The approximation properties determined by operator ideals

We introduce the notion of the right approximation property with respect to an operator ideal A and solve the duality problem for the approximation property with respect to an operator ideal A, that is, a Banach space X has the approximation property with respect to A ^d whenever X* has the right approximation property with respect to an operator ideal A. The notions of the left bounded approximation property and the left weak bounded approximation property for a Banach operator ideal are introduced and new symmetric results are obtained. Finally, the notions of the p-compact sets and the p-approximation property are extended to arbitrary Banach operator ideals. Known results of the approximation property with respect to an operator ideal and the p-approximation property are generalized.     

The <Emphasis Type="Italic">p</Emphasis>-Gelfand–Phillips property in spaces of operators and Dunford–Pettis like sets

The p-Gelfand–Phillips property (1 \({\leq}\) p < ∞) is studied in spaces of operators. Dunford–Pettis type like sets are studied in Banach spaces. We discuss Banach spaces X with the property that every p-convergent operator T:X \({\rightarrow}\) Y is weakly compact, for every Banach space Y.     

Essential spectrum of difference operators on periodic metric spaces

The paper deals with the study of Fredholm property and essential spectrum of general difference (or band) operators acting on the spaces l ^p (X) on a discrete metric space X periodic with respect to the action of a finitely generated discrete group. The Schrödinger operator on a combinatorial periodic graph is a prominent example of a band operator of this kind.     

Direct and inverse asymptotic scattering problems for Dirac-Krein systems

The asymptotic scattering matrix s _ε(λ) for a Dirac-Krein system with signature matrix J = diag{ I _p ,-I _p }, integrable potential, and the boundary condition u ₁(0, λ) = u ₂(0, λ)ε(λ) with a coefficient ε(λ) that belongs to the Schur class of holomorphic contractive p × p matrix-valued functions in the open upper half-plane is defined. The inverse asymptotic scattering problem for a given s _ε is analyzed by Krein’s method. Earlier studies by Krein and others focused on the case in which ε = I _p (or a constant unitary matrix).     

Khinchin’s inequality,Dunford-Pettis and compact operators on the space <Emphasis Type="Italic">C</Emphasis>([0, 1], <Emphasis Type="Italic">X</Emphasis>)

We prove that if X, Y are Banach spaces, Ω a compact Hausdorff space and U:C(Ω, X) → Y is a bounded linear operator, and if U is a Dunford-Pettis operator the range of the representing measure G(Σ) ? DP(X, Y) is an uniformly Dunford-Pettis family of operators and ∥G∥ is continuous at Ø. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space C([0, 1], X) with values in c ₀ or l _p, (1 ≤ p < ∞) be Dunford-Pettis and/or compact operators, in which, Khinchin’s inequality plays an important role.     

On invariant graph subspaces of a <Emphasis Type="Italic">J</Emphasis>-self-adjoint operator in the Feshbach case

We consider a J-self-adjoint 2 × 2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one main-diagonal entry of L is embedded into the absolutely continuous spectrum of the other main-diagonal entry. We work with the analytic continuation of the Schur complement of amain-diagonal entry in L?z to the unphysical sheets of the spectral parameter z plane. We present conditions under which the continued Schur complement has operator roots in the sense of Markus–Matsaev. The operator roots reproduce (parts of) the spectrum of the Schur complement, including the resonances. We, then discuss the case where there are no resonances and the associated Riccati equations have bounded solutions allowing the graph representations for the corresponding J-orthogonal invariant subspaces of L. The presentation ends with an explicitly solvable example.     

Narrow orthogonally additive operators in lattice-normed spaces

We consider a new class of narrow orthogonally additive operators in lattice-normed spaces and prove the narrowness of every C-compact norm-laterally-continuous orthogonally additive operator from a Banach–Kantorovich space V into a Banach space Y. Furthermore, every dominated Urysohn operator from V into a Banach sequence lattice Y is also narrow. We establish that the order narrowness of a dominated Urysohn operator from a Banach–Kantorovich space V into a Banach space with mixed norm W implies the order narrowness of the least dominant of the operator.     

