Tensor products of semi-closed operators and applications

Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI

In this paper we study n-inverse pairs of operators on the tensor product of Banach spaces. In particular we show that an n-inverse pair of elementary tensors of operators on the tensor product of two Banach spaces can arise only from l- and m-inverse pairs of operators on the individual spaces. This gives a converse to a result of Duggal and Muller, and proves a conjecture of the second named author. Our proof uses techniques from algebraic geometry, which generalize to other relations among operators in a tensor product. We apply this theory to obtain results for n-symmetries in a tensor product as well.

A Banach space operator T ∈ B(χ) is left polaroid if for each λ ∈ iso σa(T) there is an integer d(λ) such that asc(T - λ) = d(λ) < ∞ and (T - λ)d(λ)+1χ is closed; T is finitely left polaroid if asc(T - λ) < ∞, (T - λ)χ is closed and dim(T - λ)−1(0) < ∞ at each λ ∈ iso σa(T). The left polaroid property transfers from A and B to their tensor product A ⊗ B, hence also from A and B* to the left-right multiplication operator τAB, for Hilbert space operators; an additional condition is required for Banach space operators. The finitely left polaroid property transfers from A and B to their tensor product A⊗B if and only if 0 ∈ iso σa(A⊗B); a similar result holds for τAB for finitely left polaroid A and B*.

In this work, we introduce the notion of tensor product of spaces with a generalized 2-inner product (see Definition 5), and we establish several interesting properties (see Proposition5), thereby generalizing the classical properties of the tensor product of inner product spaces. Moreover, we equip this tensor product with a mapping that defines a generalized 2-inner product(see Theorem 3) and, consequently, endow it with a generalized 2-norm (see Theorem 1). In this context, we also define the tensor product of linear operators (see Definition 9) and prove a series of results for example, that the tensor product of two 2-bounded linear operators is again 2-bounded under the tensor product (see Proposition 10).

This chapter extends the concept of polynomially paranormal operators on Hilbert spaces and their tensor products to the broader class of polynomially *-paranormal and uncertainty-based operators.We introduce new definitions and establish sufficient conditions under which the tensor product of such operators preserves paranormality and spectral stability.Moreover, the single-valued extension property (SVEP) and spectral radius inequalities are investigated for these new operator classes.The results unify operator theoretic structures with uncertainty theory and provide a framework for future extensions to Banach spaces and C *algebras.

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complex. 23:262–295, 2007. We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approximate an operator S d given as the d-fold tensor product of a compact linear operator S 1 for d=1,2,…, with ‖S 1‖=1 and S 1 having at least two positive singular values. Let n(e,S d ) be the minimal number of information evaluations needed to approximate S d to within e∈[0,1]. We study generalized tractability by verifying when n(e,S d ) can be bounded by a multiple of a power of T(e −1,d) for all (e −1,d)∈Ω⊆[1,∞)×ℕ. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T(e −1,d) whose multiple bounds n(e,S d ). We also study weak tractability, i.e., when $\lim_{\varepsilon^{-1}+d\to\infty,(\varepsilon^{-1},d)\in \varOmega}\ln n(\varepsilon,S_{d})/(\varepsilon^{-1}+d)=0$. In our previous paper, we studied generalized tractability for proper subsets Ω of [1,∞)×ℕ, whereas in this paper we take the unrestricted domain Ω unr=[1,∞)×ℕ. We consider the three cases for which we have only finitely many positive singular values of S 1, or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S 1 decay slightly faster than logarithmically. We provide necessary and sufficient conditions on the function T such that generalized tractability holds. These conditions are obtained in terms of the singular values of S 1 and mostly asymptotic properties of T. The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T(x,y)=x 1+ln y . We also study tractability functions T of product form, T(x,y)=f 1(x)f 2(x). Assume that a i =lim inf x→∞(ln ln f i (x))/(ln ln x) is finite for i=1,2. Then generalized tractability takes place iff $$a_{i}>1\quad\mbox{and}\quad(a_{1}-1)(a_{2}-1)\ge1$$, and if (a 1−1)(a 2−1)=1 then we need to assume one more condition given in the paper. If (a 1−1)(a 2−1)>1 then the exponent of tractability is zero, and if (a 1−1)(a 2−1)=1 then the exponent of tractability is finite. It is interesting to add that for T being of the product form, the tractability conditions as well as the exponent of tractability depend only on the second singular eigenvalue of S 1 and they do not depend on the rate of their decay. Finally, we compare the results obtained in this paper for the unrestricted domain Ω unr with the results from our previous paper obtained for the restricted domain Ω res=[1,∞)×{1,2,…,d *}∪[1,e 0−1)×ℕ with d *≥1 and e 0∈(0,1). In general, the tractability results are quite different. We may have generalized tractability for the restricted domain and no generalized tractability for the unrestricted domain which is the case, for instance, for polynomial tractability T(x,y)=xy. We may also have generalized tractability for both domains with different or with the same exponents of tractability.

