On Unitary Extensions of Multiplicative Families of Partial Isometries with a Generating Subspace

We consider a two parameter family of operators on a Hilbert space given by a multiplicative family of partial isometries with a generating subspace that commutes with an unitary group. The first parameter runs on a generalized interval of an abelian ordered group and the second on an abelian group. We show that it can be extended to a two parameter group of unitary operators on a larger Hilbert space. Applications to the problem of extending generalized semigroups of isometries and locally defined positive definite functions are given.     

Ergodic semigroups of positivity preserving self-adjoint operators

We prove that an ergodic semigroup of positivity preserving self-adjoint operators is positivity improving. We also present a new proof (using Markov techniques) of the ergodicity of semigroups generated by spatially cutoff P(?)₂ Hamiltonians.     

The group of isometries of a Banach space and duality

We construct an example of a real Banach space whose group of surjective isometries has no uniformly continuous one-parameter semigroups, but the group of surjective isometries of its dual contains infinitely many of them. Other examples concerning numerical index, hermitian operators and dissipative operators are also shown.     

Representations of regularized determinants of exponentials of differential operators by functional integrals

Representations of regularized determinants of elements of one-parameter operator semigroups whose generators are second-order elliptic differential operators by Lagrangian functional integrals are obtained. Such semigroups describe solutions of inverse Kolmogorov equations for diffusion processes. For self-adjoint elliptic operators, these semigroups are often called Schrödinger semigroups, because they are obtained by means of analytic continuation from Schrödinger groups. It is also shown that the regularized determinant of the exponential of the generator (this exponential is an element of a one-parameter semigroup) coincides with the exponential of the regularized trace of the generator.     

Functional inequalities on abstract Hilbert spaces and applications

We study the essential spectrum and the semigroup property for self-adjoint operators on abstract Hilbert spaces by using functional inequalities. Some known results obtained on the L ² -space w.r.t. a measure space are generalized. The functional inequality is also used to study non-symmetric semigroups. Mathematics Subject Classification (2000): 49R20, 58F19.Research supported in part by NNSFC (10121101, 10025105), TRAPOYT and the 973-Project.     

The nisio semigroup for controlled diffusions with partial observations

A one parameter semigroup of nonlinear operators on an appropriate Banach space is constructed in the spirit of Nisio for controlled diffusions with partial observations. The method is based upon considering an equivalent problem of controlling a measure-valued process representing the conditional law of the state given past observations. The latter evolves according to the usual equations of nonlinear filtering. By considering an appropriate augmentation of the class of controls, it is shown that the “minimum cost” operators associated with this control problem indeed form a semigroup of nonlinear contractions on the space of bounded continuous real-valued functions on the state space of the above measure-valued process.     

On Semitransitive Collections of Operators

A collection F of operators on a vector space V is said to be semitransitive if for every pair of nonzero vectors x and y in V there exists a member T of F such that either Tx = y or Ty = x (or both). We study semitransitive algebras and semigroups of operators. One of the main results is that if the underlying field is algebraically closed, then every semitransitive algebra of operators on a space of dimension n contains a nilpotent element of index n. Among other results on semitransitive semigroups, we show that if the rank of nonzero members of such a semigroup acting on an n-dimensional space is a constant k, then k divides n.     

<Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Mappings. Criterion of Irreducibility

We study the operator algebra associated with a self-mapping ? on a countable set X which can be represented as a directed graph. The algebra is generated by the family of partial isometries acting on the corresponding l²(X). We study the structure of involutive semigroup multiplicatively generated by the family of partial isometries. We formulate the criterion when the algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras. In particular, we give the examples of nonisomorphic C*-algebras, which are the extensions by compact operators of the algebra of continuous functions on the unit circle.     

Local Lower Bounds on Heat Kernels

We analyze convolution semigroups on a regular measure space which satisfies the local doubling property. We assume the kernels are bounded and symmetric with the characteristic small-time, volume-dependent, singularity. Then, using a weak conservation property, we deduce local lower bounds with a comparable singularity.Applications are given to a wide range of subelliptic and strongly elliptic self-adjoint, or near self-adjoint, operators on Lie groups.     

Continuous state branching semigroups

Summary This paper is concerned with Markov processes with continuous creation where the phase space is a general separable compact metric space. The transition probabilities for such a process determine a semigroup of operators acting on a function space over the collection of bounded Borel measures on the phase space. Such a semigroup is characterized by a particular convolution condition and is called a continuous state branching semigroup. A connection is established between continuous state branching semigroups and certain semigroups of nonlinear operators and then this connection is exploited to establish an existence theorem for the former.Research associated with a project in probability at Princeton University supported by the Office of Army Research.     

