Unitary Semigroup Research Articles

The space of contractive C0-semigroups is a Baire space

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive C0-semigroups under the topology of uniform weak operator convergence on compact subsets of R+. Eisner and Serény raised in [6] and [3] the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem positively by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces. The transfer result in turn relies upon classification of topological properties via infinite games. Our approach is sufficiently general and can be applied to other contexts, e.g. the space of contractions under the ▪-topology.

A new approach to a network of congruences on an inverse semigroup

This paper enriches the list of known properties of congruence sequences starting from the universal relation and successively performing the operators lower $k$ and lower $t$. Two series of inverse semigroups, namely $\ker{\alpha_n}$-is-Clifford semigroups and $\beta_n$-is-over-$E$-unitary semigroups, are investigated. Two congruences, namely $\alpha_{n+2}$ and $\beta_{n+2}$, are found to be the least $\ker{\alpha_n}$-is-Clifford and least $\beta_n$-is-over-$E$-unitary congruences on $S$, respectively. A new system of implications is established for the quasivarieties of inverse semigroups induced by the min network.

Complex symmetric C0-semigroups on the Fock space

In this paper we introduce a new concept in dynamical systems: the complex symmetric operator C0-semigroups (groups) acting on Hilbert space. Using techniques of weighted composition operators we study these semigroups on the Fock space F2 and show, in particular, that complex symmetric operator C0-semigroups contain the familiar unitary semigroups as a proper subclass.

Large Aperiodic Semigroups

We search for the largest syntactic semigroups of star-free languages having n left quotients; equivalently, we look for the largest transition semigroups of aperiodic finite automata with n states. We first introduce unitary semigroups generated by transformations that change only one state. In particular, we study unitary-complete semigroups which have a special structure, and show that each maximal unitary semigroup is unitary-complete. For [Formula: see text] we exhibit a unitary-complete semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. We examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups.

Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator

We consider the problem of recovering the initial data (or initial state) of infinite-dimensional linear systems with unitary semigroups. It is well-known that this inverse problem is well posed if the system is exactly observable, but this assumption may be very restrictive in some applications. In this paper we are interested in systems which are not exactly observable, and in particular, where we cannot expect a full reconstruction. We propose to use the algorithm studied by Ramdani et al. in (Automatica 46:1616–1625, 2010) and prove that it always converges towards the observable part of the initial state. We give necessary and sufficient condition to have an exponential rate of convergence. Numerical simulations are presented to illustrate the theoretical results.

Models in Boundary Quantum Field Theory Associated with Lattices and Loop Group Models

In this article we give new examples of models in boundary quantum field theory, i.e. local time-translation covariant nets of von Neumann algebras, using a recent construction of Longo and Witten, which uses a local conformal net A on the real line together with an element of a unitary semigroup associated with A. Namely, we compute elements of this semigroup coming from H\"older continuous symmetric inner functions for a family of (completely rational) conformal nets which can be obtained by starting with nets of real subspaces, passing to its second quantization nets and taking local extensions of the former. This family is precisely the family of conformal nets associated with lattices, which as we show contains as a special case the level 1 loop group nets of simply connected, simply laced groups. Further examples come from the loop group net of Spin(n) at level 2 using the orbifold construction.

Boundary Quantum Field Theory on the Interior of the Lorentz Hyperboloid

We construct local, boost covariant boundary QFT nets of von Neumann algebras on the interior of the Lorentz hyperboloid LH = {x^2 - t^2 > R^2, x>0}, in the two-dimensional Minkowski spacetime. Our first construction is canonical, starting with a local conformal net on the real line, and is analogous to our previous construction of local boundary CFT nets on the Minkowski half-space. This net is in a thermal state at Hawking temperature. Then, inspired by a recent construction by E. Witten and one of us, we consider a unitary semigroup that we use to build up infinitely many nets. Surprisingly, the one-particle semigroup is again isomorphic to the semigroup of symmetric inner functions of the disk. In particular, by considering the U(1)-current net, we can associate with any given symmetric inner function a local, boundary QFT net on LH. By considering different states, we shall also have nets in a ground state, rather than in a KMS state.

Tiling Semigroups of n-Dimensional Hypercubic Tilings

In this work, we generalize to hypercubic tilings of dimension n the description of tiling semigroups as inverse semigroups associated to factorial languages and the representation of this semigroup as a P*-semigroup. In addition, we show that, in contrast with the one-dimensional case, the tiling semigroup of any n-dimensional hypercubic tiling is always infinitely presented (even as a strongly E*-unitary inverse semigroup) and give a necessary and sufficient condition for two hypercubic tiling semigroups to be isomorphic.

An Algebraic Construction of Boundary Quantum Field Theory

We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras A_V on the Minkowski half-plane M_+ starting with a local conformal net A of von Neumann algebras on the real line and an element V of a unitary semigroup E(A) associated with A. The case V=1 reduces to the net A_+ considered by Rehren and one of the authors; if the vacuum character of A is summable A_V is locally isomorphic to A_+. We discuss the structure of the semigroup E(A). By using a one-particle version of Borchers theorem and standard subspace analysis, we provide an abstract analog of the Beurling-Lax theorem that allows us to describe, in particular, all unitaries on the one-particle Hilbert space whose second quantization promotion belongs to of E(A^(0)) with A^(0) the U(1)-current net. Each such unitary is attached to a scattering function or, more generally, to a symmetric inner function. We then obtain families of models via any Buchholz-Mach-Todorov extension of A^(0). A further family of models comes from the Ising model.

