Tensor Product Of Spaces Research Articles

Properties of the Rovelli–Smolin–DePietri volume operator in the spaces of monochromatic intertwiners

We study some properties of the Rovelli–Smolin–DePietri volume operator in loop quantum gravity, which significantly simplify the diagonalization problem and shed some light on the pattern of degeneracy of the eigenstates. The operator is defined by its action in the spaces of tensor products of the irreducible SU(2) representation spaces , labelled with spins . We restrict to spaces of SU(2) invariant tensors (intertwiners) with all spins equal j 1 =⋯= j N = j. We call them spin j monochromatic intertwiners. Such spaces are important in the study of SU(2) gauge invariant states that are isotropic and can be applied to extract the cosmological sector of the theory. In the case of spin 1/2 we solve the eigenvalue problem completely: we show that the volume operator is proportional to identity and calculate the proportionality factor.

Spectra of elements in operator space tensor products of $$\hbox {C}^*$$-algebras

Spectra of elements in operator space tensor products of $$\hbox {C}^*$$-algebras

Tensor Product and Certain Solutions of Fractional Wave Type Equation

In this paper we Önd certain solutions of some fractional partial di§erential equations. Tensor product of Banach spaces is used to Önd some solutions where separation of variables does not work. We solve the fractional wave type equation using fractional Fourier Series

Routed quantum circuits

We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces. We therefore propose an extension to the framework of quantum circuits, given by routed linear maps and routed quantum circuits. We prove that this new framework allows for a consistent and intuitive diagrammatic representation in terms of circuit diagrams, applicable to both pure and mixed quantum theory, and exemplify its use in several situations, including the superposition of quantum channels and the causal decompositions of unitaries. We show that our framework encompasses the `extended circuit diagrams' of Lorenz and Barrett [arXiv:2001.07774 (2020)], which we derive as a special case, endowing them with a sound semantics.

Distinguished Property in Tensor Products and Weak* Dual Spaces

A local convex space E is said to be distinguished if its strong dual Eβ′ has the topology β(E′,(Eβ′)′), i.e., if Eβ′ is barrelled. The distinguished property of the local convex space CpX of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space CpX is distinguished if and only if any function f∈RX belongs to the pointwise closure of a pointwise bounded set in CX. The extensively studied distinguished properties in the injective tensor products CpX⊗εE and in Cp(X,E) contrasts with the few distinguished properties of injective tensor products related to the dual space LpX of CpX endowed with the weak* topology, as well as to the weak* dual of Cp(X,E). To partially fill this gap, some distinguished properties in the injective tensor product space LpX⊗εE are presented and a characterization of the distinguished property of the weak* dual of Cp(X,E) for wide classes of spaces X and E is provided.

Toeplitz Like Operators on 2-nuclear Tensor Product of Hardy Spaces

in this paper, we investigated Toeplitz like operators on vector valued Hardy spaces. Toeplitz like operators on 2-nuclear tensor product of Hardy spaces are then constructed and described using the theory of p-nuclear tensor product of Banach spaces, and their basic algebraic properties and spectrum are analyzed.

Finite Rank Solution for Conformable Degenerate First-Order Abstract Cauchy Problem in Hilbert Spaces

In this paper, we find a solution of finite rank form of fractional Abstract Cauchy Problem. The fractional derivative used is the Conformable derivative. The main idea of the proofs are based on theory of tensor product of Banach spaces.

Submodules in Polydomains and Noncommutative Varieties

Tensor product of Fock spaces is analogous to the Hardy space over the unit polydisc. This plays an important role in the development of noncommutative operator theory and function theory in the sense of noncommutative polydomains and noncommutative varieties. In this paper we study joint invariant subspaces of tensor product of full Fock spaces and noncommutative varieties. We also obtain, in particular, by using techniques of noncommutative varieties, a classification of joint invariant subspaces of n-fold tensor products of Drury–Arveson spaces.

Fractal Frames of Functions on the Rectangle

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

Modulation spaces associated with tensor products of amalgam spaces

We identify the generalised modulation spaces associated with tensor products of amalgam spaces having a large class of Banach spaces as their local component. As consequences of the main results, we describe the modulation spaces associated with tensor products of various $$L^p$$ spaces.

Constructing alternating 2-cocycles on Fourier algebras

Building on recent progress in constructing derivations on Fourier algebras, we provide the first examples of locally compact groups whose Fourier algebras support non-zero, alternating 2-cocycles; this is the first step in a larger project. Although such 2-cocycles can never be completely bounded, the operator space structure on the Fourier algebra plays a crucial role in our construction, as does the opposite operator space structure.Our construction has two main technical ingredients: we observe that certain estimates from [12] yield derivations that are “co-completely bounded” as maps from various Fourier algebras to their duals; and we establish a twisted inclusion result for certain operator space tensor products, which may be of independent interest.