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.

In this paper we provide several generalizations for tensorial and Hadamard products of positive linear operators on complex Hilbert spaces of the celebrated scalar inequalities due to Tominaga. They give both multiplicative and additive reverses of Young’s inequality for positive operators in terms of Specht’s ratio and logarithmic mean.

Two classes of random linear operators, w- and s-operators, are defined, using cylindrical measures and F-cylindrical probabilities on tensor products of Banach spaces. Various Radonifying theorems for such measures are used in order that the composition vAw of a w- or s-operator A and deterministic operators w, v be a bounded one.

I.M. Gali et al have considered a distributed control problem for hyperbolic operator with an infinite number of variables [6]. Also they established the solvability of the mixed problem for nonlinear infinite order hyperbolic equations [7]. The authors in [9] have obtained the set of inequalities defining an optimal control of a system governed by infinite tensor product of elliptic operators Ak.In the present paper, a distributed control problem for the hyperbolic operator is considered.The necessary and sufficient condition for the control to be optimal is obtained, and the set of inequalities that characterize this condition is also obtained.KeywordsHilbert SpaceInfinite NumberElliptic OperatorMixed ProblemInfinite OrderThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible n-linear versions of Grothendieck’s composition theorem in the case n ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact.

In this paper, we mainly study the frequent hypercyclicity of a scalar multiple of the tensor product of two linear fractional composition operators on , where is the weighted Dirichlet space. We will restrict the discussion to the situations that both φ and ψ are hyperbolic non automorphisms, hyperbolic automorphisms and parabolic automorphisms, respectively. Meanwhile, we give the equivalent conditions of to be frequently hypercyclic, hypercyclic, mixing and chaotic.

Erratum to the paper 'Tensor product of proper contractions, stable and posinormal operators', Publ. Math. Debrecen 71 (2007), 425-437

We describe a class of spin chains with new physical and computational properties. On the physical side, the spin chains give examples of symmetry-protected topological phases that are defined by non-onsite symmetries, i.e., symmetries that are not a tensor product of single-site operators. These phases can be detected by string-order parameters, but notably do not exhibit entanglement spectrum degeneracy. On the computational side, the spin chains represent a new class of states that can be used to deterministically teleport information across long distances, with the novel property that the necessary classical side processing is a non-linear function of the measurement outcomes. We also give examples of states that can serve as universal resources for measurement-based quantum computation, providing the first examples of such resources without entanglement spectrum degeneracy. The key tool in our analysis is a new kind of tensor network representation which we call split-index matrix product states (SIMPS). We develop the basic formalism of SIMPS, compare them to matrix product states, show how they are better equipped to describe certain kinds of non-onsite symmetries including anomalous symmetries, and discuss how they are also well-suited to describing quantum teleportation and constrained spin chains.

On the spectra of tensor product of linear operators in locally convex spaces