A limit theorem for products of free unitary operators

In this paper we study Weyl’s theorem and von Neumann’s converse of Weyl’s theorem for the classes of all operators of the form T f − 1 T g T_f^{ - 1}{T_g} and of the form T g T f − 1 {T_g}T_f^{ - 1} , where T g {T_g} and T f {T_f} are Toeplilz operators such that T f {T_f} is invertible; and we can prove that Weyl’s theorem holds for T f − 1 T g T_f^{ - 1}{T_g} and for T g T f − 1 {T_g}T_f^{ - 1} .

On the basis of a functional model one considers the scattering for operators that are close to unitary. In particular cases the presented scheme contains a series of results of L. de Branges and L. Shulman, S. N. Naboko, L. A. Sakhnovich, and H. Neidhardt. The fundamental new result is: the existence of complete local wave operators for a unitary operator and its nuclear perturbation, where the spectrum of the latter does not fill out the unit circle. In this same situation an invariance principle is obtained.

We present a general method which expresses a unitary operator by the product of operators allowed by the Hamiltonian of spin-1/2 systems. In this method, the generator of an operator is found first, and then the generator is expanded by the base operators of the product operator formalism. Finally, the base operators disallowed by the Hamiltonian, including more than two-body interaction operators, are replaced by allowed ones by the axes transformation and coupling order reduction technique. This method directly provides pulse sequences for the nuclear magnetic resonance quantum computer, and can be generally applied to other systems.

Let V be the C∗-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i1, . . . , im) with i1, . . . , im ∈ {1, . . . , k}, define a product of A1, . . . , Ak ∈ V by A1 ∗ · · · ∗ Ak = Ai1 . . . Aim . This includes the usual product A1 ∗ · · · ∗ Ak = A1 · · ·Ak and the Jordan triple product A ∗ B = ABA as special cases. Denote the numerical range of A ∈ V by W (A) = {(Ax, x) : x ∈ H, (x, x) = 1}. If there is a unitary operator U and a scalar μ satisfying μ = 1 such that φ : V→ V has the form A 7→ μU∗AU or A 7→ μU∗AtU, then φ is surjective and satisfies W (A1 ∗ · · · ∗ Ak) =W (φ(A1) ∗ · · · ∗ φ(Ak)) for all A1, . . . , Ak ∈ V. It is shown that the converse is true under the assumption that one of the terms in (i1, . . . , im) is different from all other terms. In the finite dimensional case, the converse can be proved without the surjective assumption on φ. An example is given to show that the assumption on (i1, . . . , im) is necessary. 2000 Mathematics Subject Classification. 47A12, 47B15, 47B49, 15A60, 15A04, 15A18

This paper deals with periodic pseudodifferential operators on the unit circle \mathbb T^1 = \mathbb R^1/ \mathbb Z^1 . The main results are asymptotic expansions for adjoints and products of periodic pseudodifferential operators.

The location of the zeros of discrete systems characteristic polynomials with respect to the unit circle is investigated. A new sequence of symmetric polynomials of descending degrees is defined for the characteristic polynomial and the number of zeros inside and outside the unit circle is shown to be related to a certain sign variation pattern of the polynomials in the sequence. A stability table based on this sequence is presented to obtain the sought distribution of zeros. The new table is close in appearance, size, and number of arithmetic operations to the Routh table used for continuous-time system polynomials. By comparison with the table of Jury, based on the theory of Marden, Cohn, and Schur, the new table involves about half the number of entries and a corresponding significant saving in computations. The study includes a detailed consideration of all possible cases of singular conditions so that the complete information on the number of zeros inside, outside, and on the unit circle is always obtained (including some additional information on possible reciprocal pairs of zeros). Necessary and sufficient conditions for a polynomial to have all its zeros inside the unit circle are obtained as a special outcome. Other additional necessary conditions, that are useful to shorten procedures when the table is used only for testing the stability of the system, are also given.