Four Postulates of Quantum Mechanics Are Three.

The tensor product postulate of quantum mechanics states that the Hilbert space of a composite system is the tensor product of the components' Hilbert spaces. All current formalizations of quantum mechanics that do not contain this postulate contain some equivalent postulate or assumption (sometimes hidden). Here we give a natural definition of a composite system as a set containing the component systems and show how one can logically derive the tensor product rule from the state postulate and from the measurement postulate. In other words, our Letter reduces by one the number of postulates necessary to quantum mechanics.

Numerical Index and Daugavet Property of Operator Ideals and Tensor Products

We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain space and the range space. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain space, and this result provides interesting examples. We also show that the numerical index of a projective or injective tensor product of Banach spaces is less than or equal to the numerical index of any of the factors. Finally, we show that if a projective tensor product of two Banach spaces has the Daugavet property and the unit ball of one of the factor is slicely countably determined or its dual contains a point of Fréchet differentiability of the norm, then the other factor inherits the Daugavet property. If an injective tensor product of two Banach spaces has the Daugavet property and one of the factors contains a point of Fréchet differentiability of the norm, then the other factor has the Daugavet property.

Unbounded operator algebras

In this paper, we introduce the notion of abstract local operator algebras and operator modules, and provide a representation theorem for them which extends the BRS theorem for operator algebras. Furthermore, we give a new proof for the representation theorem of local operator systems. Also, we investigate the Haagerup tensor product of local operator spaces and Morita equivalence of local operator algebras.

Positive maps and trace polynomials from the symmetric group

With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials Xα1,…,Xαr and their traces tr(Xα1,…,Xαr). Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley–Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.

Analysis of autocorrelation function of stochastic processes by F-transform of higher degree

The autocorrelation function of a stochastic process is one of the essential mathematical tools in the description of variability that is successfully applied in many scientific fields such as signal processing or financial time series analysis and forecasting. The aim of the paper is to provide an analysis of fuzzy transform of higher degree applied to stochastic processes with a focus on its autocorrelation function. We introduce a higher-degree fuzzy transform of a stochastic process and investigate its basic properties. Further, we introduce a higher-degree fuzzy transform of bivariate functions in the tensor product of polynomial spaces and demonstrate its approximation ability. In addition, we show that the bivariate higher-degree fuzzy transform of multiplicative separable functions is a product of univariate fuzzy transform of the respective functions, which is a desirable property in the processing of functions of higher dimensions. The obtained results are used to demonstrate an interesting identity between the autocorrelation function of the fuzzy transform of a stochastic process and the bivariate fuzzy transform of the autocorrelation function of the stochastic process. This identity provides two ways to determine the autocorrelation function of the fuzzy transform of a stochastic process, which can be useful, especially in a situation where its direct calculation becomes complex or even impossible, for example, the calculation of a sample autocorrelation function.

Algebraic Tensor Products Revisited: Axiomatic Approach

This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by reducing them to different but isomorphic interpretations of an abstract notion, viz. the universal property, which is based on a pair of axioms.

Tensor-Free Proximal Methods for Lifted Bilinear/Quadratic Inverse Problems with Applications to Phase Retrieval

We propose and study a class of novel algorithms that aim at solving bilinear and quadratic inverse problems. Using a convex relaxation based on tensorial lifting, and applying first-order proximal algorithms, these problems could be solved numerically by singular value thresholding methods. However, a direct realization of these algorithms for, e.g., image recovery problems is often impracticable, since computations have to be performed on the tensor-product space, whose dimension is usually tremendous. To overcome this limitation, we derive tensor-free versions of common singular value thresholding methods by exploiting low-rank representations and incorporating an augmented Lanczos process. Using a novel reweighting technique, we further improve the convergence behavior and rank evolution of the iterative algorithms. Applying the method to the two-dimensional masked Fourier phase retrieval problem, we obtain an efficient recovery method. Moreover, the tensor-free algorithms are flexible enough to incorporate a priori smoothness constraints that greatly improve the recovery results.

Tensor product technique and atomic solution of fractional Bate Man Burgers equation

Sometimes, it is not possible to find a general solution for some differential equations using some classical methods, like separation of variables. In such a case, one can try to use theory of tensor product of Banach spaces to find certain solutions, called atomic solutions. The goal of this paper is to find atomic solution for Bate Man Burgers equation.

The dual of the space of bounded operators on a Banach space

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.