Showing papers on "Shift operator published in 1992"
TL;DR: The notion of projectivity for operator spaces has been introduced independently by Paulsen and Ruan as discussed by the authors, and its significance in the theory of tensor products has already been partially explored by the aforementioned.
Abstract: The notion of a dual of an operator space which is again an operator space has been introduced independently by Vern Paulsen and the author, and by Effros and Ruan. Its significance in the theory of tensor products of operator spaces has already been partially explored by the aforementioned. Here we establish some other fundamental properties of this dual construction, and examine how it interacts with other natural categorical constructs for operator spaces. We define and study a notion of projectivity for operator spaces, and give a noncommutative version of Grothendieck' s characterization of I1 (I) spaces for a discrete set /.
179 citations
TL;DR: In this article, it was shown that certain interpolation theorems for non-commutative symmetric operator spaces can be deduced from their commutative versions from their Schmidt decomposition.
Abstract: It is shown that certain interpolation theorems for non-commutative symmetric operator spaces can be deduced from their commutative versions. A principal tool is a refinement of the notion of Schmidt decomposition of a measurable operator affiliated with a given semi-finite von Neumann algebra.
150 citations
TL;DR: In this paper, it was shown that the Fokker-planck operator describing a highly peaked scattering process in the linear transport equation is a formal asymptotic limit of the exact integral operator.
Abstract: It is shown that the Fokker-Planck operator describing a highly peaked scattering process in the linear transport equation is a formal asymptotic limit of the exact integral operator. It is also shown that such peaking is a necessary, but not sufficient, condition for the Fokker-Planck operator to be a legitimate description of such scattering. In particular, the widely used Henyey-Greenstein scattering kernel does not possess a Fokker-Planck limit.
142 citations
TL;DR: The problem is to evaluate the complexity of solving the equation to a given accuracy, i.e., given a class 2l of instances, to point out the best possible upper bound of the number of calls of the oracle sufficient to find an e-solution to each of the instances.
104 citations
TL;DR: In this article, a size-extensive formulation for an intermediate Hamiltonian H int, furnishing sizeextensive energies for the main roots, is presented, and the working model space, comprised of the main and intermediate space, is taken as complete.
84 citations
TL;DR: In this article, the algebra of q-fermion operators is re-examined and generalized q-oscillators defined for - infinity (q
Abstract: The algebra of q-fermion operators, developed earlier by two of the present authors is re-examined. It is shown that these operators represent particles that are distinct from usual spacetime fermions except in the limit q=1. It is shown that it is possible to introduce generalized q-oscillators defined for - infinity (q
53 citations
50 citations
TL;DR: In this paper, a linear rate equation in the Banach space of summable sequences, l1, is constructed based on the properties of the backward shift operator on l1 and the relationship between the linear backward shift and the nonlinear Bernoulli shift is discussed.
Abstract: We construct an example of linear rate equation in the Banach space of summable sequences, l1, that exhibits the three properties required as signature of topological chaos, namely: (i) topological transitivity, (ii) dense periodic orbits, and (iii) positive Lyapunov exponents. The example is based on the properties of the backward shift operator on the Banach space l1. Since linear chaos in the sense described above can occur only in an infinite-dimensional setting, possible finite-dimensional approximate manifestations are investigated. The relationship between the linear backward shift and the nonlinear Bernoulli shift is also discussed.
44 citations
TL;DR: It is shown that peculiarities associated with the descriptions of phase in conventional infinite Hilbert space arise from the nature of the limiting process, and that these peculiarities do not arise when the Hermitian optical phase operator is employed.
Abstract: We examine some of the attempts to describe the phase of a single field mode by a quantum operator acting in the conventional infinite Hilbert space. These operators lead to bizarre properties such as non-random phases for the number states and experience consistency difficulties when used to obtain a phase probability density. Moreover, in these approaches operator functions of phase are not simply functions of a phase operator. We show that these peculiarities do not arise when the Hermitian optical phase operator is employed. In our opinion, the problems associated with the descriptions of phase in conventional infinite Hilbert space arise from the nature of the limiting process.
41 citations
TL;DR: In this paper, a general approach to the construction of Rayleigh-Schrodinger-like perturbative expansions for Hermitian intermediate Hamiltonians using the shift technique is proposed.
Abstract: A general approach to the construction of Rayleigh-Schrodinger-like perturbative expansions for Hermitian intermediate Hamiltonians using the shift technique is proposed. The shift operator depends on the projector onto the subspace spanned by the 'main' eigenvectors of the intermediate Hamiltonian which should be determined iteratively for a given PT order. The intruder state problem can be completely eliminated by an appropriate choice of an intermediate space and shift parameters without introducing any additional perturbation. Numerical examples are reported in order to demonstrate the convergency properties and approximate size consistency of the novel method.
37 citations
TL;DR: In this article, the spectrum σ( A ) of a linear operator A can be changed upon perturbation of A with a one-dimensional operator K. In particular, for normal, self-adjoint, and unitary operators.