Brownian motion on Lie groups and open quantum systems

We study the twirling semigroups of (super) operators, namely certain quantum dynamical semigroups that are associated, in a natural way, with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group. The link connecting this class of semigroups of operators with (classical) Brownian motion is clarified. It turns out that every twirling semigroup associated with a finite-dimensional representation is a random unitary semigroup, and, conversely, every random unitary semigroup arises as a twirling semigroup. Using standard tools of the theory of convolution semigroups of measures and of convex analysis, we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite-dimensional unitary representation of a Lie group.

A nonlinear Schrödinger equation with two symmetric point interactions in one dimension

We consider a time-dependent one-dimensional nonlinear Schrödinger equation with a symmetric double-well potential represented by two Dirac's δ. Among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the originalnonlinear problem.

Unitary Numerical Semigroups and Perfect Bricks

This article completes a previous investigation of balanced and unitary numerical semigroups. The main result establishes the equivalence of unitary numerical semigroups and perfect 2 × 2 bricks.

Construction of solution for evolution equation x ttt Cx=f

The construction of solution for three-order evolution equation xttt=A3 is skillfully obtained and the semigroup of equation operator is theoretically proved, then the solution for three-order evolution equation xtttCx=f is constructed from the appropriate transformation, and the necessary and sufficient conditions of its unitary semigroup are, presented.

Rotating Singular Perturbations of the Laplacian

We study a system of a quantum particle interacting with a singular time-dependent uniformly rotating potential in 2 and 3 dimensions: in particular we consider an interaction with support on a point (rotating point interaction) and on a set of codimension 1 (rotating blade). We prove the existence of the Hamiltonians of such systems as suitable self-adjoint operators and we give an explicit expression for their unitary semigroups. Moreover we analyze the asymptotic limit of large angular velocity and we prove strong convergence of the time-dependent propagator to some one-parameter unitary group as (\omega \to \infty).

Entanglement in probability representation of quantum states and tomographic criterion of separability

Entangled and separable states of a bipartite (multipartite) system are studied in the tomographic representation of quantum states. Properties of tomograms (joint probability distributions) corresponding to entangled states are discussed. The connection with star-product quantization is presented. U(N)-tomography and spin tomography as well as the relation of the tomograms to positive and completely positive maps are considered. The tomographic criterion of separability (necessary and sufficient condition) is formulated in terms of the equality of the specific function depending on unitary group parameters and positive map semigroup parameters to unity. Generalized Werner states are used as an example.

Optimal control of systems with a unitary semigroup and with colocated control and observation

We solve the quadratic optimal control problem on an infinite time interval for a class of linear systems whose state space is a Hilbert space and whose operator semigroup is unitary. The difficulty is that the systems in this class, having unbounded control and observation operators, may be ill-posed. We show that there is a surprisingly simple solution to the problem (the optimal feedback turns out to be output feedback). Our approach is to use a change of variables which transforms the system into a one which, according to recent research, is known to be conservative. We show that, under a mild assumption, the transfer function of this conservative system is inner, and then it follows that the optimal control of this conservative system is trivial. We give an example with the wave equation on an n-dimensional domain, with Neumann control and Dirichlet observation of the velocity.

The structure of 0-bisimple strongly E *-unitary inverse monoids

We introduce a class of strongly E*-unitary inverse semigroups Si(G, P) (i = 1,2) determined by a group G and a submonoid P of G and give an embedding theorem for Si(G, P). Moreover we characterize 0-bisimple strongly E*-unitary inverse monoids and 0-bisimple strongly F*-inverse monoids by using Si(G, P).

Some notes on strongly E*-unitary inverse semigroups

In this paper, we give a direct proof that every strongly \(\user1{E}^ * --\user1{unitary}\) inverse semigroup can be embedded into a 0-semidirect product of a semilattice with zero by a group. As a corollary, we obtain a new proof of the structure theory of strongly \(\user1{E}^ * --\user1{unitary}\) inverse semigroups described in [1]. We also prove that the strongly \(\user1{E}^ * --\user1{unitary}\) inverse semigroups are precisely \(\user1{E}^ * -\user1{unitary}\) inverse semigroups equipped with a \(\user1{0 - restricted}\), idempotent pure prehomomorphism to a primitive inverse semigroup.

Inverse semigroups with zero: covers and their structure

AbstractWe obtain analogues, in the setting of semigroups with zero, of McAlister's convering theoren and the structure theorems of McAlister, O'Carroll, and Margolis and Pin. The covers come from a classCof semigroups defined by modifying one of the many characterisations ofE-unitary inverse semigroups, namely, that an inverse semigroups isE-unitary if and only if it is an inverse image of an idempotent-pure homomorphism onto a group. The classCis properly contained in the class of allE*-unitary inverse semigroups introduced by Szendrei but properly contains the class of strongly categoricalE*-unitary semigroups recently considered by Gomes and Howie.

OnP-Q ordered semigroups

In this paper,P-ordered andQ-ordered semigroups are studied. Some characterizations and properties of such semigroups are obtalned. Also the relationship between maximal (minimum) regular ordered semigroups and unitary regular semigroups is investigated.