Hubert Space Research Articles

Quantum Like Measurements in Biological Systems

We identify a living system as one that, in order to survive, continual measurements on its surroundings (and on itself). We also define a conceptual observer that reads and interprets the output of the measurements. In a different discipline of science, quantum measurements are also associated with an observer who interprets reality through the collapse procedure. However, this time, the interpretation has nothing to do with survival skills. Nevertheless, since both disciplines belong to the science of Nature, we suspect they obey similar rules. Indeed, we demonstrate that both measurements follow one fundamental principle: they both correspond with re-coherent processes. Applying a quantum-like model, such as that of a spin glass neural network, we describe brain activity with the Hubert space concept in what we refer to as quantum-like measurements.

Quantum Mechanics from Classical Logic

Although quantum mechanics is generally considered to be fundamentally incompatible with classical logic, it is argued here that the gap is not as great as it seems. Any classical, discrete, time reversible system can be naturally described using a quantum Hubert space, operators, and a Schrödinger equation. The quantum states generated this way resemble the ones in the real world so much that one wonders why this could not be used to interpret all of quantum mechanics this way. Indeed, such an interpretation leads to the most natural explanation as to why a wave function appears to "collapse" when a measurement is made, and why probabilities obey the Born rule. Because it is real quantum mechanics that we generate, Bell's inequalities should not be an obstacle.

Optimal State Estimation for Discrete-time Systems with Random Observation Delays

This paper investigates the linear minimum mean square error state estimation for discrete-time systems with Markov jump delays. In order to solve the optimal estimation problem, the single Markov delayed measurement is rewritten as an equivalent measurement with multiple constant delays, then a delay-free Markov jump linear system is obtained via state augmentation. The estimator is derived on the basis of the geometric arguments in the Hubert space, and a recursive equation of the filter is obtained by solving the Riccati equations. It is shown that the proposed state estimator is exponentially stable under standard assumptions.

Collapse of the Quantum Wave Function

Abstract The following introduction offers a broad survey of the history of quantum physics. It then outlines the position of each contributor in this Special Focus Section concerning the collapse of the quantum wavefunction and defines three important terms (Hubert space, Schrödinger’s cat, and decoherence) used in discussing this topic.

Boundedness of linear operators in the space l2

A necessary and sufficient condition for an operator to be bounded in a Hilbert space or a finite dimensional subspace of a Hubert space is derived explicitly in terms of the matrix elements of the matrix representation of the opeator. An identical condition for boundedness is shown to apply for the case of bilinear functionals in the space.

WEYL TYPE THEOREMS FOR POSINORMAL OPERATORS

Let be a bounded linear operator acting on infinite dimensional separable Hubert space H. The study of operators satisfying Weyl's theorem, Browder's theorem, the SVEP and Bishop's property is of significant interest and is currently being done by a number of mathematicians around the world. It is known that Weyl's theorem holds for M-hyponormal operators, but does not hold for dominant operators. Hence it is an interesting problem to seek a condition that implies Weyl's theorem for dominant operators. Ho Jeon et al. proved that if A is dominant and satisfies σ((A - λI)|M) = {0} ⇒ (A -λI)\M for every M ∈ Lat(A), then Weyl's theorem holds for A. Recently Cao showed that the generalized a-Weyl's theorem holds for ƒ(A), where ƒ is an analytic function defined in an open neighbourhood of σ(A) in the case where A* is p-hyponormal or M-hyponormal. Also Aiena showed that a-Weyl's theorem holds for some classes of operators. In this paper we prove that if A* is conditionally totally posinormal (with certain condition) or totally posinormal, then the generalized a-Weyl's theorem holds for A and for ƒ(A), where ƒ is an analytic function defined in an open neighbourhood of σ(A).

ON THE RANGE OF THE ELEMENTARY OPERATOR <i>X</i> ↦ <i>AXA</i> − <i>X</i>

Let L(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hubert space H into itself. Given A ∈ L(H), we define the elementary operator ∆A: L(H) → L(H) by ∆A(X) = AXA — X. In this paper, we initiate the study of the class of operator A for which $\overline {R(\Delta _A )} = \overline {R(\Delta _{A*} ),} $, where $\overline {R(\Delta _A )} $ denotes the norm closure of the range of ∆A-We call such operators quasiadjoint. We give a characterization and some basic results concerning this class of operator.

On the range of the elementary operator <i>X</i> → <i>AXA</i> – <i>X</i>

Let L(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hubert space H into itself. Given A ∈ L(H), we define the elementary operator ∆A: L(H) → L(H) by ∆A(X) = AXA — X. In this paper, we initiate the study of the class of operator A for which $\overline {R(\Delta _A )} = \overline {R(\Delta _{A*} ),} $, where $\overline {R(\Delta _A )} $ denotes the norm closure of the range of ∆A-We call such operators quasiadjoint. We give a characterization and some basic results concerning this class of operator.

