Tensor Product Of Spaces Research Articles

Laplacian eigenproblems on product regions and tensor products of Sobolev spaces

Characterizations of eigenvalues and eigenfunctions of the Laplacian on a product domain Ωp:=Ω1×Ω2 are obtained. When zero Dirichlet, Robin or Neumann conditions are specified on each factor, then the eigenfunctions on Ωp are precisely the products of the eigenfunctions on the sets Ω1, Ω2 separately. There is a related result when Steklov boundary conditions are specified on Ω2. These results enable the characterization of H1(Ωp) and H01(Ωp) as tensor products and descriptions of some orthogonal bases of the spaces. A different characterization of the trace space of H1(Ωp) is found.

Berezin transforms on noncommutative polydomains

This paper is an attempt to unify the multivariable operator model theory for ball-like domains and commutative polydiscs and extend it to a more general class of noncommutative polydomains D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) in B ( H ) n B(\mathcal {H})^n . An important role in our study is played by noncommutative Berezin transforms associated with the elements of the polydomain. These transforms are used to prove that each such polydomain has a universal model W = { W i , j } \mathbf {W}=\{\mathbf {W}_{i,j}\} consisting of weighted shifts acting on a tensor product of full Fock spaces. We introduce the noncommutative Hardy algebra F ∞ ( D q m ) F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}) as the weakly closed algebra generated by { W i , j } \{\mathbf {W}_{i,j}\} and the identity, and use it to provide a WOT-continuous functional calculus for completely non-coisometric tuples in D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) . It is shown that the Berezin transform is a completely isometric isomorphism between F ∞ ( D q m ) F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}) and the algebra of bounded free holomorphic functions on the radial part of D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) . A characterization of the Beurling type joint invariant subspaces under { W i , j } \{\mathbf {W}_{i,j}\} is also provided. It has been an open problem for quite some time to find significant classes of elements in the commutative polydisc for which a theory of characteristic functions and model theory can be developed along the lines of the Sz.-Nagy–Foias theory of contractions. We give a positive answer to this question, in our more general setting, providing a characterization for the class of tuples of operators in D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) which admit characteristic functions. The characteristic function is constructed explicitly as an artifact of the noncommutative Berezin kernel associated with the polydomain, and it is proved to be a complete unitary invariant for the class of completely non-coisometric tuples. Using noncommutative Berezin transforms and C ∗ C^* -algebra techniques, we develop a dilation theory on the noncommutative polydomain D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) .

Resolvents of operators on tensor products of Euclidean spaces

We consider the operatorwhere are matrices , means the tensor product. Norm estimates for the resolvent of that operator are derived. By these estimates, we obtain bounds for a solution of the equationand explore perturbations of that equation. The norm estimates for the resolvent of enable us to establish a bound for the distance between invariant subspaces of two matrices. Besides, the well-known Davis–Kahan result is particularly generalized. In addition, we derive a new stability test for non-linear non-autonomous ordinary differential equations.

On Bernstein classes of well approximable maps

We study the structure of families of “well approximable” elements of tensor products of Banach spaces including analogs of the classical quasianalytic classes in the sense of Bernstein and Beurling. As in the case of quasianalytic functions, we prove for members of these families variants of the Mazurkiewicz and Markushevich theorems and in some particular cases, if such elements are Banach-valued continuous maps on a compact metric space, estimate massivity of their graphs and level sets.

Extended tensor products and an operator-valued spectral mapping theorem

We introduce the notion of an extended tensor product of Banach spaces and . It is defined as a triple consisting of a Banach space and two full subalgebras and of the algebras and of all bounded linear operators on and respectively. It is assumed that is an extension of the ordinary tensor product , and the functionals on and operators on have a canonical extension from to . Every pseudo- resolvent generates a functional calculus that sends analytic - valued functions in a neighbourhood of the singular set of the pseudo- resolvent to operators on . We prove an analogue of the spectral mapping theorem for such a functional calculus.

Erratum to: A Two-step PLS Inspired Method for Linear Prediction with Group Effect

R p ⊗ R = CΣ(A)⊗ C(C′) R ⊗ C(C′)⊥ CΣ(A)⊥ ⊗ C(C′), where Rp (Rn) represent the whole space, CΣ(•) denotes column vector space with inner product defined via Σ−1, C(•) if the standard inner product is used, ⊗ denotes the tensor product of linear spaces and represents the orthogonal sum of linear spaces. The decomposition is illustrated in Fig. 1. The estimation of the mean ABC is performed through projection of the data on the subspace CΣ(A) ⊗ C(C′). The subspace of Rp ⊗ C(C′)⊥ and CΣ(A)⊥ ⊗ C(C′) are the spaces where residuals are constructed via projections, i.e.

