Symmetric Sequence Spaces Research Articles

2-Rotund norms for unconditional and symmetric sequence spaces

A reflexive Banach space with an unconditional basis admits an equivalent 1-unconditional 2R norm and embeds into a reflexive space with a 1-symmetric 2R norm. Partial results on 1-symmetric 2R renormings of spaces with a symmetric basis are obtained.

Some norm estimates for the triangular projection on the space of bounded linear operators between two symmetric sequence spaces

Some norm estimates for the triangular projection on the space of bounded linear operators between two symmetric sequence spaces

Symmetric Banach sequence spaces respect Weyl submajorization

Let E E be a symmetric Banach sequence space. We show that there exists an equivalent symmetric norm on E E which is monotone with respect to the Weyl (i.e., logarithmic) submajorization. Surprisingly, this purely commutative result is proved by a very non-commutative method.

On Arazy's problem concerning isomorphic embeddings of ideals of compact operators

The primary objective of this paper is to study the problem concerning isomorphic embeddings of ideals of compact operators due to Arazy (see [5, Problem C] and [6, Problem (B)]), which is closely related to Pełczyński's problem of uniqueness of symmetric structure in such ideals. In particular, we show that for a wide class of symmetric sequence spaces E, if CF↪SE, then F=ℓ2 and E has a subspace isomorphic to ℓ2, where CX is the symmetric ideal of compact operators on the Hilbert space ℓ2 induced by a symmetric sequence space X and SX=(⊕n=1∞Mn)CX. This extends results due to Arazy [5,6]. As a consequence, we contribute to the Arazy–Pełczyński problem on the uniqueness of symmetric structures [5,6]. In particular, we can deal with the case when E has a subspace isomorphic to ℓ2 or c0, which was left untreated in [5,6].

A characterization of $$\ell ^p$$-spaces symmetrically finitely represented in symmetric sequence spaces

For a separable symmetric sequence space X of fundamental type we identify the set $${{\mathcal {F}}}(X)$$ of all $$p\in [1,\infty ]$$ such that $$\ell ^p$$ is block finitely represented in the unit vector basis $$\{e_k\}_{k=1}^\infty$$ of X in such a way that the unit basis vectors of $$\ell ^p$$ ( $$c_0$$ if $$p=\infty$$ ) correspond to pairwise disjoint blocks of $$\{e_k\}$$ with the same ordered distribution. It turns out that $${{\mathcal {F}}}(X)$$ coincides with the set of approximate eigenvalues of the operator $$(x_k)\mapsto \sum _{k=2}^\infty x_{[k/2]}e_k$$ in X. In turn, we establish that the latter set is the interval $$[2^{\alpha _X},2^{\beta _X}]$$ , where $$\alpha _X$$ and $$\beta _X$$ are the Boyd indices of X. As an application, we find the set $${{\mathcal {F}}}(X)$$ for arbitrary Lorentz and separable Orlicz sequence spaces.

The Modulus of p-Variation and Its Applications

Let \(\nu \) be a nondecreasing concave sequence of positive real numbers and \(1\le p<\infty \). In this note, we introduce the notion of modulus of p-variation for a function of a real variable, and show that it serves in at least two important problems, namely, the uniform convergence of Fourier series and computation of certain K-functionals. Using this new tool, we first define a Banach space, denoted \(V_p[\nu ]\), that is a natural unification of the Wiener class \(BV_p\) and the Chanturiya class \(V[\nu ]\). Then we prove that \(V_p[\nu ]\) satisfies a Helly-type selection principle which enables us to characterize continuous functions in \(V_p[\nu ]\) in terms of their Fejér means. We also prove that a certain K-functional for the couple \((C,BV_p)\) can be expressed in terms of the modulus of p-variation, where C denotes the space of continuous functions. Next, we obtain equivalent optimal conditions for the uniform convergence of the Fourier series of all functions in each of the classes \(C\cap V_p[\nu ]\) and \(H^\omega \cap V_p[\nu ]\), where \(\omega \) is a modulus of continuity and \(H^\omega \) denotes its associated Lipschitz class. Finally, we establish sharp embeddings into \(V_p[\nu ]\) of various spaces of functions of generalized bounded variation. As a by-product of these latter results, we infer embedding results for certain symmetric sequence spaces.

A Note on the Symmetry of Sequence Spaces

We give a self-contained treatment of symmetric Banach sequence spaces and some of their natural properties. We are particularly interested in the symmetry of the norm and the existence of symmetric linear functionals. Many of the presented results are known or commonly accepted, but are not found in the literature.

