A characterization of bounded convex approximation properties

The characterization of bounded approximation properties defined by arbitrary operator ideals due to Oja is extended to bounded convex approximation properties. As an application, it is shown that the unique extension property of a Banach space X enables to lift the metric convex approximation property from a Banach space X to its dual space X*.     

On <Emphasis Type="Italic">p</Emphasis>-convergent Operators on Banach Lattices

The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sánchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.     

Matrix semigroups whose ring commutators have real spectra are realizable

We introduce the numerical spectrum \(\sigma _n(A)\subseteq {\mathbb {C}}\) of an (unbounded) linear operator A on a Banach space X and study its properties. Our definition is closely related to the numerical range W(A) of A and always yields a superset of W(A). In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, \(\sigma _n(A)\) is always closed, convex and contains the spectrum of A. In the paper we strongly emphasise the connection of our approach to the theory of \(C_0\)-semigroups.     

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.

     

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.     

Unbounded disjointness preserving linear functionals and operators

Let E and F be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice E, which shows that in this case the unbounded disjointness preserving operators from \({E\to F}\) separate the points of E. Then we show that every disjointness preserving operator \({T:E\to F}\) is norm bounded on an order dense ideal. In case E has order continuous norm, this implies that every unbounded disjointness preserving map \({T:E\to F}\) has a unique decomposition T = R + S, where R is a bounded disjointness preserving operator and S is an unbounded disjointness preserving operator, which is zero on a norm dense ideal. For the case that E = C(X), with X a compact Hausdorff space, we show that every disjointness preserving operator \({T:C(X)\to F}\) is norm bounded on a norm dense sublattice algebra of C(X), which leads then to a decomposition of T into a bounded disjointness preserving operator and a finite sum of unbounded disjointness preserving operators, which are zero on order dense ideals.     

Efficiency of weak greedy algorithms for m-term approximations

We investigate the efficiency of weak greedy algorithms for m-term expansional approximation with respect to quasi-greedy bases in general Banach spaces.We estimate the corresponding Lebesgue constants for the weak thresholding greedy algorithm(WTGA) and weak Chebyshev thresholding greedy algorithm.Then we discuss the greedy approximation on some function classes.For some sparse classes induced by uniformly bounded quasi-greedy bases of L_p,1p∞,we show that the WTGA realizes the order of the best m-term approximation.Finally,we compare the efficiency of the weak Chebyshev greedy algorithm(WCGA) with the thresholding greedy algorithm(TGA) when applying them to quasi-greedy bases in L_p,1≤p∞,by establishing the corresponding Lebesgue-type inequalities.It seems that when p2 the WCGA is better than the TGA.     

Fredholmness of Toeplitz operators on generalized Hardy spaces over the polydisc

Let A be a bounded linear operator and P a bounded linear projection on a Banach space X. We show that the operator semigroup \({(e^{t(A-kP)})_{t \ge 0}}\) converges to a semigroup on a subspace of X as \({k \to \infty}\) and we compute the limit semigroup.     

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.     

The best <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> approximation of the Laplace operator by linear bounded operators in the classes of functions of two and three variables

Close two-sided estimates are obtained for the best approximation in the space L _p (? ^m ), m = 2 and 3, 1 ≤ p ≤ ∞, of the Laplace operator by linear bounded operators in the class of functions for which the second power of the Laplace operator belongs to the space L _p (? ^m ). We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class given with an error. We present an operator whose deviation from the Laplace operator is close to the best.     

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.     

