Symmetric Operator Spaces Research Articles

Narrow operators on [formula omitted]-complete lattice-normed spaces

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C-complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C-compact orthogonally additive operator T:X→Y from an atomless C-complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator TN:E(X)→E(X) to be C-compact. We also show that the sum of two orthogonally additive operators G and T, where G is narrow and T is laterally-to-norm continuous and C-compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space CE.

Hermitian operators and isometries on symmetric operator spaces

Let \mathcal{M} be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space \mathcal{H} equipped with a semifinite faithful normal trace \tau . Let E(\mathcal{M},\tau) be a symmetric operator space affiliated with \mathcal{M} , whose norm is order continuous and is not proportional to the Hilbertian norm {\parallel{\cdot}\parallel}_2 on L_2(\mathcal{M},\tau) . We obtain a general description of all bounded hermitian operators on E(\mathcal{M},\tau) . This is the first time that the description of hermitian operators on a symmetric operator space (even for a noncommutative L_p -space) is obtained in the setting of general (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standing open problem concerning the description of isometries raised in the 1980s, which generalizes and unifies numerous earlier results.

Distributional inequalities for noncommutative martingales

We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distributional versions of the noncommutative Stein, dual Doob, martingale transform and Burkholder-Gundy inequalities. Our proof relies upon new and powerful extrapolation theorems. As an application, we obtain some new martingale inequalities in symmetric quasi-Banach operator spaces and some interesting endpoint estimates. Our main approach demonstrates a method to build the noncommutative and classical probabilistic inequalities in an entirely operator theoretic way.

Free Rosenthal Inequalities on Vector-Valued Symmetric Operator Spaces

Abstract Let $\mathcal{M}$ be a finite von Neumann algebra and let $E$ be an interpolation space between $L_1(0,1)$ and $L_{\infty }(0,1)$. For every $q\in [1,\infty ]$, we introduce a general notion of a vector-valued symmetric operator space $E(\mathcal{M};\ell _q)$ via $K$-functional of the couple $(L_1(\mathcal M;\ell _q),L_{\infty }(\mathcal M;\ell _q))$. We then establish free Rosenthal inequalities on $E(\mathcal{M};\ell _q)$, which provide an equivalent characterization of the quantity $\|(x_k)_{k\geq 0}\|_{E(\mathcal{M};\ell _q)},$ where $(x_k)_{k\geq 0}\subset E(\mathcal{M})$ is a sequence of freely independent random variables.

Tree Martingales in Noncommutative Probability Spaces

We introduce the notion of a tree martingale in a noncommutative probability space and prove the Burkholder–Gundy inequalities for tree martingales in symmetric operator spaces. In particular, we establish some inequalities in this setting via an approach based on the concept of generalized singular valued functions of noncommutative random valuables.

On subspaces spanned by freely independent random variables in noncommutative Lp-spaces

This paper deals with new phenomena appearing in the structure of Banach spaces arising in noncommutative integration theory. Let E be a fully symmetric Banach function space on [0, 1] and M be a finite von Neumann algebra. Let [xk] be the closed subspace spanned by a sequence (xk) of freely independent mean zero random variables from E(ℳ). The subspace [xk] is complemented in E(ℳ) if and only if the closed subspace spanned by the pairwise orthogonal sequence (xk ⊗ ek) is complemented in a certain symmetric operator space $$Z_E^2\left({{\cal M}\overline \otimes {\ell_\infty}} \right)$$ . We obtain noncommutative (free) analogues of classical results of Dor and Starbird as well as those of Kadec and Pelczynski. We show that [xk] is complemented in L1(ℳ)provided (xk) is equivalent in L1(ℳ) to the standard basis of l2, while this never happens in the classical case. We prove that a sequence of freely independent copies of a mean zero random variable x in Lp(ℳ), 1 ≤ p ≤ 2, is equivalent to the standard basis in some Orlicz sequence space ℳΦ and give a precise description of the connection between the Orlicz function Φ and the distribution of the given random variable x. Finally, we prove that [xk] spanned by a sequence of freely independent copies of a mean zero random variable is complemented in E(ℳ) if and only if (xk) is equivalent in E(ℳ) to the standard basis of l2.

M-embedded symmetric operator spaces and the derivation problem

AbstractLet ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

Positive Linear Isometries in Symmetric Operator Spaces

Let $$(\mathcal {M},\tau )$$ and $$(\mathcal {N},\nu )$$ be semifinite von Neumann algebras equipped with faithful normal semifinite traces and let $$E(\mathcal {M},\tau )$$ and $$F(\mathcal {N},\nu )$$ be symmetric operator spaces associated with these algebras. We provide a sufficient condition on the norm of the space $$F(\mathcal {N},\nu )$$ guaranteeing that every positive linear isometry $$T:E(\mathcal {M},\tau ){\mathop {\longrightarrow }\limits ^{into}} F(\mathcal {N},\nu )$$ is “disjointness preserving” in the sense that $$T(x)T(y)=0$$ provided that $$xy=0$$ , $$0\le x,y\in E(\mathcal {M},\tau )$$ . This fact, in turn, allows us to describe the general form of such isometries.

Derivations with values in ideals of semifinite von Neumann algebras

Let M be a semifinite von Neumann algebra with a faithful semifinite normal trace τ and let A be an arbitrary C⁎-subalgebra of M. Assume that E is a fully symmetric function space on (0,∞) having Fatou property and order continuous norm and E(M,τ) is the corresponding symmetric operator space. We prove that every derivation δ:A→E(M,τ):=E(M,τ)∩M is inner, strengthening earlier results by Kaftal and Weiss [28]. In the case when M is a semifinite non-finite factor, we show that our assumptions on E(0,∞) are sharp.

