Transfer functions of regular linear systems Part II: The system operator and the Lax-Phillips semigroup

This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as ``Part I'. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by , would be the -dependent matrix . In the general case, is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks and , as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of where the right lower block is the feedthrough operator of the system. Using , we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the ``initial time' is . We also introduce the Lax-Phillips semigroup induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of and also the points where is not invertible, in terms of the spectrum of the generator of (for various values of ). The system is dissipative if and only if (with index zero) is a contraction semigroup.

     

The oscillation of the Poisson semigroup associated to parabolic Hermite operator

Let O(P_τ~L) be the oscillation of the Possion semigroup associated with the parabolic Hermite operator L = ?_t-?+|x|~2. We show that O(P_τ~L) is bounded from L~p(R~(n+1))into itself for 1 p ∞, bounded from L~1(R~(n+1)) into weak-L~1(R~(n+1)) and bounded from L_c~∞(R~(n+1)) into BMO(R~(n+1)). In the case p = ∞ we show that the range of the image of the operator O(P_τ~L) is strictly smaller than the range of a general singular operator.     

Lax-Phillips Scattering for Atomorphic Functions Based on the Eisenstein Transform

We construct a Lax-Phillips scattering system on the arithmetic quotient space of the Poincaré upper half-plane by the full modular group, based on the Eisenstein transform. We identify incoming and outgoing subspaces in the ambient space of all functions with finite energy-form for the non-Euclidean wave equation. The use of the Eisenstein transform along with some properties of the Eisenstein series of two variables enables one to work only on the space corresponding to the continuous spectrum of the Laplace-Beltrami operator. It is shown that the scattering matrix is the complex function appearing in the the functional equation of the Eisenstein series of two variables. We obtain a compression operator constructed from the Laplace-Beltrami operator, whose spectrum consists of eigenvalues that coincide, counted with multiplicities, with the non-trivial zeros of the Riemann zeta-function. For this purpose we construct and use a scattering model on the one-dimensional Euclidean space.      

Loewner chains associated with the generalized Roper-Suffridge extension operator

In this paper, we introduce two classes of generalized Roper-Suffridge extension operators and prove that they can be embedded in Loewner chains. In particular, our proof shows that these two classes of operators preserve starlikeness and spirallikeness of type α on two important classes of Reinhardt domains in Cn, respectively. Finally, some other related results are given.     

On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations

For a singular perturbation , n ≤ ∞, of a positive self-adjoint operator A ₀ with Lebesgue spectrum, the spectral analysis of the corresponding self-adjoint operator realizations A _T is carried out and the scattering matrix is calculated in terms of parameters t _ij under some additional restrictions on singular elements ψ _j . The results obtained enable one to apply the Lax-Phillips approach in scattering theory. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 679–688, May, 2005.     

Lototsky-Schnabl operators associated with a strictly elliptic differential operator and their corresponding Feller semigroup

Given an open bounded convex subset of ^p , a strictly elliptic differential operatorL and a continuous function , and denoted withT _L the Dirichlet operator associated withL, the Lototsky-Schnabl operators associated withT _L and are investigated. In particular, conditions are established which ensure the existence of a Feller semigroup represented by limit of powers of these operators. Then the analytic expression of the infinitesimal generator is determined and some properties of the semigroup are deduced. Finally, the saturation class of Lototsky-Schnabl operators is determined.Work supported by a C.N.R. Research Grant (n. 201.19.1, November 30, 1994)     

Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator

By analysing some explicit examples we investigate the positivity and the non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator associated with the operator \(\varDelta +\lambda I\) as \(\lambda \) varies. It is known that the semigroup is positive if \(\lambda <\lambda _1\) , where \(\lambda _1\) is the principal eigenvalue of \(-\varDelta \) with Dirichlet boundary conditions. We show that it is possible for the semigroup to be non-positive, eventually positive or positive and irreducible depending on \(\lambda >\lambda _1\) .     

Perturbation of the generating operator of an analytic semigroup

Voronezh Military Aviation Engineering Academy. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 1, pp. 62–63, January–March, 1988.     

Loewner chains associated with the generalized Roper-Suffridge extension operator on some domains

In this paper, we consider the generalized Roper-Suffridge extension operator defined by

     

Slowly changing vectors and the asymptotic finite-dimensionality of an operator semigroup

Siberian Mathematical Journal - Let X be a Banach space and let T: X → X be a linear power bounded operator. Put X 0 = {x ∈ X | T n x → 0}. We prove that if X 0 ≠ X then...     

Schrödinger equation and the oscillatory semigroup for the Hermite operator

We discuss the regularity of the oscillatory semigroup e^itH, where H=-Δ+|x|² is the n-dimensional Hermite operator. The main result is a Strichartz-type estimate for the oscillatory semigroup e^itH in terms of the mixed L^p spaces. The result can be interpreted as the regularity of solution to the Schrödinger equation with potential V(x)=|x|².     

Operators associated with the Jacobi semigroup

The aim of this paper is to introduce some operators induced by the Jacobi differential operator and associated with the Jacobi semigroup, where the Jacobi measure is considered in the multidimensional case.In this context, we introduce potential operators, fractional integrals, fractional derivates, Bessel potentials and give a version of Carleson measures.We establish a version of Meyer’s multiplier theorem and by means of this theorem, we study fractional integrals and fractional derivates.Potential spaces related to Jacobi expansions are introduced and using fractional derivates, we give a characterization of these spaces. A version of Calderon’s Reproduction Formula and a version of Fefferman’s theorem are given.Finally, we present a definition of Triebel–Lizorkin spaces and Besov spaces in the Jacobi setting.     

广义算子半群与广义分布参数系统的适定性

首先,针对广义分布参数系统的求解问题,提出了由Hilbert空间中有界线性算子所引导的广义算子半群和广义积分半群;其次,讨论了广义预解算子的性质、广义算子半群与广义积分半群的性质;最后,研究了广义分布参数系统的适定性问题.     

Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator

The paper is devoted to a careful analysis of the shape-preserving properties of the strongly continuous semigroup generated by a particular second-order differential operator, with particular emphasis on the preservation of higher order convexity and Lipschitz classes. In addition, the asymptotic behaviour of the semigroup is investigated as well. The operator considered is of interest, since it is a unidimensional Black-Scholes operator so that our results provide qualitative information on the solutions of classical problems in option pricing theory in Mathematical Finance. The paper is dedicated to Professor Luigi Albano on the occasion of his 70th birthday.     

Construction of some quantum stochastic operator cocycles by the semigroup method

A new method for the construction of Fock-adapted quantum stochastic operator cocycles is outlined, and its use is illustrated by application to a number of examples arising in physics and probability. The construction uses the Trotter-Kato theorem and a recent characterisation of such cocycles in terms of an associated family of contraction semigroups. In celebration of Kalyan Sinha’s sixtieth birthday