Linear Dynamical Systems.- Semigroups, Generators, and Resolvents.- Perturbation and Approximation of Semigroups.- Spectral Theory for Semigroups and Generators.- Asymptotics of Semigroups.- Semigroups Everywhere.- A Brief History of the Exponential Function.

https://www.springer.com/productFlyer_978-0-387-98463-6.pdf?SGWID=0-0-1297-1590487-0

One-Parameter Semigroups for Linear Evolution Equations

General Theory of Banach Algebra. By C.E. Rickart. Pp. 394. 79s. 1960. (Van Nostrand, London)

function as a particular Banach space (whose norm satisfies the parallelogram law), but rather as an inner-product space. It is in terms of the innerproduct space structure that most of the terminology and techniques are developed. On the other hand, this type of Hilbert space considerations find no real parallel in the general Banach space setting. Some time ago, while trying to carry over a Hilbert space argument to a general Banach space situation, we were led to use a suitable mapping from a Banach space into its dual in order to make up for the lack of an innerproduct. Our procedure suggested the existence of a general theory which it seemed should be useful in the study of operator (normed) algebras by providing better insight on known facts, a more adequate language to "classify" special types of operators, as well as new techniques. These ideas evolved into a theory of semi-inner-product spaces which is presented in this paper (together with certain applications)(1). We shall consider vector spaces on which instead of a bilinear form there is defined a form [x, y] which is linear in one component only, strictly positive, and satisfies a Schwarz inequality. Such a form induces a norm, by setting |x| = ([x, x])112; and for every normed space one can construct at least one such form (and, in general, infinitely many) consistent with the

/pdf/semi-inner-product-spaces-3rzdaso8du.pdf

Semi-inner-product spaces

Matrix formulations are presented for the discrete-ordinate and the matrix-operator methods of solving the transfer of solar radiation in a plane-parallel scattering atmosphere. Eigenspace transformations of symmetric matrices are introduced into the method of Stamnes and Swanson instead of using the decomposition of an asymmetric matrix. The computational stability is considerably improved by this algorithm, especially for single-precision calculations. Representations of the reflection and transmission matrices in the matrix-operator method are also given, in terms of the indicated formulations, by considering a boundary-value problem of the discrete-ordinate method. The solutions of the discrete-ordinate method for inhomogeneous atmospheres are given by combining discrete-ordinate solutions for respective homogeneous sublayers through the addition technique of the matrix-operator method.

Matrix formulations for the transfer of solar radiation in a plane-parallel scattering atmosphere.

This paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation. To obtain this result, we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size. The paper also contains some simple numerical examples and some new results for the Milne problem. Introduction. The computation of the critical size and the diffusion approximation for the transport equation have been closely related and this is due to the following facts. First, the computation of the critical size is much easier for the diffusion approximation than for the original transport equation. Second, for the critical size one can consider a host media X = nXQ with X0 given and 17 a positive number. The problem of the critical size is then reduced to the computation of the parameter zj. It turns out that when the transport operator is almost conservative, the critical value of the parameter 17 is large and it is exactly for this range of value that the diffusion approximation is accurate. On the other hand the "physical" boundary condition for the diffusion approxi- mation is of the form

/pdf/diffusion-approximation-and-computation-of-the-critical-size-2g57u5jgcy.pdf

Diffusion approximation and computation of the critical size

Spectra of perturbed semigroups with applications to transport theory

Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator

Hier bedeuten X, Y beliebige Operatoren und ;t,/~ komplexe Zahlen. Wie bekannt, haben I .M. GELFAND und M. A. NEUMARK bewiesen [1], d a b die Postulate 1 ~ bis 5 ~ die Unteralgebren dieser Algebra vollst~ndig charakterisieren: Jede BANACHsche Algebra mit einer Involution, die den Bedingungen t ~ bis 5 ~ geniigt, lliBt sich isometrisch-isomorph auf eine gieichm~iBig abgeschlossene Algebra von Operatoren "eines HILBERTSchen Raumes abbilden. Das ftinfte Postnlat erwies sich iibrigens als iiberfliissig, weil es aus den vier anderen Postulaten folgt (s. [6], S. 884). Wenn X * = X, so nennt man den Operator X selbstadjungiert. Das Ziel dieser Arbeit ist nun, ein anderes System yon Postulaten aufzustellen, das die genannten Algebren mittels einer metrischen Kennzeichnung der in ihnen enthaltenen selbstadjungierten Operatoren charakterisiert. Sei also ~ eine BANACHsche Algebra mit Einheitselement, das wir mit I bezeichnen wollen. Unser Ziel ist der Beweis des folgenden Satzes: I. Es sei Heine Untermenge yon ~, die ]olgenden drei Bedingungen geni~gt: A. Jedes Element x E ~ lgft sick mindestens in einer Weise in der Form x = h + i g schreiben, wo h, gE H'und i2= --1. B. Geh6rt h zu H und ist ~ eine reelle Zahl, so gilt ] [ l+ i~h] l~ t+o(~) , wenn ~ gegen Null strebt. C. Fiir jedes hE H existiert eine Zerlegung hZ= u + iv, u, v E H, so, daft u v = v u ist. Dann hat die Algebra ?3 die ]olgenden Eigenscha]ten: t. Die Zerlegung x = h + ig ist ]iir jedes x E ~ eindeutig. 2. Wird x * = h i g geset.zt, ]alls x '--h + i g gilt, so ist die Abbildung x--->x* eine Involution in ~, die den Postulaten t ~ 2 ~ 3 ~ und 5 ~ geniigt. Far die Elemente hE H gilt auch 4 ~ also ][h2][=llh[] ~. 3][xilo = lix* xil ~ ist eine Norm in f3, die der urs priinglichen [lxi] ~quivalent ist.

Ivan Vidav

Papers

Spectra of perturbed semigroups with applications to transport theory

Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator

Eine metrische Kennzeichnung der selbstadjungierten Operatoren

On the spectrum of relaxation lengths of neutron distributions in a moderator

On Operator-Valued Analytic Functions With Constant Norm*