Invariance of the order and type of a sequence of operators

In the paper, the invariance property of characteristics (the order and type) of an operator and of a sequence of operators with respect to a topological isomorphism is proved. These characteristics give precise upper and lower bounds for the expressions ‖A_n(x)‖_p and enable one to state and solve problems of operator theory in locally convex spaces in a general setting. Examples of such problems are given by the completeness problem for the set of values of a vector function in a locally convex space, the structure problem for a subspace invariant with respect to an operator A, the problem of applicability of an operator series to a locally convex space, the theory of holomorphic operator-valued functions, the theory of operator and differential-operator equations in nonnormed spaces, and so on. However, the immediate evaluation of characteristics of operators (and of sequences of operators) directly by definition is practically unrealizable in spaces with more complicated structure than that of countably normed spaces, due to the absence of an explicit form of seminorms or to their complicated structure. The approach that we use enables us to find characteristics of operators and sequences of operators using the passage to the dual space, by-passing the definition, and makes it possible to obtain bounds for the expressions ‖A_n(x)‖_p even if an explicit form of seminorms is unknown.     

On the positive commutator in the radical

In this paper we prove that a positive commutator between a positive compact operator A and a positive operator B is in the radical of the Banach algebra generated by A and B. Furthermore, on every at least three-dimensional Banach lattice we construct finite rank operators A and B satisfying \(AB\ge BA\ge 0\) such that the commutator \(AB-BA\) is not contained in the radical of the Banach algebra generated by A and B. These two results now completely answer to two open questions published in (Bra?i? et al., Positivity 14:431–439, 2010). We also obtain relevant results in the case of the Volterra and the Donoghue operator.     

The Corona problem on a complete ultrametric algebraically closed field

Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the ”open” unit disk D of IK provided with the Gauss norm. Let Mult(A, ‖. ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Mult _m (A, ‖. ‖) be the subset of the ? ∈ Mult(A, ‖. ‖) whose kernel is amaximal ideal and let Mult ₁(A, ‖. ‖) be the subset of the ? ∈ Mult(A, ‖. ‖) whose kernel is a maximal ideal of the form (x ? a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem the question whether Mult ₁(A, ‖. ‖) is dense in Mult _m (A, ‖. ‖). In a previous paper, it was proved that when IK is spherically complete, the answer is yes. Here we generalize this result to any algebraically closed complete ultrametric field, which particularly applies to ? _p . On the other hand, we also show that the continuous multiplicative seminorms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of Mult ₁(A, ‖. ‖), which suggest that Mult ₁(A, ‖. ‖)might be dense in Mult(A, ‖. ‖).     

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.     

Generalized Drazin invertibility and local spectral theory

If \(A\in B(\mathcal{X})\) is an upper triangular Banach space operator with diagonal \((A_1,A_2)\), \(A_1\) invertible and \(A_2\) quasinilpotent, then \(A_1^{-1}\oplus A_2\) satisfies either of the single-valued extension property, Dunford’s condition (C), Bishop’s property \((\beta )\), decomposition property \((\delta )\) or is decomposable if and only if \(A_1\) has the property. The operator \(A^{-1}_1\oplus 0\) is subscalar (resp., left polaroid, right polaroid) if and only if \(A_1\) is subscalar (resp., left polaroid, right polaroid). For Drazin invertible operators A, with Drazin inverse B, this implies that B satisfies any one of these properties if and only if A satisfies the property.     

Density of characters of bounded <Emphasis Type="Italic">p</Emphasis>-adic analytic functions in the topological dual

Let K be an ultrametric complete algebraically closed field, let D be a disk {x ∈ K ‖x| < R} (with R in the set of absolute values of K) and let A be the Banach algebra of bounded analytic functions in D. The vector space generated by the set of characters of A is dense in the topological dual of A if and only if K is not spherically complete. Let H(D) be the Banach algebra of analytic elements in D. The vector space generated by the set of characters of H(D) is never dense in the topological dual of H(D).     

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.     

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.     

On perturbation theory for points of discrete spectrum of dissipative operator functions

We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α ₀ ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α ₀ and R is a bounded positive operator; moreover, the point α ₀ is a normal eigenvalue of the operator function L(α, θ ₀) for some θ ₀ ∈ ?, and the number θ ₀ is a normal eigenvalue of the operator function L(α ₀ θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α ₀ and θ ₀, respectively, and on the representation of the corresponding eigenfunctions by series.