Nonlinear Frames and Sparse Reconstructions in Banach Spaces

In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps F between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames, p-frames, Banach frames, g-frames and fusion frames. In general Banach space setting, stable algorithms to reconstruct a signal x from its noisy measurement \(F(x)+\epsilon \) may not exist. In this paper, we establish exponential convergence of two iterative reconstruction algorithms when F is not too far from some bounded below linear operator with bounded pseudo-inverse, and when F is a well-localized map between two Banach spaces with dense Hilbert subspaces. The crucial step to prove the latter conclusion is a novel fixed point theorem for a well-localized map on a Banach space. In the second part of this paper, we consider stable reconstruction of sparse signals in a union \(\mathbf{A}\) of closed linear subspaces of a Hilbert space \(\mathbf{H}\) from their nonlinear measurements. We introduce an optimization framework called a sparse approximation triple \((\mathbf{A}, \mathbf{M}, \mathbf{H})\), and show that the minimizer

$$\begin{aligned} x^*=\mathrm{argmin}_{\hat{x}\in {\mathbf M}\ \mathrm{with} \ \Vert F(\hat{x})-F(x^0)\Vert \le \epsilon } \Vert \hat{x}\Vert _{\mathbf M} \end{aligned}$$

provides a suboptimal approximation to the original sparse signal \(x^0\in \mathbf{A}\) when the measurement map F has the sparse Riesz property and the almost linear property on \({\mathbf A}\). The above two new properties are shown to be satisfied when F is not far away from a linear measurement operator T having the restricted isometry property.

     

Strictly singular operators in pairs of <Emphasis Type="Italic">L</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript> space

Let E and F be Banach spaces. A linear operator from E to F is said to be strictly singular if, for any subspace Q ? E, the restriction of A to Q is not an isomorphism. A compactness criterion for any strictly singular operator from L_p to L_q is found. There exists a strictly singular but not superstrictly singular operator on L_p, provided that p ≠ 2.     

On the Classification of <Emphasis Type="Italic">p</Emphasis>-Adic UHF Banach Algebras

For any prime number p let Ω_p be the p-adic counterpart of the complex numbers C. In this paper we investigate the class of p-adic UHF Banach algebras. A p-adic UHF Banach algebra is any unital p-adic Banach algebra A of the form \(A = \overline {U{M_n}} \), where (M_n) is an increasing sequence of p-adic Banach subalgebras of M such that each M_n is generated over Ω_p by an algebraic system of matrix units {e _ij ^{( n)} | 1 ≤ i, j ≤ p_n }. The main result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra.     

On the functions of one selfadjoint integral operator

We construct a selfadjoint Akhiezer integral operator S on L ₂(?1, 1) with the spectrum {?1; 0; 1} and the property that every function φ(S) that is not a multiple of S fails to be an Akhiezer integral operator.     

Operator ultra-amenability

Extending M. Daws’ definition of ultra-amenable Banach algebras, we introduce the notion of operator ultra-amenability for completely contractive Banach algebras. For a locally compact group G, we show that the operator ultra-amenability of A(G) imposes severe restrictions on G. In particular, it forces G to be a discrete, amenable group with no infinite abelian subgroups. For various classes of such groups, this means that G is finite.     

C*-algebras have a quantitative version of Pełczyński’s property (V)

A Banach space X has Pe?czyński’s property (V) if for every Banach space Y every unconditionally converging operator T: X → Y is weakly compact. H.Pfitzner proved that C*-algebras have Pe?czyński’s property (V). In the preprint (Kruli?ová, (2015)) the author explores possible quantifications of the property (V) and shows that C(K) spaces for a compact Hausdorff space K enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover, we prove that in dual Banach spaces a quantitative version of the property (V) implies a quantitative version of the Grothendieck property.     

On the Second Parameter of an (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">p</Emphasis>)-Isometry

A bounded linear operator T on a Banach space X is called an (m, p)-isometry if it satisfies the equation \({\sum_{k=0}^{m}(-1)^{k} {m \choose k}\|T^{k}x\|^{p}=0}\) , for all \({x \in X}\) . In this paper we study the structure which underlies the second parameter of (m, p)-isometric operators. We concentrate on determining when an (m, p)-isometry is a (μ, q)-isometry for some pair (μ, q). We also extend the definition of (m, p)-isometry, to include p = ∞ and study basic properties of these (m, ∞)-isometries.     