Existence and nonexistence of hypercyclic semigroups

In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from--and considerably shorter than--the one recently given by Bermúdez, Bonilla and Martinón. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Fréchet space. This complements recent results due to Bès and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.

     

Regularization for Ill-posed Cauchy problems associated with generators of analytic semigroups

This paper is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. Our main result is that if −A is the generator of an analytic semigroup, then there exists a family of regularizing operators for such an ill-posed Cauchy problem by using the quasi-reversibility method, fractional powers and semigroups of linear operators. The applications to ill-posed partial differential equations are also given.     

A Pettis-type integral and applications to transition semigroups

Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we give conditions ensuring that a semigroup consisting of kernel operators has a Laplace transform which also consists of kernel operators. We also provide conditions under which a semigroup is uniquely determined by its Laplace transform.     

Continuity properties of Markov semigroups and their restrictions to invariant <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-spaces

We consider Markov semigroups on the cone of positive finite measures on a complete separable metric space. Such a semigroup extends to a semigroup of linear operators on the vector space of measures that typically fails to be strongly continuous for the total variation norm. First we characterise when the restriction of a Markov semigroup to an invariant L ¹-space is strongly continuous. Aided by this result we provide several characterisations of the subspace of strong continuity for the total variation norm. We prove that this subspace is a projection band in the Banach lattice of finite measures, and consequently obtain a direct sum decomposition.     

Rotenberg模型中迁移半群的本质谱

在Lp（1（≤）p＜＋∞）空间中,本文运用半群理论研究了Rotenberg模型中具光滑边界条件的迁移半群的本质谱.采用半群方法和比较算子等方法,证明了对任意的t＞0,s＞0,算子[UH（t）-U0（t）]U0（s）[UH（t）-U0（t）]在Lp（1＜p＜＋∞）在空间中紧和在L1空间弱紧,得到迁移半群VH（t）与V0（t）有相同的本质谱型.     

Non-Differentiable Skew Convolution Semigroups and Related Ornstein–Uhlenbeck Processes

It is proved that a general non-differentiable skew convolution semigroup associated with a strongly continuous semigroup of linear operators on a real separable Hilbert space can be extended to a differentiable one on the entrance space of the linear semigroup. A càdlàg strong Markov process on an enlargement of the entrance space is constructed from which we obtain a realization of the corresponding Ornstein–Uhlenbeck process. Some explicit characterizations of the entrance spaces for special linear semigroups are given.     

Partially isometric dilations of noncommuting -tuples of operators

Given a row contraction of operators on a Hilbert space and a family of projections on the space that stabilizes the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries that satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold decomposition for partial isometries to describe the models for these dilations, and we discuss how the basic properties of a dilation depend on the row contraction.

     

Product formulas,nonlinear semigroups,and accretive operators

A special case of our main theorem, when combined with a known result of Brezis and Pazy, shows that in reflexive Banach spaces with a uniformly Gâteaux differentiable norm, resolvent consistency is equivalent to convergence for nonlinear contractive algorithms. (The linear case is due to Chernoff.) The proof uses ideas of Crandall, Liggett, and Baillon. Other applications of our theorem include results concerning the generation of nonlinear semigroups (e.g., a nonlinear Hille-Yosida theorem for “nice” Banach spaces that includes the familiar Hilbert space result), the geometry of Banach spaces, extensions of accretive operators, invariance criteria, and the asymptotic behavior of nonlinear semigroups and resolvents. The equivalence between resolvent consistency and convergence for nonlinear contractive algorithms seems to be new even in Hilbert space. Our nonlinear Hille-Yosida theorem is the first of its kind outside Hilbert space. It establishes a biunique correspondence between m-accretive operators and semigroups on nonexpansive retracts of “nice” Banach spaces and provides affirmative answers to two questions of Kato.     

An analogue of Hunt's representation theorem in quantum probability

In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of, instruments on groups and the associated semigroups of probability operators, which now are defined on spaces of functions with values in a von Neumann algebra. We consider a semigroup of probability operators with a continuity property weaker than uniform continuity, and we succeed in characterizing its infinitesimal generator under the additional hypothesis that twice differentiable functions belong to the domain of the generator. Such hypothesis can be proved in some particular cases. In this way a partial quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained. Our result provides also a closed characterization of generators of a new class of not norm continuous quantum dynamical semigroups.     

Spectra and semigroup smoothing for non-elliptic quadratic operators

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.