TL;DR: In this article, a self-adjoint time operator for massless relativistic systems in terms of the generators of the Poincare group was constructed, and the transformation properties of the internal time and position operator under Lorentz boosts were investigated.
Abstract: We construct a self-adjoint time operator for massless relativistic systems in terms of the generators of the Poincare group The Lie algebra generated by the time operator and the generators of the Poincare group turns out to be an infinitedimensional extension of the Poincare algebra The internal time operator generates two new entities, namely the velocity operator and the internal position operator The transformation properties of the internal time and position operator under Lorentz boosts are different from what one would expect from relativity theory This difference reflects the fact that the time concept associated with the internal time operator is radically different from the time coordinate of Minkowski space, due to the nonlocality of the time operator The spectral projections of the time operator allow us to construct incoming subspaces for the wave equation without invoking Huygens' principle, as in two and one spatial dimensions where Huygens' principle does not hold
TL;DR: In this article, the authors prove that the period doubling operator has an expanding direction at the fixed point and use a sequence of linear operators with finite ranks to study this induced operator.
Abstract: We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a “Perron-Frobenius type operator,” to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to study this induced operator. The proof is constructive. One can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point.
TL;DR: In this article, the authors derived a relation between the anomalous dimensions of the composite operators and the unintegrable part of the operator product coefficients by imposing consistency conditions, and then gave a formula for the derivatives of a correlation function of composite operators with respect to the parameters (i.e., the strong fine structure constant and the quark mass) of QCD in four-dimensional euclidean space.
TL;DR: In this article, a non-canonical Hermitian operator corresponding to phase angle, based on the Weyl correspondence rule, has been proposed and the matrix elements of this operator coincide, in the correspondence limit, with those of a phase operator proposed by Barnett and Pegg, when the dimension of their defining space becomes infinite.
Abstract: We suggest a candidate for a non-canonical Hermitian operator corresponding to phase angle, based on the Weyl correspondence rule. The matrix elements of this operator, for harmonic oscillator number states, coincide, in the correspondence limit, with those of a phase operator proposed by Barnett and Pegg, when the dimension of their defining space becomes infinite.
Journal Article•
TL;DR: In this article, the authors studied the negative definiteness of an operator-valued Hermitian form defined on a space of operators and investigated the biholomorphic linear fractional transformations between them.
Abstract: 0. Introduction Our object is to study domains which are the region of negative definiteness of an operator-valued Hermitian form defined on a space of operators and to investigate the biholomorphic linear fractional transformations between them. This is a unified setting in which to consider operator balls, operator half-planes, strictly J-contractive operators, strictly J-dissipative operators, etc., and the biholomorphic images of these domains under linear fractional transformations. Our approach is close in spirit to that of Potapov [28], Krein and Smuljan [27] and Smuljan [33]. At the same time, because we consider subspaces of operators, our circular domains include the matrix balls which E. Cartan [6] obtained as the classical bounded symmetric domains and they include the Siegel domains of genus 2 and 3 which Pyatetskii-Shapiro [29] associates with these domains as well as the infinite
TL;DR: The differential operator associated with the prolate spheroidal functions commutes with a certain integral operator as discussed by the authors, whose kernel is the reproducing kernel of the Paley-Wiener space restricted to a bounded interval.
Abstract: The differential operator associated with the prolate spheroidal functions commutes with a certain integral operator. Its kernel is the reproducing kernel of the Paley-Wiener space restricted to a bounded interval. This “lucky accident” occurs because both the differential operator and its Fourier transform commute with multiplication by the characteristic function of an interval. It is the only operator of its type to do so.
01 Jan 1992
TL;DR: In this article, the relative operator entropy is introduced by generalizing the Kubo-Ando theory of operator means, which does not include the logarithm and the entropy function which are operator monotone and often used in information theory.
Abstract: The notion of operator monotone functions was introduced by Lowner and that of operator concave functions by Kraus who is his student. Operator means were introduced by Ando and the general theory of them was established by Kubo and Ando himself. By their theory, a nonnegative operator monotone function is now considered as a variation of an operator mean. However this theory does not include the logarithm and the entropy function which are operator monotone and often used in information theory. These functions are operator concave and satisfy Jensen’s inequality. So, considering operator means from the historical viewpoint, we shall introduce the relative operator entropy by generalizing the Kubo-Ando theory. Though its definition is derived from the Kubo-Ando theory of operator means, it can be constructed also in some ways. The relative operator entropy has of course some entropy-like properties.
TL;DR: In this article, the energy operator for particles obeying infinite statistics defined by aq-deformation of the Heisenberg algebra was constructed for a class of particles. But this operator is not applicable to all particles.
Abstract: We construct the energy operator for particles obeying infinite statistics defined by aq-deformation of the Heisenberg algebra.
24 Aug 1992
TL;DR: A one-state linear operator algorithm (OLOA) with modulus m, denned in the paper, operates with one non-negative integer x in the following manner.