The dual Radon–Nikodym property for finitely generated Banach <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-modules

We extend the well-known criterion of Lotz for the dual Radon–Nikodym property (RNP) of Banach lattices to finitely generated Banach C(K)-modules and Banach C(K)-modules of finite multiplicity. Namely, we prove that if X is a Banach space from one of these classes then its Banach dual \(X^\star \) has the RNP iff X does not contain a closed subspace isomorphic to \(\ell ^1\).     

On Causal Invertibility with Respect to a Cone of Integral-Difference Operators in Vector Function Spaces

Let \(\mathbb{S}\) be a cone in ? ⁿ . A bounded linear operator T: L _p (? ⁿ ) → L _p (? ⁿ ) is said to be causal with respect to \(\mathbb{S}\) if the implication x(s) = 0 (s ε W ? \(\mathbb{S}\)) ? (Tx) (s) = 0 (s ε W ? \(\mathbb{S}\)) is valid for any x ε L _p (? ⁿ ) and any open subset W\(\subseteq\) ? ⁿ . The set of all causal operators is a Banach algebra. We describe the spectrum of the operator

$(Tx)(t) = \sum\limits_{n = 1}^\infty {a_n x(t - t_n )} + \int {\mathbb{S}g(s)x(t - s)ds,} \quad t \in \mathbb{R}^n ,$

in this algebra. Here x ranges in a Banach space \(\mathbb{E}\), the a _n are bounded linear operators in \(\mathbb{E}\), and the function g ranges in the set of bounded operators in \(\mathbb{E}\).

     

Equivalence of <Emphasis Type="Italic">K</Emphasis>-functionals and modulus of smoothness constructed by generalized Dunkl translations

In a Hilbert space L _2,α := L ₂(?, |x|^2α+1 dx), α > ? 1/2, we study the generalized Dunkl translations constructed by the Dunkl differential-difference operator. Using the generalized Dunkl translations, we define generalized modulus of smoothness in the space L _2,α . Based on the Dunkl operator we define Sobolev-type spaces and K-functionals. The main result of the paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness.     

Numerical ranges for pairs of operators,duality mappings with gauge function,and spectra of nonlinear operators

We define and study numerical ranges for pairs of nonlinear operators F and J which act between some Banach space X and its dual X*, with respect to some increasing gauge function φ. Connections with spectra for certain classes of nonlinear operators introduced recently in the literature are also established. As a sample example, we consider the case when F is the duality map of the Lebesgue space L ^p (Ω), J is the duality map of the corresponding Sobolev space W ₀ ^1,p (Ω), and φ(t)=t ^p?1 (1<p<∞). This leads to existence, uniqueness, and perturbation results for a homogeneous eigenvalue problem involving the p-Laplace operator.     

On the Dunford Property (<Emphasis Type="Italic">C</Emphasis>) for Bounded Linear Operators <Emphasis Type="Italic">RS</Emphasis> and <Emphasis Type="Italic">SR</Emphasis>

In this paper we show that if \({S\in L(X,Y)}\) and \({R\in L(Y,X),}\) X and Y complex Banach spaces, then the products RS and SR share the Dunford property (C). We also study property (C) for R, S, RS and \({SR \in L(X)}\) in the case that R and S satisfies the operator equations RSR = R ² and SRS = S ².     

Reflexivity and the Grothendieck property for positive tensor products of Banach lattices-I

Let X be a Banach lattice and p, p′ be real numbers such that 1 < p, p′<∞ and 1/p + 1/p′ = 1. Then \({\ell_p\hat{\otimes}_FX}\) (respectively, \({\ell_p\tilde{\otimes}_{i}X}\)), the Fremlin projective (respectively, the Wittstock injective) tensor product of ? _p and X, has reflexivity or the Grothendieck property if and only if X has the same property and each positive linear operator from ? _p (respectively, from ? _p′) to X* (respectively, to X**) is compact.