MATRICES AS SEMIGROUP GENERATORS - THE 2X2 CASE
No full textMany systems of linear evolution equations can be written as a single equation
\dot u(t)=\s A u(t),\eqno(*)
where is a function with values in a product space and
\s A =(A_{ij})_{n\times n} is a matrix whose entries are
linear operators on . In order to prove the well-posedness of
one shows that \s A generates a strongly continuous semigroup on .
In this paper we consider the case where the are polynomials
with respect to a single (unbounded) operator on and
restrict our attention to the case of matrices
POLYNOMIAL OPERATOR MATRICES AS GENERATORS - THE GENERAL-CASE
No full textOf concern are systems of linear evolution equations
\dot u(t)=\s A u(t),\qquad u(0)=u_0,\leqno({\rm ACP})
where is a function with values in a product Banach space \s E :=E^n and
\s A=(p_{ij}(A)) is a matrix whose entries are polynomials in
a fixed linear, possibly unbounded operator on . In this paper we will study
the well-posedness of , i.e., we will characterize those polynomial operator
matrices \s A generating a strongly continuous semigroup on \s E
The Laplacian on with generalized Wentzell boundary conditions
No full textIn this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain Omega in R-m defined by
(1) Af := Deltaf, D(A) := {f is an element of C-n(1) ((Omega) over bar) : Deltaf is an element of C((Omega) over bar); Deltaf + betapartial derivativef/partial derivativen + gammaf = 0 on partial derivativeOmega}
generates an analytic semigroup of angle pi/2 on C((Omega) over bar) for every beta > 0 and gamma is an element of C (partial derivativeOmega) (for the 2 definition of C-n(1) ((Omega) over bar) cf. (1.3))
Positivity and stability for one-sided coupled operator matrices
No full textMany evolutionary systems can be described by an abstract Cauchy problem governed by an operator matrix. Assuming this problem to be "one-sided coupled" and "well-posed" we study in this paper the positivity and the stability of the associated matrix semigroup. The abstract results are illustrated by several examples
Spectral theory and generator property for one-sided coupled operator matrices
No full textMany initial value problems like Volterra equations, delay equations or wave equations can be reduced to an abstract Cauchy problem governed by an opercator matrix. We introduce a new class of unbounded operator matrices corresponding to these equations and study the spectral theory, compute the adjoint and analyze the generator property of its elements. The abstract results are illustrated by a series of applications
A spectral mapping theorem for polynomial operator matrices
No full textSystems of linear evolution equations can be written as a single equation (∗) u ̇(t)=Au(t), where u is a function with values in a product space En and A=(Aij)n×n is an operator matrix. Often the entries Aij are polynomials pij(A) with respect to a single (unbounded) operator A on E. In order to solve (∗) one has to determine the properties of the operator matrix A. In particular, one has to find an appropriate domain D(A) such that A is closed. This is discussed in the first part of this paper. Then it is important to compute the spectrum σ(A) of A. One expects a kind of spectral mapping theorem based on the spectrum σ(A) of A and the structure of the matrix (pij). We show in Part 2 in which sense such a spectral mapping theorem holds. An application to stability theory, i.e., the computation of an estimate for the spectral bound s(A), concludes this paper
ON PERTURBATIONS OF LINEAR M-ACCRETIVE OPERATORS ON REFLEXIVE BANACH-SPACES
No full textIn this note we prove some results on the m-accretivity of sums and products of linear operators. In particular we obtain the following theorem: Let A, B be two m-accretive operators on a reflexive Banach space. If A is invertible and (A')B--1' is accretive then BA(-1) and A + B are m-accretive
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