Abstract: A one-state linear operator algorithm (OLOA) with modulus m, denned in the paper, operates with one non-negative integer x in the following manner. According to the value r=x MOD m either the computation is halted, or x is replaced by (ax+b) DIV c, where a,b,c are constants dependent only on the r, and the operation is repeated with the new value gained. The notion of a universal OLOA is defined, and a universal OLOA with modulus 396 is constructed in the paper.
TL;DR: In the theory of linear operators in a Hilbert space, an important role is played by the so-called operator-value-R-functions as discussed by the authors, i.e., functions T(1), analytic in the upper half-plane iC+, whose values are bounded operators, acting in a Euclidean space H and having positive imaginary part (Im T(%) ~ 0, Im% > 0).
Abstract: In the theory of linear operators in a Hilbert space an important role is played by the so-called operator-value~ R-functions. By the latter we mean [i] functions T(1), analytic in the upper half-plane iC+, whose values are bounded operators, acting in a Hilbert space H and having positive imaginary part (Im T(%) ~ 0, Im% > 0). We elucidate briefly the manner in which this object arises in the theory of perturbations of self-adjoint (and also nonself-adjoint) operators. Namely, assume that in a Hilbert space H there are given two self-adjoint operators A, A + V, where V ~ 0 is a bounded operator. Then from Hilbert's identity we obtain the following representation for the "bordered" resolvent of the "perturbed" operator A + V:
TL;DR: In this article, the Lagrange inversion formula has been used to define higher-order derivative identities and identities among Bell polynomials, which specialize to a large variety of interesting identities among binomial coefficients and classical orthogonal polynomial coefficients.
Abstract: A natural differential operator series is one that commutes with every function. The only linear examples are the formal series operators $e^{\alpha zD} $ representing translations. This paper discusses a surprising natural nonlinear “normally ordered” differential operator series, arising from the Lagrange inversion formula. The operator provides a wide range of new higher-order derivative identities and identities among Bell polynomials. These identities specialize to a large variety of interesting identities among binomial coefficients and classical orthogonal polynomials, a number of which are new.
TL;DR: In this paper, the authors constructed the scattering operator for a spinor field in a time dependent background by the Dyson expansion and showed that the restriction of the operator to the positive spectral subspace (with respect to a reference Hamiltonian) is Fredholm.
Abstract: We construct the scattering operator for a spinor field in a time dependent background by the Dyson expansion. Then we show that the restriction of the scattering operator to the positive spectral subspace (with respect to a reference Hamiltonian) is Fredholm. The computation of the index of this restriction is reduced to the index computation for an elliptic pseudodifferential operator of order zero. We obtain the index in terms of a cohomological formula by means of the Atiyah-Singer index theorem.
TL;DR: The zeros of predictor polynomials are shown to belong to the numerical range of a shift operator associated with the particular prediction problem under consideration.
Abstract: The zeros of predictor polynomials are shown to belong to the numerical range of a shift operator associated with the particular prediction problem under consideration. The numerical range consists of the classical field of values of the shift operator when the setting is Hilbert space, but a new definition is necessary when the setting is a general normed space. It is shown that a predictor polynomial is not stable in general. Nevertheless, for predictor polynomials in l/sub p/ spaces, it is shown that their zeros belong to the open circular disk with radius 2. >
TL;DR: In this article, it was shown that for non-semi-Fredholm operators, given such an operator, we can find a compact operator with arbitrarily small norm such that the perturbed operator has infinite dimensional kernel and cokernel.
Abstract: A classical result of Fredholm theory states that perturbing a semi-Fredholm operator T, (that is, an operator with closed range R(T) and finite dimensional kernel N(T) or cokernel Y/R(T)) by an operator A with norm ||A|| < j(T), the minimum modulus of T, or A compact, we obtain a semi-Fredholm operator T + A with dimN{T + A) < dimN(T) and dimY/R(T + A) ^ dimY/R(T). Hence, for semiFredholm operators the finite dimension of the kernel or the cokernel is stable under small or compact perturbations. However, for non-semi-Fredholm operators, we have the opposite case: Given such an operator we can find a compact operator with arbitrarily small norm such that the perturbed operator has infinite dimensional kernel and cokernel [6], (see also [7]).
TL;DR: In this paper, an application of a certain integral operator for analytic functions in the unit disk is presented, where the integral operator is applied to a set of analytic functions, and the object of the present paper is to show the application of this integral operator to analytic functions.
TL;DR: In this paper, a unified theory for the determination of McMillan degree and Kronecker observability indices of systems represented by polynomial matrices in the two shift operators, considering the MFD and ARMA forms as special cases.
Abstract: Linear time-invariant, finite-dimensional discrete time systems are very often specified in a polynomial form representation in either a forward or backward shift operator (sometimes called MFD and ARMA form). In this paper it is shown that there exists a unifying theory for the determination of McMillan degree and Kronecker observability indices of systems represented by polynomial matrices in the two shift operators, considering the MFD and ARMA forms as special cases. Treating dynamical systems in terms of their behaviour, i.e. the set of admissible signal trajectories, the notions of McMillan degree and observability indices can be generalized to non-controllable as well as to non-causal systems.