Lack of isomorphic embeddings of symmetric function spaces into operator ideals

Let E(0,1) be a symmetric space on (0,1) and CF be a symmetric ideal of compact operators on the Hilbert space ℓ2 associated with a symmetric sequence space F. We give several criteria for E(0,1) and F so that E(0,1) does not embed into the ideal CF, extending the result for the case when E(0,1)=Lp(0,1) and F=ℓp, 1≤p<∞, due to Arazy and Lindenstrauss [5].

A class of a non-Banach F-lattices E such that every orthomorphism on E is central with applications to Musielak–Orlicz sequence spaces

Let E=(E,‖⋅‖) be a non-Banach F-lattice non-containing a subspace linearly homeomorphic to the F-lattice ω:=RN. The symbol Orth(E) denotes the linear lattice of all orthomorphisms on E, and Z(E) denotes the principal ideal of Orth(E) generated by the identity IE on E. In this paper, we give a sufficient condition for the equality Orth(E)=Z(E) to hold. In particular, this identity holds true for every symmetric sequence space E. We also obtain a useful series-criterion for the class of discrete F-lattices E with order continuous norm for the identity Orth(E)=Z(E) to hold. These results apply non-trivially for the class of Musielak-Orlicz sequence spaces. We give a few examples illustrating our main results. At the end of this paper, we set a few open questions about orthomorphisms.

Rosenthal’s inequalities: ${\Delta }$-norms and quasi-Banach symmetric sequence spaces

Let $X$ be a symmetric quasi-Banach function space with the Fatou property and let $E$ be an arbitrary symmetric quasi-Banach sequence space. Suppose that $(f_k)_{k\geq 0}\subset X$ is a sequence of independent random variables. We present a necessary and

The optimal symmetric quasi-Banach range of the discrete Hilbert transform

We identify symmetric quasi-Banach range of the discrete Calder\'{o}n operator and Hilbert transform acting on a symmetric quasi-Banach sequence space. As an application we present an example of optimal range in the case when the domain of those operators is the weak-$\ell_{1}$ space of sequences.

Ergodic theorems in symmetric sequence spaces

Ergodic theorems in symmetric sequence spaces

The uniqueness of the symmetric structure in ideals of compact operators

Let Hbe a separable infinite-dimensional complex Hilbert space, let L(H) be the C∗-algebra of bounded linear operators acting in H, and let K(H) be the two-sided ideal of compact linear operators in L(H). Let (E,∥⋅∥E) be a symmetric sequence space, and let CE:={x∈K(H):{sn(x)}∞n=1∈E} be the proper two-sided ideal in L(H), where {sn(x)}∞n=1 are the singular values of a compact operator x. It is known that CE is a Banach symmetric ideal with respect to the norm ∥x∥CE=∥{sn(x)}∞n=1∥E. A symmetric ideal CE is said to have a unique symmetric structure if CE=CF, that is E=F, modulo norm equivalence, whenever (CE,∥⋅∥CE) is isomorphic to another symmetric ideal (CF,∥⋅∥CF). At the Kent international conference on Banach space theory and its applications (Kent, Ohio, August 1979), A. Pelczynsky posted the following problem: (P) Does every symmetric ideal have a unique symmetric structure? This problem has positive solution for Schatten ideals Cp, 1≤p&lt;∞ (J. Arazy and J. Lindenstrauss, 1975). For arbitrary symmetric ideals problem (P) has not yet been solved. We consider a version of problem (P) replacing an isomorphism U:(CE,∥⋅∥CE)→(CF,∥⋅∥CF) by a positive linear surjective isometry. We show that if F is a strongly symmetric sequence space, then every positive linear surjective isometry U:(CE,∥⋅∥CE)→(CF,∥⋅∥CF) is of the form U(x)=u∗xu, x∈CE, where u∈L(H) is a unitary or antiunitary operator. Using this description of positive linear surjective isometries, it is established that existence of such an isometry U:CE→CF implies that (E,∥⋅∥E)=(F,∥⋅∥F).

On Extrapolation Properties of Schatten–von Neumann Classes

For a certain special class of symmetric sequence spaces, we give an explicit relation between the interpolation and extrapolation representations. This relation is carried over to symmetrically normed ideals of compact operators.

Isometries and Hermitian operators on complex symmetric sequence spaces

We consider a complex symmetric sequence space E that possesses the Fatou property and is different from l2. We prove that, for every surjective linear isometry V on E, there exist λ n ∈ ℂ with |λ n | = 1 and a bijective mapping π on the set ℕ of natural numbers such that $$V\left( {\left\{ {\xi _n } \right\}_{n \in \mathbb{N}} } \right) = \left\{ {\lambda _n \xi _{\pi (n)} } \right\}_{n \in \mathbb{N}}$$ for every {ξ n {n∈ℕ ∈ E.

Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals

This is a survey article of geometric properties of noncommutative symmetric spaces of measurable operators $E(\mathcal{M},\tau)$, where $\mathcal{M}$ is a semifinite von Neumann algebra with a faithful, normal, semifinite trace $\tau$, and $E$ is a symmetric function space. If $E\subset c_0$ is a symmetric sequence space then the analogous properties in the unitary matrix ideals $C_E$ are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value function, submajorization in the sense of Hardy, Littlewood and Polya, Kothe duality, the spaces $L_p(\mathcal{M},\tau)$, $1\le p<\infty$, the identification between $C_E$ and $G(B(H), \rm{tr})$ for some symmetric function space $G$, the commutative case when $E$ is identified with $E(\mathcal{N}, \tau)$ for $\mathcal{N}$ isometric to $L_\infty$ with the standard integral trace, trace preserving $*$-isomorphisms between $E$ and a $*$-subalgebra of $E(\mathcal{M},\tau)$, and a general method of removing the assumption of non-atomicity of $\mathcal{M}$. The main results on geometric properties are given in separate sections. We present the results on (complex) extreme points, (complex) strict convexity, strong extreme points and midpoint local uniform convexity, $k$-extreme points and $k$-convexity, (complex or local) uniform convexity, smoothness and strong smoothness, (strongly) exposed points, (uniform) Kadec-Klee properties, Banach-Saks properties, Radon-Nikodým property and stability in the sense of Krivine-Maurey. We also state some open problems.

ON MULTIPLIERS BETWEEN SOME SPACES OF HOLOMORPHIC FUNCTIONS

We study function multipliers between spaces of holomorphic functions on the unit disc of the complex plane generated by symmetric sequence spaces. In the case of sequence$\ell ^{p}$spaces we recover Nikol’skii’s results [‘Spaces and algebras of Toeplitz matrices operating on$\ell ^{p}$’,Sibirsk. Mat. Zh.7(1966), 146–158].

Bohnenblust–Hille inequalities for Lorentz spaces via interpolation

We prove that the Lorentz sequence space l2m∕(m+1),1 is, in a precise sense, optimal among all symmetric Banach sequence spaces satisfying a Bohnenblust–Hille-type inequality for m-linear forms or m-homogeneous polynomials on ℂn. Motivated by this result we develop methods for dealing with subtle Bohnenblust–Hille-type inequalities in the setting of Lorentz spaces. Based on an interpolation approach and the Blei–Fournier inequalities involving mixed-type spaces, we prove multilinear and polynomial Bohnenblust–Hille-type inequalities in Lorentz spaces with subpolynomial and subexponential constants. An application to the theory of Dirichlet series improves a deep result of Balasubramanian, Calado and Queffelec.

Summing multi-norms defined by Orlicz spaces and symmetric sequence space

We develop the notion of the \((X_1,X_2)\)-summing power-norm based on a~Banach space \(E\), where \(X_1\) and \(X_2\) are symmetric sequence spaces. We study the particular case when \(X_1\) and \(X_2\) are Orlicz spaces \(\ell_\Phi\) and \(\ell_\Psi\) respectively and analyze under which conditions the \((\Phi, \Psi)\)-summing power-norm becomes a~multinorm. In the case when \(E\) is also a~symmetric sequence space \(L\), we compute the precise value of \(\|(\delta_1,\cdots,\delta_n)\|_n^{(X_1,X_2)}\) where \((\delta_k)\) stands for the canonical basis of \(L\), extending known results for the \((p,q)\)-summing power-norm based on the space \(\ell_r\) which corresponds to \(X_1=\ell_p\), \(X_2=\ell_q\), and \(E=\ell_r\).

Banach envelopes in symmetric spaces of measurable operators

We study Banach envelopes for commutative symmetric sequence or function spaces, and noncommutative symmetric spaces of measurable operators. We characterize the class (HC) of quasi-normed symmetric sequence or function spaces E for which their Banach envelopes \(\widehat{E}\) are also symmetric spaces. The class of symmetric spaces satisfying (HC) contains but is not limited to order continuous spaces. Let \(\mathcal {M}\) be a non-atomic, semifinite von Neumann algebra with a faithful, normal, \(\sigma \)-finite trace \(\tau \) and E be as symmetric function space on \([0,\tau (\mathbf 1 ))\) or symmetric sequence space. We compute Banach envelope norms on \(E(\mathcal {M},\tau )\) and \(C_E\) for any quasi-normed symmetric space E. Then we show under assumption that \(E\in (HC)\) that the Banach envelope \(\widehat{E(\mathcal {M},\tau )}\) of \(E(\mathcal {M},\tau )\) is equal to \(\widehat{E}\left( \mathcal {M},\tau \right) \) isometrically. We also prove the analogous result for unitary matrix spaces \(C_E\).