On the Polynomials with All Their Zeros on the Unit Circle

A constantK 0 (m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK 0(h) (indicated by the change in module of the quadrilaterals with vertices on the circle) as a special case. A necessary and sufficient condition is established forK 0 (m) (h) =K 1(h). It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property thatK 0 (m) (h)<K 1(h), wherek 1(h) is the maximal dilatation ofh.

In this paper, we study the computation of the moments associated to rational weight functions given as a power spectrum with known or unknown poles of any order in the interior of the unit disc. A recursive algebraic procedure is derived that computes the moments in a finite number of steps. We also study the associated interpolatory quadrature formulas with equidistant nodes on the unit circle. Explicit expressions are given for the positive quadrature weights in the case of a polynomial weight function. For rational weight functions with simple poles, mostly real or uniformly distributed on a circle in the open unit disc, we also obtain expressions for the quadrature weights and sufficient conditions that guarantee that they are positive. The Poisson kernel is a simple example of a rational weight function, and in the last section, we derive an asymptotic expansion of the quadrature error.

It is known that entangling capacities of bipartite unitary operations may be used to place upper bounds on communication capacities. Here we present alternative definitions of entangling capacities for unitary operations and use them to place lower bounds on the communication capacities of arbitrary bipartite unitaries. Thus entangling capacities may be used to place both upper and lower bounds on the communication capacities for arbitrary bipartite unitaries, allowing them to be more easily estimated. We elucidate the connection between these results and the standard formula for the entanglement assisted capacity of channels. In addition, we present more general methods for producing ensembles for communication from initial states for entanglement creation, and thereby derive sufficient conditions such that the usual forms of entangling capacity provide lower bounds on communication capacities.

In this paper we discuss about the image of Gabor frame under a unitary operator and derive a sufficient condition under which a unitary operator from $L^2 (\mathbb R)$ to $l^2 (\mathbb Z)$ maps Gabor frame in $L^2 (\mathbb R)$ to a Gabor frame in $l^2 (\mathbb Z)$.

A sufficiency condition for a periodic process with free period to be a weak local minimum is given. By making detailed use of the properties of the monodromy matrix, the transition matrix calculated over a closed orbit, previous results are corrected, extended, and clarified. A necessary condition for a real valued periodic Riccati matrix to exist is that there be no distinct eigenvalues of the monodromy matrix on the unit circle. For local sufficiency, it is required that no eigenvalues of the monodromy matrix, except a double root at unity, lie on the unit circle.

Let be a unitary operator on a Hubert space H. Then in particular, (i) is a contraction, i.e. || T 1; and (ii) The spectrum of is a subset of the unit circle, i.e. Sp (T)cC, where C denotes the set of complex numbers of absolute value one. Call an arbitrary operator a unimodular contraction if it satisfies conditions (i) and (ii) above. Then several questions immediately come to mind. Do there exist nonunitary unimodular contractions? If so, what is the nature of their spectra, e.g. what subsets of the unit circle arise as spectra of nonunitary unimodular contractions; when does the spectrum contain point, residual, or continuous spectrum? Under what conditions is a unimodular contraction unitary? What is the nature of operator algebras containing nonunitary unimodular contractions? In this paper examples are given of nonunitary unimodular contractions. It is shown (Theorem 2) that such exist with arbitrarily prescribed spectrum, which however can contain no residual spectrum. It is also shown (Theorem 1) that nonunitary unimodular contractions exist only in infinite von Neumann algebras. This result is applied to a mapping problem of operator algebras.

In a recent paper by Jiang (1988), a necessary and sufficient condition is given for a set of real interval matrices to have all their eigenvalues in the unit circle. This result is shown by counter-example to be false and the flaw in the proof is pointed out.

We study two harmonic oscillators with a kinetic coupling system. By taking a unitary transformation approach, we turn the system into the Fock space of two independent harmonic oscillators and derive the density matrix for it. The corresponding unitary operator is characteristic of including frequency-jump squeezing transformation. By virtue of the technique of the integration within an ordered product of operators, we manifestly show that the ground state of the system is a squeezed state.