Practical estimation of Volterra filters of arbitrary degree

Generalized Fock spaces (GFS) were proposed some time ago as a general framework for non-linear system identification and signal analysis. Corresponding to a particular class of reproducing kernel Hubert spaces (RKHS) they provide a natural framework for the estimation of continuous- and discrete-time Volterra filters (VF). Whilst both finite and infinite degree VF are treated in a single framework using GFS an open question is the practical estimation of infinite degree VF. In this paper this problem is resolved and it is shown how VF of arbitrary degree (including infinite) can be computed with finite resources. The solutions are global best approximations to the entire VF based on the available data. Use is made of recent advances in the machine learning community, such as the support vector machine, which make extensive use of RKHS ideas. This paper also highlights the importance of this work to the system identification community. A number of particular contributions are made including a derivation of a particular GFS for VF which permits a simply computable reproducing kernel, derivations of the associated mappings between Volterra parameter space and RKHS parameter space which permit the extraction of the Volterra parameters from the RKHS approximation and solve a set of “free” parameters, extending the GFS framework to include regularization and a detailed simulation-based comparison of the estimation of VF.

ON THE RIESZ IDEMPOTENT FOR A CLASS OF OPERATORS

A result from Stampili says that, for hyponormal operators on Hubert space, the Riesz idempotent at an isolated point of a spectrum is self-adjoint, with its range given by null space of both the operator and its adjoint. In this note, we extend this result to operators that are algebraically class H(q). The arguments involve local spectral theory and extensions of a-Browder's theorem.

ON THE RIESZ IDEMPOTENT FOR A CLASS OF OPERATORS

A result from Stampili says that, for hyponormal operators on Hubert space, the Riesz idempotent at an isolated point of a spectrum is self-adjoint, with its range given by null space of both the operator and its adjoint. In this note, we extend this result to operators that are algebraically class H(q). The arguments involve local spectral theory and extensions of a-Browder's theorem.

Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hubert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Holder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.

The Analytic Algebras Of Higher Rank Graphs

We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hubert space and creation operators that are partial isometries acting on the space. We call the weak operator topology closed algebra generated by these operators a 'higher rank semigroupoid algebra'. A number of examples are discussed in detail, including the single vertex case and higher rank cycle graphs. In particular, the cycle graph algebras are identified as matricial multivariable function algebras. We obtain reflexivity for a wide class of graphs and characterize semisimplicity in terms of the underlying graph.

THE ANALYTIC ALGEBRAS OF HIGHER RANK GRAPHS

We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hubert space and creation operators that are partial isometries acting on the space. We call the weak operator topology closed algebra generated by these operators a ' higher rank semigroupoid algebra'. A number of examples are discussed in detail, including the single vertex case and higher rank cycle graphs. In particular, the cycle graph algebras are identified as matricial multivariable function algebras. We obtain reflexivity for a wide class of graphs and characterize semisimplicity in terms of the underlying graph.

Isometric multipliers ofL p (G, X)

Let G be a locally compact group with a fixed right Haar measure andX a separable Banach space. LetL p (G, X) be the space of X-valued measurable functions whose norm-functions are in the usualL p . A left multiplier ofL p (G, X) is a bounded linear operator onB p (G, X) which commutes with all left translations. We use the characterization of isometries ofL p (G, X) onto itself to characterize the isometric, invertible, left multipliers ofL p (G, X) for 1 ≤p ∞,p ≠ 2, under the assumption thatX is not thel p -direct sum of two non-zero subspaces. In fact we prove that ifT is an isometric left multiplier ofL p (G, X) onto itself then there existsa y ∃ G and an isometryU ofX onto itself such thatTf(x) = U (R y f)(x). As an application, we determine the isometric left multipliers of L1 ∩L p (G, X) and L1 ∩C 0 (G, X) whereG is non-compact andX is not the lp-direct sum of two non-zero subspaces. If G is a locally compact abelian group andH is a separable Hubert space, we define $$A^p (G,H) = \{ f \in L^1 (G,H):\hat f \in L^p (\Gamma ,H)\} $$ where г is the dual group of G. We characterize the isometric, invertible, left multipliers ofA p (G, H), provided G is non-compact. Finally, we use the characterization of isometries ofC(G, X) for G compact to determine the isometric left multipliers ofC(G, X) providedX * is strictly convex.