Tensor Products of Approximation Spaces Associated with Regular Elliptic Operators

We prove an interpolation theorem for the tensor products of approximation spaces associated with regular elliptic operators and show their applicability to the problem of approximation of elements of the tensor products of L p -spaces.

Modification of Multiple Knot B-Spline Wavelet for Solving (Partially) Dirichlet Boundary Value Problem

A construction of multiple knot B-spline wavelets has been given in (C. K. Chui and E. Quak, Wavelet on a bounded interval, In: D. Braess and L. L. Schumaker, editors. Numerical methods of approximation theory. Basel: Birkhauser Verlag; (1992), pp. 57-76). In this work, we first modify these wavelets to solve the elliptic (partially) Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods. We generalize this construction to two dimensional case by Tensor product space. In addition, the solution of the system discretized by Galerkin method with modified multiple knot B-spline wavelets is discussed. We also consider a nonlinear partial differential equation for unsteady flows in an open channel called Saint-Venant. Since the solving of this problem by some methods such as finite difference and finite element produce unsuitable approximations spe- cially in the ends of channel, it is solved by multiple knot B-spline wavelet method that yields a very well approximation. Finally, some numerical examples are given to support our theoretical results.

The Hilbert tensor product and inductive limits

In the present paper we prove that given two inductive limits of Hilbert spaces \(E= \mathrm{ind}_{n\in \mathbb {N}}E_n\) and \(F=\mathrm{ind}_{n\in \mathbb {N}} F_n\) the complete Hilbert tensor product \(E \tilde{\otimes }_{\sigma } F\) of E and F is topologically isomorphic to the inductive limit of the inductive spectrum \(\left( E_n\tilde{\otimes }_{\sigma }F_n\right) _{n\in \mathbb {N}}\). To this end we consider the Hilbert tensor product for the tensor product of spaces equipped with Hilbertian semi norms, spaces, that we call semi-unitary. We conclude with two consequences, first the positive solution of Grothendieck’s probleme des topologies for Frechet–Hilbert spaces and the complete Hilbert tensor product and second the computation of tensor products where at least one space is not Schwartz, e.g. the tensor product of the space of Schwartz distributions \(\fancyscript{D}'\left( \varOmega \right) \) with the space \(\fancyscript{D}_{\mathrm{L}_2}\) of all smooth functions all the derivatives of which are square integrable or its strong dual, which has its application in parameter dependence problems.

On the Nuclear Norm and the Singular Value Decomposition of Tensors

Finding the rank of a tensor is a problem that has many applications. Unfortunately, it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of the rank of a tensor. We determine the nuclear norm of various tensors of interest. Along the way, we also do a systematic study various measures of orthogonality in tensor product spaces and we give a new generalization of the singular value decomposition to higher-order tensors.

Bilinear ideals in operator spaces

We introduce a concept of bilinear ideal of jointly completely bounded mappings between operator spaces. In particular, we study the bilinear ideals N of completely nuclear, I of completely integral and E of completely extendible bilinear mappings. We also consider the multiplicatively bounded bilinear mappings MB and its symmetrization SMB. We prove some basic properties of them, one of which is the fact that I is naturally identified with the ideal of (linear) completely integral mappings on the injective operator space tensor product.

Canonical forms of [formula omitted] tensors over the real field, algebraically closed fields, and finite fields

We classify the orbits of elements of the tensor product spaces F2⊗F3⊗F3 for all finite, real, and algebraically closed fields under the action of two natural groups. The result can also be interpreted as the classification of the (GL(F2)×GL(F3)×GL(F3))-orbits in the 17-dimensional projective space of 2×3×3 tensors, which is the natural home of the Segre product of a projective line and two projective planes. The proof is geometric in nature, relies on properties of the Segre embedding, and uses the terminology of projective spaces.

Dissecting weak discernibility of quanta

In this paper I critically examine latest attempts to formalize quantum-mechanical relations that are supposed to weakly discern elementary particles. I argue that all of them make illegitimate and unavoidable reference to numerical identity, and therefore cannot be used as a means to ground (or derive) quantitative facts of identity/distinctness in the qualitative characteristics of quantum systems. I compare my criticism of weak discernibility with the general circularity objection known from the literature, and I show that my argument is more specific, as it is based on a particular criterion which differentiates between legitimate and illegitimate uses of identity. In the end I suggest that we should reevaluate the role of permutation invariance in expressing the facts of qualitative differences between particles. Taking into account the inevitable symmetrization requirement applied to operators in tensor product spaces, it may be claimed that particles of the same type can be absolutely discerned in some accessible states.