Johnson–Schechtman inequalities for noncommutative martingales

In this paper we study Johnson–Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space E(M) of a finite von Neumann algebra M without making any assumption on the Boyd indices of E. We show that we can obtain Johnson–Schechtman inequalities for arbitrary martingale difference sequences and that, in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables, the inequalities are one-sided except when E=L2(0,1). As an application, we partly resolve a problem stated by Randrianantoanina and Wu in [46]. We also show that we can obtain sharp Φ-moment analogues for Orlicz functions satisfying p-convexity and q-concavity for 1≤p≤2, q=2 and p=2, 2<q<∞. This is new even for the classical case. We also extend and strengthen the noncommutative Burkholder–Gundy inequalities in symmetric spaces and in the Φ-moment case.

Hölder inequality for symmetric operator spaces and trace property of K-cycles

Hölder inequality for symmetric operator spaces and trace property of K-cycles

Riesz outer product Hilbert space frames: Quantitative bounds, topological properties, and full geometric characterization

Outer product frames are important objects in Hilbert space frame theory. But very little is known about them. In this paper, we make the first detailed study of the family of outer product frames induced directly by vector sequences. We are interested in both the quantitative attributes of these outer product sequences (in particular, their Riesz and frame bounds) and their independence and spanning properties. We show that Riesz sequences of vectors yield Riesz sequences of outer products with the same (or better) Riesz bounds. Equiangular tight frames are shown to produce Riesz sequences with optimal Riesz bounds for outer products. We provide constructions of frames which produce Riesz outer product bases with “good” Riesz bounds. We show that the family of unit norm frames which yield independent outer product sequences is open and dense (in a Euclidean-analytic sense) within the topological space ⊗i=1MSN−1 where M is less than or equal to the dimension of the space of symmetric operators on HN; that is to say, almost every frame with such a bound on its cardinality will induce a set of independent outer products. Thus, this would mean that finding the necessary and sufficient conditions such that the induced outer products are dependent is a more interesting question. For the coup de grâce, we give a full analytic and geometric classification of such sequences which produce dependent outer products.

Φ-moment inequalities for independent and freely independent random variables

This paper is devoted to the study of Φ-moments of sums of independent/freely independent random variables. More precisely, let (fk)k=1n be a sequence of positive (symmetrically distributed) independent random variables and let Φ be an Orlicz function with Δ2-condition. We provide an equivalent expression for the quantity E(Φ(∑k=1nfk)) in term of the sum of disjoint copies of the sequence (fk)k=1n. We also prove an analogous result in the setting of free probability. Furthermore, we provide an equivalent characterization of τ(Φ(sup1≤k≤n+⁡xk)) for positive freely independent random variables and also present some new results on free Johnson–Schechtman inequalities in the quasi-Banach symmetric operator space.

Positive semigroups and algebraic Riccati equations in Banach spaces

We generalize Wonham’s theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to $$A^*P+PA-PBB^*P+C^*C=0$$ when (A, B) is exponentially stabilizable and (C, A) is exponentially detectable. The proof is based on a new approach that treats the linear part of the equation as the generator of a positive semigroup on the space of symmetric operators from a Banach space to its dual, and the quadratic part as an order concave map. A direct analog of global Newton’s iteration for concave functions is then used to approximate the solution, the approximations converge in the strong operator topology, and the convergence is monotone. The linearized equations are the well-known Lyapunov equations of the form $$A^*P+PA=-Q$$ , and semigroup stability criterion in terms of them is also generalized.

Hölder's inequality for roots of symmetric operator spaces

We prove a version of Hölder's inequality with a constant for $p$th roots of symmetric operator spaces of operators affiliated to a semifinite von Neumann algebra factor, and with constant equal to $1$ for strongly symmetric operator spaces.

Sums and intersections of symmetric operator spaces

Let L(H) be the algebra of all bounded operators on a separable Hilbert space H. We completely describe symmetric operator ideals E in L(H), which can be represented as a sum (or, an intersection) of two other (distinct from E) symmetric operator ideals in L(H). We also present a version of our results for symmetric operator spaces affiliated with a semifinite atomless von Neumann algebra M.

Completeness of quasi-normed symmetric operator spaces

We show that (generalized) Calkin correspondence between quasi-normed symmetric sequence spaces and symmetrically quasi-normed ideals of compact operators on an infinite-dimensional Hilbert space preserves completeness. We also establish a semifinite version of this result.

Johnson–Schechtman inequalities in the free probability theory

We introduce and study a “free Kruglov operator”. As an application of this study, we prove an analogue of the Johnson–Schechtman inequalities in the setting of free probability theory and an arbitrary symmetric operator space. This extends and complements earlier Lp-results of Junge et al. (2007) [12].

On the space of symmetric operators with multiple ground states

We study the homology structure of the filtration of the space of self-adjoint operators by the multiplicity of the ground state. We consider only operators acting on a finite-dimensional complex or real Hilbert space, but infinite-dimensional generalizations are easy to guess.

Generalized bicircular projections via rank preserving maps on the spaces of symmetric and antisymmetric operators

We study several related maps from the space of symmetric (respectively, antisymmetric) operators, acting on a complex Hilbert space, to itself: rank preserving linear maps (more precisely, maps preserving rank one operators in the symmetric case, and maps preserving rank two operators in the antisymmetric case), surjective linear isometries and generalized bicircular projections.