An Uncertainty Principle Related to the Euclidean Motion Group

We show that a well-known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group. 1. Uncertainty principles related to Lie group representations Let G be a Lie group with Lie algebra g, and let (n, H) be a unitary representation of G. Then each element Xe g generates a closed, skew-adjoint operator n(X) on H by n(exptX)x-x n(X)x = hm '-o t with domain D(n(X)) consisting of all xe H for which the limit exists. The uncertainty principle related to n says that for operators generated by X, Y and [X, Y] the following holds ''n(X)x''''n(Y)x''>^'(n([X,Y])x,x)' for all xeD(n(X))nD(n(Y))nD(n([X, Y])). We would like to advocate this as a natural way to achieve uncertainty principles. It was first proposed in Kraus [4] for Lie groups with three dimensions or fewer, and the dimension constraint was recently removed by Christensen in [2]. For example, the classical Heisenberg uncertainty principle for functions on Un is easily derived in this way from the Schrodinger representation of the Heisenberg group (see [3, p. 212]). It is the purpose of this note to point out that the uncertainty principle for the circle, which was motivated by Breitenberger [1] and further discussed in [5; 6; 7; 8; 9; 10] is obtained similarly from the principal series representation of the Euclidean motion group of U2. * Corresponding author; e-mail: vepjan@math.ku.dk Mathematical Proceedings of the Royal Irish Academy, 104A (2), 249-252 (2004) © Royal Irish Academy This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms 250 Mathematical Proceedings of the Royal Irish Academy 2. The Euclidean motion group and a unitary representation Let G be the Euclidean motion group G=j(r,z)=(^or ^|rER,zecj Its Lie algebra is fl={(j j)|reR,zec} Let H be the Hubert space H = L2(J) of square integrable functions on the circle T = {s e C| 's' = 1}, with inner product (/, g) = JT f(t)g(t)dt. As in [1 1, chapter V] the following defines a unitary representation of G on ^i: 7Tfl(r, z)/(?) = ^Re(z^/(^-^), (j e T) where a e C. For simplicity we assume in the following that a = l, which is sufficient for our purpose. The representation n' will be denoted n. 3. Operators generated from the representation We now generate three operators from elements of the Lie algebra g. Let X,YU r2egbe *-G s), .-(s i) ^(o ?) then exp(ijr)=^ J), exParl)=^ i) and QxV(tY2)=(^ ^ and we then get 7t(X)f(s) = lim i-+0 / i->0 ? where /'(a) = jtf{eus). This operator has domain {/eL2(T)|?h-»/V0 absolutely continuous with /7 e L2(T)} (3.1) Also rv^^ v hm 1*0, t)f(s) f(s) hm e^s)fis) fis) . ,_, . 7t( rv^^ Fj)/^) = v hm i-o t t-+o t and /VUM r hm 40, iY)/(j) /(j) hm r e^fis) fis) . . . _. _ . . ni /VUM Y2)fis) = hm r t^o i /->o t when s=e10 . Both of these operators are defined on H. This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms Christensen and Schlichtkrull Euclidean motion group 251 4. The uncertainty principle Since [X,YX] = Y21 [X,Y2]=-Yl the uncertainty principle gives 1(7*7, )/,/>| = '(n([X, Y2])fJ)' The first author would like to thank the people at NUI Galway for the help and friendliness he received while writing his thesis.

Some Variants of Weber's Theorem

Weber's theorem says that if A : H → H is bounded and linear on a separable Hubert space, then any operator that is compact, commutes with A and lies in the weak closure of the range of the inner derivation induced by A must also be quasinilpotent. In this note we consider related problems for generalised inner derivations associated with operators A and B on H.

GENERALISED P-SYMMETRIC OPERATORS

Let H denote a complex Hubert space, L(H) the algebra of all bounded linear operators on H and C₁(H), the trace class operators. We study the pairs of operators A, B εL(H) with the property that AT= TB and T εC₁ (H) implies B*T = TA*. The main result is that this property is equivalent to the self-adjointness of the ultraweak closure of the range of a generalised derivation.

SOME VARIANTS OF WEBER'S THEOREM

Weber's theorem says that if A : H → H is bounded and linear on a separable Hubert space, then any operator that is compact, commutes with A and lies in the weak closure of the range of the inner derivation induced by A must also be quasinilpotent. In this note we consider related problems for generalised inner derivations associated with operators A and B on H.

Generalised P-Symmetric Operators

Let H denote a complex Hubert space, L(H) the algebra of all bounded linear operators on H and C₁(H), the trace class operators. We study the pairs of operators A, B εL(H) with the property that AT= TB and T εC₁ (H) implies B*T = TA*. The main result is that this property is equivalent to the self-adjointness of the ultraweak closure of the range of a generalised derivation.