A nearly-optimal algorithm for the Fredholm problem of the second kind over a non-tensor product Sobolev space

In a previous paper, the authors showed that the information complexity of the Fredholm problem of the second kind is essentially the same as that of the approximation problems over the spaces of kernels and right-hand sides. This allowed us to give necessary and sufficient conditions for the Fredholm problem to exhibit a particular level of tractability (for information complexity) over weighted tensor product (\textsc{wtp}) spaces, as well as over an important class of \textit{not} necessarily tensor product weighted Sobolev spaces. Furthermore, we addressed the overall complexity of this Fredholm problem for the case in which the kernels and right-hand sides belong to a \textsc{wtp} space. For this case, we showed that a nearly-minimal-error interpolatory algorithm is easily implementable, with cost very close (to within a logarithmic factor) to the information cost. As a result, tractability results, which had previously only held for the information complexity, now hold for the overall complexity--provided that our kernels and right-hand sides belong to \textsc{wtp} spaces. This result does not hold for the weighted Sobolev spaces mentioned above, since they are not necessarily tensor product spaces. In this paper, we close this gap. We exhibit an easily implementable iterative approximation to a nearly minimal error interpolatory algorithm for this family of weighted Sobolev spaces. This algorithm exhibits the same good properties as the algorithm presented in the previous paper.

Multiresolution analysis for compactly supported interpolating tensor product wavelets

We construct multidimensional interpolating tensor product multiresolution analyses (MRA's) of the function spaces C0(ℝn, K), K = ℝ or K = ℂ, consisting of real or complex valued functions on ℝn vanishing at infinity and the function spaces Cu(ℝn, K) consisting of bounded and uniformly continuous functions on ℝn. We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence from the one-dimensional case to our n-dimensional construction.

Octahedral norms in spaces of operators

We study octahedral norms in the space of bounded linear operators between Banach spaces. In fact, we prove that L(X,Y) has octahedral norm whenever X⁎ and Y have octahedral norm. As a consequence the space of operators L(ℓ1,X) has octahedral norm if, and only if, X has octahedral norm. These results also allow us to get the stability of strong diameter 2 property for projective tensor products of Banach spaces, which is an improvement of the known results about the size of nonempty relatively weakly open subsets in the unit ball of the projective tensor product of Banach spaces.

Additivity rates and PPT property for random quantum channels

Inspired by Montanaro’s work, we introduce the concept of additivity rates of a quantum channel L, which give the first order (linear) term of the minimum output p-Rényi entropies of L ⊗r as functions of r. We lower bound the additivity rates of arbitrary quantum channels using the operator norms of several interesting matrices including partially transposed Choi matrices. As a direct consequence, we obtain upper bounds for the classical capacity of the channels. We study these matrices for random quantum channels defined by random subspaces of a bipartite tensor product space. A detailed spectral analysis of the relevant random matrix models is performed, and strong convergence towards free probabilistic limits is shown. As a corollary, we compute the threshold for random quantum channels to have the positive partial transpose (PPT) property. We then show that a class of random PPT channels violate generically additivity of the p-Rényi entropy for all p≥30.95.

Note on Arens regularity of symmetric tensor products

We investigate symmetric regularity of sums of symmetric tensor products of Banach spaces and Arens regularity of symmetric tensor products of Banach algebras. An example for the Hilbert space is obtained.

Bell inequalities from group actions of single-generator groups

We study a method of generating Bell inequalities by using group actions of single-generator Abelian groups. Two parties, Alice and Bob, each make one of $M$ possible measurements on a system, with each measurement having $K$ possible outcomes. The probabilities for the outcomes of these measurements are $P({a}_{j}=k,{b}_{{j}^{\ensuremath{'}}}={k}^{\ensuremath{'}})$, where $j,{j}^{\ensuremath{'}}\ensuremath{\in}{1,2,...,M}$ and $k,{k}^{\ensuremath{'}}\ensuremath{\in}{0,1,...,K\ensuremath{-}1}$. The sums of some subsets of these probabilities have upper bounds when the probabilities result from a local, realistic theory that can be violated if the probabilities come from quantum mechanics. In our case the subsets of probabilities are generated by a group action, in particular, a representation of a single-generator group acting on product states in a tensor-product Hilbert space. We show how this works for several cases, including $M=2$, $K=3$, and general $M$, $K=2$. We also discuss the resulting inequalities in terms of nonlocal games.

Essential spectrum of a fermionic quantum field model

An interaction system of a fermionic quantum field is considered. The state space is defined by a tensor product space of a fermion Fock space and a Hilbert space. It is assumed that the total Hamiltonian is a self-adjoint operator on the state space and bounded from below. Then it is proven that a subset of real numbers is the essential spectrum of the total Hamiltonian. It is applied to a Yukawa interaction system, which is a system of a Dirac field coupled to a Klein–Gordon, and the HVZ theorem is obtained.